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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Performance Analysis of Decentralized Federated Learning Deployments</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2503.11828v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S1" title="In Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">I </span><span class="ltx_text ltx_font_smallcaps">Introduction</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S2" title="In Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II </span><span class="ltx_text ltx_font_smallcaps">Background Concepts</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S3" title="In Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III </span><span class="ltx_text ltx_font_smallcaps">Related Works</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4" title="In Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IV </span><span class="ltx_text ltx_font_smallcaps">Convergence Rate Analysis</span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4.SS1" title="In IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-A</span> </span><span class="ltx_text ltx_font_italic">Problem Formulation</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4.SS2" title="In IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-B</span> </span><span class="ltx_text ltx_font_italic">Convergence Analysis</span></span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4.SS2.SSS1" title="In IV-B Convergence Analysis ‣ IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-B</span>1 </span>Continuous Linear</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4.SS2.SSS2" title="In IV-B Convergence Analysis ‣ IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-B</span>2 </span>Continuous Ring</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4.SS2.SSS3" title="In IV-B Convergence Analysis ‣ IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-B</span>3 </span>Aggregate Linear</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4.SS2.SSS4" title="In IV-B Convergence Analysis ‣ IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-B</span>4 </span>Aggregate Ring</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4.SS2.SSS5" title="In IV-B Convergence Analysis ‣ IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-B</span>5 </span>Aggregate Star</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4.SS2.SSS6" title="In IV-B Convergence Analysis ‣ IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">IV-B</span>6 </span>Aggregate Mesh</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S5" title="In Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">V </span><span class="ltx_text ltx_font_smallcaps">Implementation Details</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S6" title="In Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">VI </span><span class="ltx_text ltx_font_smallcaps">Baseline and DFL Evaluations</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S7" title="In Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">VII </span><span class="ltx_text ltx_font_smallcaps">Evaluation Results</span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S7.SS1" title="In VII Evaluation Results ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">VII-A</span> </span><span class="ltx_text ltx_font_italic">Impact of Topologies on the Convergence Rate</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S7.SS2" title="In VII Evaluation Results ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">VII-B</span> </span><span class="ltx_text ltx_font_italic">Impact of Non-IID data Distribution</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S8" title="In Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">VIII </span><span class="ltx_text ltx_font_smallcaps">Conclusion</span></span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">Performance Analysis of Decentralized Federated Learning Deployments</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> Chengyan Jiang1, Jiamin Fan2, Talal Halabi3, Israat Haque1 </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation"> 1<span class="ltx_text ltx_font_italic" id="id1.1.id1">Dalhousie University, Canada</span>, 3<span class="ltx_text ltx_font_italic" id="id2.2.id2">University of Victoria, Canada</span>, 2<span class="ltx_text ltx_font_italic" id="id3.3.id3">Laval University, Canada</span> </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id4.id1">The widespread adoption of smartphones and smart wearable devices has led to the widespread use of Centralized Federated Learning (CFL) for training powerful machine learning models while preserving data privacy. However, CFL faces limitations due to its overreliance on a central server, which impacts latency and system robustness. Decentralized Federated Learning (DFL) is introduced to address these challenges. It facilitates direct collaboration among participating devices without relying on a central server. Each device can independently connect with other devices and share model parameters. This work explores crucial factors influencing the convergence and generalization capacity of DFL models, emphasizing network topologies, non-IID data distribution, and training strategies. We first derive the convergence rate of different DFL model deployment strategies. Then, we comprehensively analyze various network topologies (e.g., linear, ring, star, and mesh) with different degrees of non-IID data and evaluate them over widely adopted machine learning models (e.g., classical, deep neural networks, and Large Language Models) and real-world datasets. The results reveal that models converge to the optimal one for IID data. However, the convergence rate is inversely proportional to the degree of non-IID data distribution. Our findings will serve as valuable guidelines for designing effective DFL model deployments in practical applications.</p> </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">I </span><span class="ltx_text ltx_font_smallcaps" id="S1.1.1">Introduction</span> </h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">The advent of the Internet of Things (IoT), their applications (e.g., connected autonomous vehicles (CAVs), industrial IoT), and their volume of network and application data open up opportunities for developing data-driven decision making. However, collecting this data at cloud servers for further processing brings challenges like adequate transmission bandwidth and acceptable response time to end users. Edge computing is a paradigm of processing users’ requests closer to them, e.g., in edge servers, to reduce resource demands while improving response time. Furthermore, federated learning emerged to offer data privacy with reduced bandwidth demand <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib1" title="">1</a>]</cite>. Specifically, end devices do not share their data over the network to a central server to process them. Instead, the server shares the initial set of parameters with end devices to train a model on their local data. After completing the training, the devices share their trained parameters with the server to aggregate these parameters. The server shares the updated model parameters again, and the process continues until the model is globally trained with the coordination of the server. This methodology optimizes computational resources and accelerates developing and deploying intelligent network functions and services while preserving data privacy <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib2" title="">2</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">The above server-coordinated approach faces limitations like single points of failure, communication bottlenecks, and trust issues. To overcome these issues, Decentralized Federated Learning (DFL) facilitates the direct collaboration among devices, eliminating the need for a central server and enhancing robustness, trust, and resource distributions <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib3" title="">3</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib4" title="">4</a>]</cite>. Each device performs both computation and communication to serve as a server or a client. The decentralized approach enhances system fault tolerance and robustness and better protects data privacy by avoiding the security risks associated with centralized data storage.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">The effectiveness and reliability of DFL systems hinge upon critical factors such as network topology, non-IID data distributions, and training strategies <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib4" title="">4</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib3" title="">3</a>]</cite>. The standard topologies include linear, ring, and star, each having its strength in resource usage and model convergence rates. Non-IID data distributions refer to scenarios with devices having data with different distributions, characteristics, and statistical properties. Finally, DFL training strategies can be <span class="ltx_text ltx_font_italic" id="S1.p3.1.1">continuous</span> and <span class="ltx_text ltx_font_italic" id="S1.p3.1.2">aggregation</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib3" title="">3</a>]</cite>. In the former, the model parameters flow from one device to another in a sequential fashion, while the latter scheme deploys periodic aggregation of model parameters from multiple participating devices. Thus, depending on the application, users may choose a combination of network topology, degree of non-IID data distribution, and training strategy to meet their demand.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">Therefore, a thorough analysis is essential to understand how variations in network topology, degree of non-IID data, and different training strategies impact DFL performance. While existing studies have addressed some aspects of these challenges, notable gaps persist. Sheller <span class="ltx_text ltx_font_italic" id="S1.p4.1.1">et al.</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib5" title="">5</a>]</cite> compare convergence performance and accuracy across devices using continual training strategies with linear and ring topologies in DFL. Similarly, Chen <span class="ltx_text ltx_font_italic" id="S1.p4.1.2">et al.</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib6" title="">6</a>]</cite> introduce a novel DFL solution utilizing a mesh topology and aggregate training strategy. However, neither study investigated the impact of the degree of non-IID data distribution, a crucial consideration in real-world applications. Furthermore, in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib7" title="">7</a>]</cite>, the authors focus on the convergence of Centralized Federated Learning (CFL) under varying degrees of non-IID data distributions using convex local models, leaving unexplored the efficacy of non-convex models. These gaps highlight the need for a comprehensive assessment to bridge the understanding of how DFL systems perform under diverse conditions, including both convex and non-convex models.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">This paper rigorously analyzes the performance of DFL over the three factors mentioned above to train classical machine learning models (e.g., SVM, logic regression), neural networks (e.g., ResNet, DistilBERT), and large language models (LLMs) (e.g., MiniGPT-4) to shed light on their usage in emerging IoT applications. Specifically, we consider the combination of standard topologies with DFL training strategies along with IID and non-IID data distributions. First, we conduct a theoretical analysis of the convergence and generalization capabilities of convex models. Then, we extensively evaluate their performance to show alignment with the theoretical analysis. The results show that as the degree of non-IID increases, the actual loss value diverges from the ideal one, indicating that non-IID data distribution significantly affects the convergence of the DFL model. Furthermore, we evaluate non-convex models (ResNet, DistilBERT, and MiniGPT-4) to show their usage in DFL-based deployments. Our main contribution involves:</p> </div> <div class="ltx_para" id="S1.p6"> <ul class="ltx_itemize" id="S1.I1"> <li class="ltx_item" id="S1.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i1.p1"> <p class="ltx_p" id="S1.I1.i1.p1.1">This work provides a comprehensive analysis of the factors influencing the convergence efficiency and generalization capacity of DFL models, covering both theoretical foundations and practical considerations.</p> </div> </li> <li class="ltx_item" id="S1.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i2.p1"> <p class="ltx_p" id="S1.I1.i2.p1.1">We rigorously establish the convergence of different network structures in DFL through convex optimization. This theoretical insight provides valuable guidance for researchers aiming to structure their DFL models effectively.</p> </div> </li> <li class="ltx_item" id="S1.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i3.p1"> <p class="ltx_p" id="S1.I1.i3.p1.1">To analyze the convergence performance of different model configurations in DFL under non-IID data, we employ various setups, including Continuous Linear, Continuous Ring, Aggregate Ring, Aggregate Linear, Aggregate Mesh, and Aggregate Star. The degree of non-IID data is quantified using a label imbalance method.</p> </div> </li> <li class="ltx_item" id="S1.I1.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i4.p1"> <p class="ltx_p" id="S1.I1.i4.p1.1">We implement and assess five models: two convex (SVM and Logistic Regression) and three non-convex (ResNet, DistilBERT, and MiniGPT-4). Evaluation results show that all models across different topologies and training strategies converge under IID data. In the case of non-IID data, the performance is inversely proportional to the degree of distribution. </p> </div> </li> </ul> </div> <div class="ltx_para" id="S1.p7"> <p class="ltx_p" id="S1.p7.1">The remainder of the paper is structured as follows. Section <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S2" title="II Background Concepts ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">II</span></a> describes some background concepts. Section <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S3" title="III Related Works ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">III</span></a> discusses related work. Section <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4" title="IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">IV</span></a> presents our convergence rate analysis. Section <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S5" title="V Implementation Details ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">V</span></a> describes our implementation details. Section <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S6" title="VI Baseline and DFL Evaluations ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">VI</span></a> presents the evaluation setup. Section <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S7" title="VII Evaluation Results ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">VII</span></a> discussed the evaluation results. Finally, Section <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S8" title="VIII Conclusion ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">VIII</span></a> concludes the paper.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">II </span><span class="ltx_text ltx_font_smallcaps" id="S2.1.1">Background Concepts</span> </h2> <figure class="ltx_figure" id="S2.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="315" id="S2.F1.g1" src="extracted/6279016/images/cross_device.png" width="419"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S2.F1.2.1.1" style="font-size:90%;">Figure 1</span>: </span><span class="ltx_text" id="S2.F1.3.2" style="font-size:90%;">Cross silo vs Cross device</span></figcaption> </figure> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">This section presents the necessary background information to follow the proposed work.</p> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.1">In Decentralized Federated Learning (DFL), model updates and parameter aggregation are carried out through direct communication among participating devices without having any central control. Thus, each device independently trains a model on its local data and exchanges model updates with other devices. They form different network topologies, e.g., linear, ring, star, and mesh <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib3" title="">3</a>]</cite>. DFL excels in enabling the secure sharing of privacy-preserving data between real and virtual entities. This capability proves particularly advantageous in fostering task coordination and asynchronous knowledge exchange within domains such as robotics, energy, and utility sectors, with significant applications in Industry 4.0 and mobile services <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib4" title="">4</a>]</cite>. As shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S2.F1" title="Figure 1 ‣ II Background Concepts ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">1</span></a>, there are two types of DFL: <span class="ltx_text ltx_font_italic" id="S2.p2.1.1">cross-silo</span> and <span class="ltx_text ltx_font_italic" id="S2.p2.1.2">cross-device</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib3" title="">3</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib4" title="">4</a>]</cite>. The former enables distributed learning among organizations or data centers, i.e., a relatively small number of participating nodes. On the other hand, cross-device DFL involves a larger number of nodes, each with a modest amount of data and limited computational power. Our analysis applies to both cross-silo and cross-device DFL, i.e., to scenarios with varying numbers of devices .</p> </div> <div class="ltx_para" id="S2.p3"> <p class="ltx_p" id="S2.p3.1"><span class="ltx_text ltx_font_bold" id="S2.p3.1.1">Network Topology:</span> Participating devices in DFL training usually form linear, ring, star, and mesh network topologies, each with unique strengths. For instance, in ring and linear topologies, devices pass model parameters to downstream devices after finishing their turn, i.e., enable sequential parameter transmission without significant bandwidth demands. In the case of the star topology, the central device is responsible for parameter transmission and aggregation among all devices <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib8" title="">8</a>]</cite>. Note that unlike CFL, where the central device is a server, the central device in star topology is a participating node (server or client) with additional responsibilities of parameter processing. Thus, users may select a device with the most robust communication capabilities and the highest computational power as a central node. All devices in mesh topology are interconnected to offer the highest level of reliability in the presence of communication failures, where devices directly exchange parameters.</p> </div> <div class="ltx_para" id="S2.p4"> <p class="ltx_p" id="S2.p4.1">The participating nodes in DFL can be categorized as a trainer, aggregator, proxy, and idle <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib4" title="">4</a>]</cite>. The <span class="ltx_text ltx_font_italic" id="S2.p4.1.1">trainer and aggregator</span> nodes process data to train a model locally and aggregate parameters, respectively. The <span class="ltx_text ltx_font_italic" id="S2.p4.1.2">proxy</span> aggregates and forwards data between devices. Finally, <span class="ltx_text ltx_font_italic" id="S2.p4.1.3">idle</span> nodes are the ones who temporarily refrain from participating in the current training, e.g., maybe due to resource limitations. Thus, each node acts as a trainer in the continuous ring and linear topologies. Nodes are both trainer and aggregator in the aggregate ring, aggregate linear, and mesh topologies. Finally, every node except the central one serves as a trainer in the star topology, where the central node serves as the trainer and the aggregator. We do not consider proxy or idle nodes in our evaluation, as our focus is to assess the impact of topologies and training strategies on the performance of DFL.</p> </div> <div class="ltx_para" id="S2.p5"> <p class="ltx_p" id="S2.p5.1"><span class="ltx_text ltx_font_bold" id="S2.p5.1.1">Training Strategy:</span> DFL has two training strategies: <span class="ltx_text ltx_font_italic" id="S2.p5.1.2">aggregate</span> and <span class="ltx_text ltx_font_italic" id="S2.p5.1.3">continuous</span>. The former is similar to FedAvg <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib9" title="">9</a>]</cite>, i.e., participating devices collect parameters from others and aggregate them along with their own parameters. In a continuous scheme, participating devices start training from one of them and then sequentially pass parameters to subsequent ones to update their models. This scheme focuses on continuously learning and adapting to new data considering previously learned information, i.e., useful in scenarios where data distribution changes over time. Thus, if we combine the standard four topologies with training strategies, we get six deployment options for DFL training. Note that we exclude continuous star and continuous mesh as such combinations are not legitimate due to the ordering on message exchanges. Thus, we analyze the performance of these six deployments (Continuous Linear, Continuous Ring, Aggregate Linear, Aggregate Ring, Aggregate Star, and Aggregate Mesh) both mathematically and experimentally to show their applicability in different applications.</p> </div> <div class="ltx_para" id="S2.p6"> <p class="ltx_p" id="S2.p6.1"><span class="ltx_text ltx_font_bold" id="S2.p6.1.1">Non-IID Data:</span> Non-IID data refers to the uneven or random statistical distribution of data on different clients. In federated learning, this means that the data on each device may vary significantly, which can adversely impact the distributed training process <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib11" title="">11</a>]</cite>. Zhao <span class="ltx_text ltx_font_italic" id="S2.p6.1.2">et al.</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib10" title="">10</a>]</cite> have demonstrated that the accuracy of FedAvg <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib9" title="">9</a>]</cite>, a fundamental algorithm in federated learning that averages the model parameters from each client weighted by the number of samples on the device to form a global model, diminishes notably with increasing data heterogeneity. Hangyu <span class="ltx_text ltx_font_italic" id="S2.p6.1.3">et al.</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib11" title="">11</a>]</cite> have showcased various types of non-IID data in real-life scenarios, categorized into <span class="ltx_text ltx_font_italic" id="S2.p6.1.4">attribute</span> skew and <span class="ltx_text ltx_font_italic" id="S2.p6.1.5">label</span> skew. The former often associated with Horizontal Federated Learning (HFL), involves variations in the number of attributes on different devices. Notably, our study predominantly explores label skew due to its prevalence in Vertical Federated Learning (VFL), which is more relevant to our study on DFL. Label skew refers to differences in the number of labels for each device. In our investigation, we quantify the degree of non-IID Data in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S7" title="VII Evaluation Results ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">VII</span></a>.</p> </div> <div class="ltx_para" id="S2.p7"> <p class="ltx_p" id="S2.p7.1">In this study, we focus on Horizontal Federated Learning (HFL), the most common form of federated learning, which finds widespread applications in real-world scenarios, such as user device behavior data and regional medical records. By evaluating the performance of DFL under different network topologies within an HFL context, our research ensures broader applicability and practical relevance. Additionally, we emphasize the importance of Label Skew, a prevalent form of non-IID data in HFL, where clients’ datasets often exhibit imbalanced label distributions. This imbalance can cause local models to overfit specific categories, thereby degrading the generalization capability of the global model. Label skew is widely observed in practice and significantly impacts the convergence and parameter aggregation of DFL. Investigating how DFL performs under different topologies with label skew enhances our understanding of optimizing decentralized federated systems, improving robustness and efficiency in non-IID data scenarios.</p> </div> <div class="ltx_para" id="S2.p8"> <p class="ltx_p" id="S2.p8.9"><span class="ltx_text ltx_font_bold" id="S2.p8.1.1">Strongly Convex and <math alttext="L" class="ltx_Math" display="inline" id="S2.p8.1.1.m1.1"><semantics id="S2.p8.1.1.m1.1a"><mi id="S2.p8.1.1.m1.1.1" xref="S2.p8.1.1.m1.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S2.p8.1.1.m1.1b"><ci id="S2.p8.1.1.m1.1.1.cmml" xref="S2.p8.1.1.m1.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.1.1.m1.1c">L</annotation><annotation encoding="application/x-llamapun" id="S2.p8.1.1.m1.1d">italic_L</annotation></semantics></math>-smooth:</span> A general assumption in federated learning-based training is that the models are strongly convex <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib7" title="">7</a>]</cite>, which facilitates the stability and convergence of algorithms. This assumption ensures that the function has a unique global minimum without being trapped in local minima when searching for the optimal solution, thereby enhancing the stability and reliability of the algorithm. A function is <span class="ltx_text ltx_font_italic" id="S2.p8.9.2">strongly convex</span> if there is a constant <math alttext="\mu>0" class="ltx_Math" display="inline" id="S2.p8.2.m1.1"><semantics id="S2.p8.2.m1.1a"><mrow id="S2.p8.2.m1.1.1" xref="S2.p8.2.m1.1.1.cmml"><mi id="S2.p8.2.m1.1.1.2" xref="S2.p8.2.m1.1.1.2.cmml">μ</mi><mo id="S2.p8.2.m1.1.1.1" xref="S2.p8.2.m1.1.1.1.cmml">></mo><mn id="S2.p8.2.m1.1.1.3" xref="S2.p8.2.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p8.2.m1.1b"><apply id="S2.p8.2.m1.1.1.cmml" xref="S2.p8.2.m1.1.1"><gt id="S2.p8.2.m1.1.1.1.cmml" xref="S2.p8.2.m1.1.1.1"></gt><ci id="S2.p8.2.m1.1.1.2.cmml" xref="S2.p8.2.m1.1.1.2">𝜇</ci><cn id="S2.p8.2.m1.1.1.3.cmml" type="integer" xref="S2.p8.2.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.2.m1.1c">\mu>0</annotation><annotation encoding="application/x-llamapun" id="S2.p8.2.m1.1d">italic_μ > 0</annotation></semantics></math> such that for all points <math alttext="x" class="ltx_Math" display="inline" id="S2.p8.3.m2.1"><semantics id="S2.p8.3.m2.1a"><mi id="S2.p8.3.m2.1.1" xref="S2.p8.3.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.p8.3.m2.1b"><ci id="S2.p8.3.m2.1.1.cmml" xref="S2.p8.3.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.3.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.p8.3.m2.1d">italic_x</annotation></semantics></math> and <math alttext="y" class="ltx_Math" display="inline" id="S2.p8.4.m3.1"><semantics id="S2.p8.4.m3.1a"><mi id="S2.p8.4.m3.1.1" xref="S2.p8.4.m3.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S2.p8.4.m3.1b"><ci id="S2.p8.4.m3.1.1.cmml" xref="S2.p8.4.m3.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.4.m3.1c">y</annotation><annotation encoding="application/x-llamapun" id="S2.p8.4.m3.1d">italic_y</annotation></semantics></math> in its domain, the function lies above a quadratic curve with curvature <math alttext="\mu" class="ltx_Math" display="inline" id="S2.p8.5.m4.1"><semantics id="S2.p8.5.m4.1a"><mi id="S2.p8.5.m4.1.1" xref="S2.p8.5.m4.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.p8.5.m4.1b"><ci id="S2.p8.5.m4.1.1.cmml" xref="S2.p8.5.m4.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.5.m4.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.p8.5.m4.1d">italic_μ</annotation></semantics></math>. This property ensures the function has a unique global minimum, leading to faster convergence rates <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib12" title="">12</a>]</cite>. A function is called <span class="ltx_text ltx_font_italic" id="S2.p8.9.3">L-smooth</span> if its gradient is L-Lipschitz continuous, i.e., the rate of change of the function is constrained by <math alttext="L" class="ltx_Math" display="inline" id="S2.p8.6.m5.1"><semantics id="S2.p8.6.m5.1a"><mi id="S2.p8.6.m5.1.1" xref="S2.p8.6.m5.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S2.p8.6.m5.1b"><ci id="S2.p8.6.m5.1.1.cmml" xref="S2.p8.6.m5.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.6.m5.1c">L</annotation><annotation encoding="application/x-llamapun" id="S2.p8.6.m5.1d">italic_L</annotation></semantics></math>; in other words, the change in the gradient is bounded by <math alttext="L" class="ltx_Math" display="inline" id="S2.p8.7.m6.1"><semantics id="S2.p8.7.m6.1a"><mi id="S2.p8.7.m6.1.1" xref="S2.p8.7.m6.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S2.p8.7.m6.1b"><ci id="S2.p8.7.m6.1.1.cmml" xref="S2.p8.7.m6.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.7.m6.1c">L</annotation><annotation encoding="application/x-llamapun" id="S2.p8.7.m6.1d">italic_L</annotation></semantics></math> times the distance between <math alttext="x" class="ltx_Math" display="inline" id="S2.p8.8.m7.1"><semantics id="S2.p8.8.m7.1a"><mi id="S2.p8.8.m7.1.1" xref="S2.p8.8.m7.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.p8.8.m7.1b"><ci id="S2.p8.8.m7.1.1.cmml" xref="S2.p8.8.m7.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.8.m7.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.p8.8.m7.1d">italic_x</annotation></semantics></math> and <math alttext="y" class="ltx_Math" display="inline" id="S2.p8.9.m8.1"><semantics id="S2.p8.9.m8.1a"><mi id="S2.p8.9.m8.1.1" xref="S2.p8.9.m8.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S2.p8.9.m8.1b"><ci id="S2.p8.9.m8.1.1.cmml" xref="S2.p8.9.m8.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.9.m8.1c">y</annotation><annotation encoding="application/x-llamapun" id="S2.p8.9.m8.1d">italic_y</annotation></semantics></math>. This property ensures that optimal learning does not overshoot the minimum. Both functions guide the convergence rate derivation process to determine the convergence speed; thus, they are incorporated into our mathematical analysis.</p> </div> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">III </span><span class="ltx_text ltx_font_smallcaps" id="S3.1.1">Related Works</span> </h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.1">This section presents the analysis of CFL and DFL similar to our study in order to position our contribution to the field.</p> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.1"><span class="ltx_text ltx_font_bold" id="S3.p2.1.1">Centralized Federated Learning:</span> Hangyu <span class="ltx_text ltx_font_italic" id="S3.p2.1.2">et al.</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib11" title="">11</a>]</cite> evaluate various degrees of non-IID data in real-world scenarios and show that the degree of such data significantly impacts the prediction accuracy of the global model. Similarly, Zhao <span class="ltx_text ltx_font_italic" id="S3.p2.1.3">et al.</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib10" title="">10</a>]</cite> show as data heterogeneity increases, the accuracy of the FedAvg algorithm decreases significantly. Their findings show that when data distribution differs greatly among different clients, the effectiveness of centralized federated learning is significantly compromised. Li <span class="ltx_text ltx_font_italic" id="S3.p2.1.4">et al.</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib7" title="">7</a>]</cite> mathematically prove the above claim and provide a solid foundation for understanding the impact of non-IID data on the convergence of federated learning. However, their work lacks analysis of different network topologies and only focuses on the FedAvg training strategy without exploring other training strategies.</p> </div> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.1"><span class="ltx_text ltx_font_bold" id="S3.p3.1.1">Decentralized Federated Learning:</span> Sheller <span class="ltx_text ltx_font_italic" id="S3.p3.1.2">et al.</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib5" title="">5</a>]</cite> are the first to analyze the performance comparison between continuous linear, continuous ring, and centralized federated learning. This study offers valuable insights into the impact of different topologies on DFL performance. However, the analysis lacks other standard topologies like star and mesh, the effect of non-IID data, and theoretical analysis. A similar study is performed in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib13" title="">13</a>]</cite> without analyzing topological impact. Angelia <span class="ltx_text ltx_font_italic" id="S3.p3.1.3">et al.</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib14" title="">14</a>]</cite> analyze the network topologies of DFL using undirected and directed graphs. They apply the Metropolis algorithm to update directed graphs and the Push-sum algorithm for undirected graphs, analyzing the computational complexity and convergence speed of each method. However, the study does not show the impact of non-IID data over various network topologies.</p> </div> <div class="ltx_para" id="S3.p4"> <p class="ltx_p" id="S3.p4.1">In summary, although existing studies have made some progress in exploring DFL, there are still unresolved issues. Particularly in handling non-IID data and complex network topologies, the performance and convergence of DFL require further analysis. We explore the performance of DFL through both theoretical <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4" title="IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">IV</span></a> and experimental <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S7" title="VII Evaluation Results ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">VII</span></a> analysis. We first define six different DFL configurations based on two different training strategies and network topologies and conduct convergence analysis on them <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4" title="IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">IV</span></a>. Then, we define the required non-IID settings in the experiments <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S7" title="VII Evaluation Results ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">VII</span></a> and measure the performance of the six different DFL configurations under varying degrees of non-IID data.</p> </div> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">IV </span><span class="ltx_text ltx_font_smallcaps" id="S4.1.1">Convergence Rate Analysis</span> </h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">This section develops optimization problems to derive the convergence rate of six different DFL deployments to shed light on their expected behavior. We use the notations in Table <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4.T1" title="TABLE I ‣ IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">I</span></a> throughout this section.</p> </div> <figure class="ltx_table" id="S4.T1"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S4.T1.24.1.1" style="font-size:90%;">TABLE I</span>: </span><span class="ltx_text" id="S4.T1.25.2" style="font-size:90%;">Parameters used in our analysis</span></figcaption> <table class="ltx_tabular ltx_centering ltx_align_middle" id="S4.T1.22"> <tr class="ltx_tr" id="S4.T1.22.23"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.22.23.1"><span class="ltx_text ltx_font_bold" id="S4.T1.22.23.1.1">Notation</span></td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S4.T1.22.23.2"><span class="ltx_text ltx_font_bold" id="S4.T1.22.23.2.1">Description</span></td> </tr> <tr class="ltx_tr" id="S4.T1.1.1"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.1.1.1"><math alttext="\mathcal{L}oss(\cdot;\cdot)" class="ltx_Math" display="inline" id="S4.T1.1.1.1.m1.2"><semantics id="S4.T1.1.1.1.m1.2a"><mrow id="S4.T1.1.1.1.m1.2.3" xref="S4.T1.1.1.1.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.T1.1.1.1.m1.2.3.2" xref="S4.T1.1.1.1.m1.2.3.2.cmml">ℒ</mi><mo id="S4.T1.1.1.1.m1.2.3.1" xref="S4.T1.1.1.1.m1.2.3.1.cmml">⁢</mo><mi id="S4.T1.1.1.1.m1.2.3.3" xref="S4.T1.1.1.1.m1.2.3.3.cmml">o</mi><mo id="S4.T1.1.1.1.m1.2.3.1a" xref="S4.T1.1.1.1.m1.2.3.1.cmml">⁢</mo><mi id="S4.T1.1.1.1.m1.2.3.4" xref="S4.T1.1.1.1.m1.2.3.4.cmml">s</mi><mo id="S4.T1.1.1.1.m1.2.3.1b" xref="S4.T1.1.1.1.m1.2.3.1.cmml">⁢</mo><mi id="S4.T1.1.1.1.m1.2.3.5" xref="S4.T1.1.1.1.m1.2.3.5.cmml">s</mi><mo id="S4.T1.1.1.1.m1.2.3.1c" xref="S4.T1.1.1.1.m1.2.3.1.cmml">⁢</mo><mrow id="S4.T1.1.1.1.m1.2.3.6.2" xref="S4.T1.1.1.1.m1.2.3.6.1.cmml"><mo id="S4.T1.1.1.1.m1.2.3.6.2.1" stretchy="false" xref="S4.T1.1.1.1.m1.2.3.6.1.cmml">(</mo><mo id="S4.T1.1.1.1.m1.1.1" lspace="0em" rspace="0em" xref="S4.T1.1.1.1.m1.1.1.cmml">⋅</mo><mo id="S4.T1.1.1.1.m1.2.3.6.2.2" rspace="0em" xref="S4.T1.1.1.1.m1.2.3.6.1.cmml">;</mo><mo id="S4.T1.1.1.1.m1.2.2" lspace="0em" rspace="0em" xref="S4.T1.1.1.1.m1.2.2.cmml">⋅</mo><mo id="S4.T1.1.1.1.m1.2.3.6.2.3" stretchy="false" xref="S4.T1.1.1.1.m1.2.3.6.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.1.1.1.m1.2b"><apply id="S4.T1.1.1.1.m1.2.3.cmml" xref="S4.T1.1.1.1.m1.2.3"><times id="S4.T1.1.1.1.m1.2.3.1.cmml" xref="S4.T1.1.1.1.m1.2.3.1"></times><ci id="S4.T1.1.1.1.m1.2.3.2.cmml" xref="S4.T1.1.1.1.m1.2.3.2">ℒ</ci><ci id="S4.T1.1.1.1.m1.2.3.3.cmml" xref="S4.T1.1.1.1.m1.2.3.3">𝑜</ci><ci id="S4.T1.1.1.1.m1.2.3.4.cmml" xref="S4.T1.1.1.1.m1.2.3.4">𝑠</ci><ci id="S4.T1.1.1.1.m1.2.3.5.cmml" xref="S4.T1.1.1.1.m1.2.3.5">𝑠</ci><list id="S4.T1.1.1.1.m1.2.3.6.1.cmml" xref="S4.T1.1.1.1.m1.2.3.6.2"><ci id="S4.T1.1.1.1.m1.1.1.cmml" xref="S4.T1.1.1.1.m1.1.1">⋅</ci><ci id="S4.T1.1.1.1.m1.2.2.cmml" xref="S4.T1.1.1.1.m1.2.2">⋅</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.1.1.1.m1.2c">\mathcal{L}oss(\cdot;\cdot)</annotation><annotation encoding="application/x-llamapun" id="S4.T1.1.1.1.m1.2d">caligraphic_L italic_o italic_s italic_s ( ⋅ ; ⋅ )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S4.T1.1.1.2">User-specified loss function</td> </tr> <tr class="ltx_tr" id="S4.T1.3.3"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.2.2.1"><math alttext="p_{k}" class="ltx_Math" display="inline" id="S4.T1.2.2.1.m1.1"><semantics id="S4.T1.2.2.1.m1.1a"><msub id="S4.T1.2.2.1.m1.1.1" xref="S4.T1.2.2.1.m1.1.1.cmml"><mi id="S4.T1.2.2.1.m1.1.1.2" xref="S4.T1.2.2.1.m1.1.1.2.cmml">p</mi><mi id="S4.T1.2.2.1.m1.1.1.3" xref="S4.T1.2.2.1.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.T1.2.2.1.m1.1b"><apply id="S4.T1.2.2.1.m1.1.1.cmml" xref="S4.T1.2.2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.T1.2.2.1.m1.1.1.1.cmml" xref="S4.T1.2.2.1.m1.1.1">subscript</csymbol><ci id="S4.T1.2.2.1.m1.1.1.2.cmml" xref="S4.T1.2.2.1.m1.1.1.2">𝑝</ci><ci id="S4.T1.2.2.1.m1.1.1.3.cmml" xref="S4.T1.2.2.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.2.2.1.m1.1c">p_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.T1.2.2.1.m1.1d">italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S4.T1.3.3.2">Weights of the <math alttext="k" class="ltx_Math" display="inline" id="S4.T1.3.3.2.m1.1"><semantics id="S4.T1.3.3.2.m1.1a"><mi id="S4.T1.3.3.2.m1.1.1" xref="S4.T1.3.3.2.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.T1.3.3.2.m1.1b"><ci id="S4.T1.3.3.2.m1.1.1.cmml" xref="S4.T1.3.3.2.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.3.3.2.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.T1.3.3.2.m1.1d">italic_k</annotation></semantics></math>-th device</td> </tr> <tr class="ltx_tr" id="S4.T1.4.4"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.4.4.1"><math alttext="N" class="ltx_Math" display="inline" id="S4.T1.4.4.1.m1.1"><semantics id="S4.T1.4.4.1.m1.1a"><mi id="S4.T1.4.4.1.m1.1.1" xref="S4.T1.4.4.1.m1.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S4.T1.4.4.1.m1.1b"><ci id="S4.T1.4.4.1.m1.1.1.cmml" xref="S4.T1.4.4.1.m1.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.4.4.1.m1.1c">N</annotation><annotation encoding="application/x-llamapun" id="S4.T1.4.4.1.m1.1d">italic_N</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S4.T1.4.4.2">Number of devices</td> </tr> <tr class="ltx_tr" id="S4.T1.5.5"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.5.5.1"><math alttext="F(w)" class="ltx_Math" display="inline" id="S4.T1.5.5.1.m1.1"><semantics id="S4.T1.5.5.1.m1.1a"><mrow id="S4.T1.5.5.1.m1.1.2" xref="S4.T1.5.5.1.m1.1.2.cmml"><mi id="S4.T1.5.5.1.m1.1.2.2" xref="S4.T1.5.5.1.m1.1.2.2.cmml">F</mi><mo id="S4.T1.5.5.1.m1.1.2.1" xref="S4.T1.5.5.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.T1.5.5.1.m1.1.2.3.2" xref="S4.T1.5.5.1.m1.1.2.cmml"><mo id="S4.T1.5.5.1.m1.1.2.3.2.1" stretchy="false" xref="S4.T1.5.5.1.m1.1.2.cmml">(</mo><mi id="S4.T1.5.5.1.m1.1.1" xref="S4.T1.5.5.1.m1.1.1.cmml">w</mi><mo id="S4.T1.5.5.1.m1.1.2.3.2.2" stretchy="false" xref="S4.T1.5.5.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.5.5.1.m1.1b"><apply id="S4.T1.5.5.1.m1.1.2.cmml" xref="S4.T1.5.5.1.m1.1.2"><times id="S4.T1.5.5.1.m1.1.2.1.cmml" xref="S4.T1.5.5.1.m1.1.2.1"></times><ci id="S4.T1.5.5.1.m1.1.2.2.cmml" xref="S4.T1.5.5.1.m1.1.2.2">𝐹</ci><ci id="S4.T1.5.5.1.m1.1.1.cmml" xref="S4.T1.5.5.1.m1.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.5.5.1.m1.1c">F(w)</annotation><annotation encoding="application/x-llamapun" id="S4.T1.5.5.1.m1.1d">italic_F ( italic_w )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S4.T1.5.5.2">Function for Decentralized Federated Learning (DFL)</td> </tr> <tr class="ltx_tr" id="S4.T1.7.7"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.6.6.1"><math alttext="F^{\ast}" class="ltx_Math" display="inline" id="S4.T1.6.6.1.m1.1"><semantics id="S4.T1.6.6.1.m1.1a"><msup id="S4.T1.6.6.1.m1.1.1" xref="S4.T1.6.6.1.m1.1.1.cmml"><mi id="S4.T1.6.6.1.m1.1.1.2" xref="S4.T1.6.6.1.m1.1.1.2.cmml">F</mi><mo id="S4.T1.6.6.1.m1.1.1.3" xref="S4.T1.6.6.1.m1.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.T1.6.6.1.m1.1b"><apply id="S4.T1.6.6.1.m1.1.1.cmml" xref="S4.T1.6.6.1.m1.1.1"><csymbol cd="ambiguous" id="S4.T1.6.6.1.m1.1.1.1.cmml" xref="S4.T1.6.6.1.m1.1.1">superscript</csymbol><ci id="S4.T1.6.6.1.m1.1.1.2.cmml" xref="S4.T1.6.6.1.m1.1.1.2">𝐹</ci><ci id="S4.T1.6.6.1.m1.1.1.3.cmml" xref="S4.T1.6.6.1.m1.1.1.3">∗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.6.6.1.m1.1c">F^{\ast}</annotation><annotation encoding="application/x-llamapun" id="S4.T1.6.6.1.m1.1d">italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S4.T1.7.7.2">The optimal <math alttext="F" class="ltx_Math" display="inline" id="S4.T1.7.7.2.m1.1"><semantics id="S4.T1.7.7.2.m1.1a"><mi id="S4.T1.7.7.2.m1.1.1" xref="S4.T1.7.7.2.m1.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S4.T1.7.7.2.m1.1b"><ci id="S4.T1.7.7.2.m1.1.1.cmml" xref="S4.T1.7.7.2.m1.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.7.7.2.m1.1c">F</annotation><annotation encoding="application/x-llamapun" id="S4.T1.7.7.2.m1.1d">italic_F</annotation></semantics></math> under ideal conditions</td> </tr> <tr class="ltx_tr" id="S4.T1.9.9"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.8.8.1"><math alttext="F_{k}" class="ltx_Math" display="inline" id="S4.T1.8.8.1.m1.1"><semantics id="S4.T1.8.8.1.m1.1a"><msub id="S4.T1.8.8.1.m1.1.1" xref="S4.T1.8.8.1.m1.1.1.cmml"><mi id="S4.T1.8.8.1.m1.1.1.2" xref="S4.T1.8.8.1.m1.1.1.2.cmml">F</mi><mi id="S4.T1.8.8.1.m1.1.1.3" xref="S4.T1.8.8.1.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.T1.8.8.1.m1.1b"><apply id="S4.T1.8.8.1.m1.1.1.cmml" xref="S4.T1.8.8.1.m1.1.1"><csymbol cd="ambiguous" id="S4.T1.8.8.1.m1.1.1.1.cmml" xref="S4.T1.8.8.1.m1.1.1">subscript</csymbol><ci id="S4.T1.8.8.1.m1.1.1.2.cmml" xref="S4.T1.8.8.1.m1.1.1.2">𝐹</ci><ci id="S4.T1.8.8.1.m1.1.1.3.cmml" xref="S4.T1.8.8.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.8.8.1.m1.1c">F_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.T1.8.8.1.m1.1d">italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S4.T1.9.9.2">Objective for the <math alttext="k" class="ltx_Math" display="inline" id="S4.T1.9.9.2.m1.1"><semantics id="S4.T1.9.9.2.m1.1a"><mi id="S4.T1.9.9.2.m1.1.1" xref="S4.T1.9.9.2.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.T1.9.9.2.m1.1b"><ci id="S4.T1.9.9.2.m1.1.1.cmml" xref="S4.T1.9.9.2.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.9.9.2.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.T1.9.9.2.m1.1d">italic_k</annotation></semantics></math>-th device</td> </tr> <tr class="ltx_tr" id="S4.T1.11.11"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.10.10.1"><math alttext="F_{k}^{\ast}" class="ltx_Math" display="inline" id="S4.T1.10.10.1.m1.1"><semantics id="S4.T1.10.10.1.m1.1a"><msubsup id="S4.T1.10.10.1.m1.1.1" xref="S4.T1.10.10.1.m1.1.1.cmml"><mi id="S4.T1.10.10.1.m1.1.1.2.2" xref="S4.T1.10.10.1.m1.1.1.2.2.cmml">F</mi><mi id="S4.T1.10.10.1.m1.1.1.2.3" xref="S4.T1.10.10.1.m1.1.1.2.3.cmml">k</mi><mo id="S4.T1.10.10.1.m1.1.1.3" xref="S4.T1.10.10.1.m1.1.1.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.T1.10.10.1.m1.1b"><apply id="S4.T1.10.10.1.m1.1.1.cmml" xref="S4.T1.10.10.1.m1.1.1"><csymbol cd="ambiguous" id="S4.T1.10.10.1.m1.1.1.1.cmml" xref="S4.T1.10.10.1.m1.1.1">superscript</csymbol><apply id="S4.T1.10.10.1.m1.1.1.2.cmml" xref="S4.T1.10.10.1.m1.1.1"><csymbol cd="ambiguous" id="S4.T1.10.10.1.m1.1.1.2.1.cmml" xref="S4.T1.10.10.1.m1.1.1">subscript</csymbol><ci id="S4.T1.10.10.1.m1.1.1.2.2.cmml" xref="S4.T1.10.10.1.m1.1.1.2.2">𝐹</ci><ci id="S4.T1.10.10.1.m1.1.1.2.3.cmml" xref="S4.T1.10.10.1.m1.1.1.2.3">𝑘</ci></apply><ci id="S4.T1.10.10.1.m1.1.1.3.cmml" xref="S4.T1.10.10.1.m1.1.1.3">∗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.10.10.1.m1.1c">F_{k}^{\ast}</annotation><annotation encoding="application/x-llamapun" id="S4.T1.10.10.1.m1.1d">italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S4.T1.11.11.2">The optimal <math alttext="F_{k}" class="ltx_Math" display="inline" id="S4.T1.11.11.2.m1.1"><semantics id="S4.T1.11.11.2.m1.1a"><msub id="S4.T1.11.11.2.m1.1.1" xref="S4.T1.11.11.2.m1.1.1.cmml"><mi id="S4.T1.11.11.2.m1.1.1.2" xref="S4.T1.11.11.2.m1.1.1.2.cmml">F</mi><mi id="S4.T1.11.11.2.m1.1.1.3" xref="S4.T1.11.11.2.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.T1.11.11.2.m1.1b"><apply id="S4.T1.11.11.2.m1.1.1.cmml" xref="S4.T1.11.11.2.m1.1.1"><csymbol cd="ambiguous" id="S4.T1.11.11.2.m1.1.1.1.cmml" xref="S4.T1.11.11.2.m1.1.1">subscript</csymbol><ci id="S4.T1.11.11.2.m1.1.1.2.cmml" xref="S4.T1.11.11.2.m1.1.1.2">𝐹</ci><ci id="S4.T1.11.11.2.m1.1.1.3.cmml" xref="S4.T1.11.11.2.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.11.11.2.m1.1c">F_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.T1.11.11.2.m1.1d">italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> under ideal conditions</td> </tr> <tr class="ltx_tr" id="S4.T1.13.13"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.12.12.1"><math alttext="\xi_{k}" class="ltx_Math" display="inline" id="S4.T1.12.12.1.m1.1"><semantics id="S4.T1.12.12.1.m1.1a"><msub id="S4.T1.12.12.1.m1.1.1" xref="S4.T1.12.12.1.m1.1.1.cmml"><mi id="S4.T1.12.12.1.m1.1.1.2" xref="S4.T1.12.12.1.m1.1.1.2.cmml">ξ</mi><mi id="S4.T1.12.12.1.m1.1.1.3" xref="S4.T1.12.12.1.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.T1.12.12.1.m1.1b"><apply id="S4.T1.12.12.1.m1.1.1.cmml" xref="S4.T1.12.12.1.m1.1.1"><csymbol cd="ambiguous" id="S4.T1.12.12.1.m1.1.1.1.cmml" xref="S4.T1.12.12.1.m1.1.1">subscript</csymbol><ci id="S4.T1.12.12.1.m1.1.1.2.cmml" xref="S4.T1.12.12.1.m1.1.1.2">𝜉</ci><ci id="S4.T1.12.12.1.m1.1.1.3.cmml" xref="S4.T1.12.12.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.12.12.1.m1.1c">\xi_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.T1.12.12.1.m1.1d">italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S4.T1.13.13.2">SGD randomly chosen sample from <math alttext="k" class="ltx_Math" display="inline" id="S4.T1.13.13.2.m1.1"><semantics id="S4.T1.13.13.2.m1.1a"><mi id="S4.T1.13.13.2.m1.1.1" xref="S4.T1.13.13.2.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.T1.13.13.2.m1.1b"><ci id="S4.T1.13.13.2.m1.1.1.cmml" xref="S4.T1.13.13.2.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.13.13.2.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.T1.13.13.2.m1.1d">italic_k</annotation></semantics></math>-th device</td> </tr> <tr class="ltx_tr" id="S4.T1.15.15"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.14.14.1"><math alttext="w_{k}" class="ltx_Math" display="inline" id="S4.T1.14.14.1.m1.1"><semantics id="S4.T1.14.14.1.m1.1a"><msub id="S4.T1.14.14.1.m1.1.1" xref="S4.T1.14.14.1.m1.1.1.cmml"><mi id="S4.T1.14.14.1.m1.1.1.2" xref="S4.T1.14.14.1.m1.1.1.2.cmml">w</mi><mi id="S4.T1.14.14.1.m1.1.1.3" xref="S4.T1.14.14.1.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.T1.14.14.1.m1.1b"><apply id="S4.T1.14.14.1.m1.1.1.cmml" xref="S4.T1.14.14.1.m1.1.1"><csymbol cd="ambiguous" id="S4.T1.14.14.1.m1.1.1.1.cmml" xref="S4.T1.14.14.1.m1.1.1">subscript</csymbol><ci id="S4.T1.14.14.1.m1.1.1.2.cmml" xref="S4.T1.14.14.1.m1.1.1.2">𝑤</ci><ci id="S4.T1.14.14.1.m1.1.1.3.cmml" xref="S4.T1.14.14.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.14.14.1.m1.1c">w_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.T1.14.14.1.m1.1d">italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S4.T1.15.15.2">Weights for <math alttext="k" class="ltx_Math" display="inline" id="S4.T1.15.15.2.m1.1"><semantics id="S4.T1.15.15.2.m1.1a"><mi id="S4.T1.15.15.2.m1.1.1" xref="S4.T1.15.15.2.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.T1.15.15.2.m1.1b"><ci id="S4.T1.15.15.2.m1.1.1.cmml" xref="S4.T1.15.15.2.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.15.15.2.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.T1.15.15.2.m1.1d">italic_k</annotation></semantics></math>-th device</td> </tr> <tr class="ltx_tr" id="S4.T1.17.17"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.16.16.1"><math alttext="n_{k}" class="ltx_Math" display="inline" id="S4.T1.16.16.1.m1.1"><semantics id="S4.T1.16.16.1.m1.1a"><msub id="S4.T1.16.16.1.m1.1.1" xref="S4.T1.16.16.1.m1.1.1.cmml"><mi id="S4.T1.16.16.1.m1.1.1.2" xref="S4.T1.16.16.1.m1.1.1.2.cmml">n</mi><mi id="S4.T1.16.16.1.m1.1.1.3" xref="S4.T1.16.16.1.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.T1.16.16.1.m1.1b"><apply id="S4.T1.16.16.1.m1.1.1.cmml" xref="S4.T1.16.16.1.m1.1.1"><csymbol cd="ambiguous" id="S4.T1.16.16.1.m1.1.1.1.cmml" xref="S4.T1.16.16.1.m1.1.1">subscript</csymbol><ci id="S4.T1.16.16.1.m1.1.1.2.cmml" xref="S4.T1.16.16.1.m1.1.1.2">𝑛</ci><ci id="S4.T1.16.16.1.m1.1.1.3.cmml" xref="S4.T1.16.16.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.16.16.1.m1.1c">n_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.T1.16.16.1.m1.1d">italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S4.T1.17.17.2">Number of samples associated with the <math alttext="k" class="ltx_Math" display="inline" id="S4.T1.17.17.2.m1.1"><semantics id="S4.T1.17.17.2.m1.1a"><mi id="S4.T1.17.17.2.m1.1.1" xref="S4.T1.17.17.2.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.T1.17.17.2.m1.1b"><ci id="S4.T1.17.17.2.m1.1.1.cmml" xref="S4.T1.17.17.2.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.17.17.2.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.T1.17.17.2.m1.1d">italic_k</annotation></semantics></math>-th device</td> </tr> <tr class="ltx_tr" id="S4.T1.18.18"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.18.18.1"><math alttext="L" class="ltx_Math" display="inline" id="S4.T1.18.18.1.m1.1"><semantics id="S4.T1.18.18.1.m1.1a"><mi id="S4.T1.18.18.1.m1.1.1" xref="S4.T1.18.18.1.m1.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S4.T1.18.18.1.m1.1b"><ci id="S4.T1.18.18.1.m1.1.1.cmml" xref="S4.T1.18.18.1.m1.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.18.18.1.m1.1c">L</annotation><annotation encoding="application/x-llamapun" id="S4.T1.18.18.1.m1.1d">italic_L</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S4.T1.18.18.2">Parameter for L-smoothness</td> </tr> <tr class="ltx_tr" id="S4.T1.20.20"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.19.19.1"><math alttext="\mu" class="ltx_Math" display="inline" id="S4.T1.19.19.1.m1.1"><semantics id="S4.T1.19.19.1.m1.1a"><mi id="S4.T1.19.19.1.m1.1.1" xref="S4.T1.19.19.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.T1.19.19.1.m1.1b"><ci id="S4.T1.19.19.1.m1.1.1.cmml" xref="S4.T1.19.19.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.19.19.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.T1.19.19.1.m1.1d">italic_μ</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S4.T1.20.20.2">Parameter for <math alttext="\mu" class="ltx_Math" display="inline" id="S4.T1.20.20.2.m1.1"><semantics id="S4.T1.20.20.2.m1.1a"><mi id="S4.T1.20.20.2.m1.1.1" xref="S4.T1.20.20.2.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.T1.20.20.2.m1.1b"><ci id="S4.T1.20.20.2.m1.1.1.cmml" xref="S4.T1.20.20.2.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.20.20.2.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.T1.20.20.2.m1.1d">italic_μ</annotation></semantics></math>-strong convexity</td> </tr> <tr class="ltx_tr" id="S4.T1.21.21"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.21.21.1"><math alttext="Z" class="ltx_Math" display="inline" id="S4.T1.21.21.1.m1.1"><semantics id="S4.T1.21.21.1.m1.1a"><mi id="S4.T1.21.21.1.m1.1.1" xref="S4.T1.21.21.1.m1.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S4.T1.21.21.1.m1.1b"><ci id="S4.T1.21.21.1.m1.1.1.cmml" xref="S4.T1.21.21.1.m1.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.21.21.1.m1.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S4.T1.21.21.1.m1.1d">italic_Z</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S4.T1.21.21.2">The level for the NON-IID</td> </tr> <tr class="ltx_tr" id="S4.T1.22.22"> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.22.22.1"><math alttext="NC" class="ltx_Math" display="inline" id="S4.T1.22.22.1.m1.1"><semantics id="S4.T1.22.22.1.m1.1a"><mrow id="S4.T1.22.22.1.m1.1.1" xref="S4.T1.22.22.1.m1.1.1.cmml"><mi id="S4.T1.22.22.1.m1.1.1.2" xref="S4.T1.22.22.1.m1.1.1.2.cmml">N</mi><mo id="S4.T1.22.22.1.m1.1.1.1" xref="S4.T1.22.22.1.m1.1.1.1.cmml">⁢</mo><mi id="S4.T1.22.22.1.m1.1.1.3" xref="S4.T1.22.22.1.m1.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.22.22.1.m1.1b"><apply id="S4.T1.22.22.1.m1.1.1.cmml" xref="S4.T1.22.22.1.m1.1.1"><times id="S4.T1.22.22.1.m1.1.1.1.cmml" xref="S4.T1.22.22.1.m1.1.1.1"></times><ci id="S4.T1.22.22.1.m1.1.1.2.cmml" xref="S4.T1.22.22.1.m1.1.1.2">𝑁</ci><ci id="S4.T1.22.22.1.m1.1.1.3.cmml" xref="S4.T1.22.22.1.m1.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.22.22.1.m1.1c">NC</annotation><annotation encoding="application/x-llamapun" id="S4.T1.22.22.1.m1.1d">italic_N italic_C</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_border_b ltx_border_r ltx_border_t" id="S4.T1.22.22.2">No coverage</td> </tr> </table> </figure> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S4.SS1.4.1.1">IV-A</span> </span><span class="ltx_text ltx_font_italic" id="S4.SS1.5.2">Problem Formulation</span> </h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1">We consider a homogeneous environment (e.g., industrial edge nodes perform similar tasks) composed of devices with the same communication and computing capabilities. We do not distinguish between wired and wireless communication; instead, we assume an abstract mode of communication with a focus on the effect of network topologies on DFL performance. We develop distributed optimization problems for DFL deployment as shown in Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4.E1" title="In IV-A Problem Formulation ‣ IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">1</span></a>, where the function focuses on optimizing the global model parameters by minimizing the global loss function <math alttext="F(w)" class="ltx_Math" display="inline" id="S4.SS1.p1.1.m1.1"><semantics id="S4.SS1.p1.1.m1.1a"><mrow id="S4.SS1.p1.1.m1.1.2" xref="S4.SS1.p1.1.m1.1.2.cmml"><mi id="S4.SS1.p1.1.m1.1.2.2" xref="S4.SS1.p1.1.m1.1.2.2.cmml">F</mi><mo id="S4.SS1.p1.1.m1.1.2.1" xref="S4.SS1.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p1.1.m1.1.2.3.2" xref="S4.SS1.p1.1.m1.1.2.cmml"><mo id="S4.SS1.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S4.SS1.p1.1.m1.1.2.cmml">(</mo><mi id="S4.SS1.p1.1.m1.1.1" xref="S4.SS1.p1.1.m1.1.1.cmml">w</mi><mo id="S4.SS1.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S4.SS1.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.1b"><apply id="S4.SS1.p1.1.m1.1.2.cmml" xref="S4.SS1.p1.1.m1.1.2"><times id="S4.SS1.p1.1.m1.1.2.1.cmml" xref="S4.SS1.p1.1.m1.1.2.1"></times><ci id="S4.SS1.p1.1.m1.1.2.2.cmml" xref="S4.SS1.p1.1.m1.1.2.2">𝐹</ci><ci id="S4.SS1.p1.1.m1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.1.m1.1c">F(w)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.1.m1.1d">italic_F ( italic_w )</annotation></semantics></math>. Different network topologies determine how information is transmitted between devices with the same objective. This objective is to collaboratively compute local loss functions and share information to jointly optimize the global model.</p> </div> <div class="ltx_para" id="S4.SS1.p2"> <table class="ltx_equation ltx_eqn_table" id="S4.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\min_{w}\left\{F(w)=\sum_{k=1}^{N}p_{k}F_{k}(w)\right\}" class="ltx_Math" display="block" id="S4.E1.m1.4"><semantics id="S4.E1.m1.4a"><mrow id="S4.E1.m1.4.4.2" xref="S4.E1.m1.4.4.3.cmml"><munder id="S4.E1.m1.3.3.1.1" xref="S4.E1.m1.3.3.1.1.cmml"><mi id="S4.E1.m1.3.3.1.1.2" xref="S4.E1.m1.3.3.1.1.2.cmml">min</mi><mi id="S4.E1.m1.3.3.1.1.3" xref="S4.E1.m1.3.3.1.1.3.cmml">w</mi></munder><mo id="S4.E1.m1.4.4.2a" xref="S4.E1.m1.4.4.3.cmml">⁡</mo><mrow id="S4.E1.m1.4.4.2.2" xref="S4.E1.m1.4.4.3.cmml"><mo id="S4.E1.m1.4.4.2.2.2" xref="S4.E1.m1.4.4.3.cmml">{</mo><mrow id="S4.E1.m1.4.4.2.2.1" xref="S4.E1.m1.4.4.2.2.1.cmml"><mrow id="S4.E1.m1.4.4.2.2.1.2" xref="S4.E1.m1.4.4.2.2.1.2.cmml"><mi id="S4.E1.m1.4.4.2.2.1.2.2" xref="S4.E1.m1.4.4.2.2.1.2.2.cmml">F</mi><mo id="S4.E1.m1.4.4.2.2.1.2.1" xref="S4.E1.m1.4.4.2.2.1.2.1.cmml">⁢</mo><mrow id="S4.E1.m1.4.4.2.2.1.2.3.2" xref="S4.E1.m1.4.4.2.2.1.2.cmml"><mo id="S4.E1.m1.4.4.2.2.1.2.3.2.1" stretchy="false" xref="S4.E1.m1.4.4.2.2.1.2.cmml">(</mo><mi id="S4.E1.m1.1.1" xref="S4.E1.m1.1.1.cmml">w</mi><mo id="S4.E1.m1.4.4.2.2.1.2.3.2.2" stretchy="false" xref="S4.E1.m1.4.4.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="S4.E1.m1.4.4.2.2.1.1" rspace="0.111em" xref="S4.E1.m1.4.4.2.2.1.1.cmml">=</mo><mrow id="S4.E1.m1.4.4.2.2.1.3" xref="S4.E1.m1.4.4.2.2.1.3.cmml"><munderover id="S4.E1.m1.4.4.2.2.1.3.1" xref="S4.E1.m1.4.4.2.2.1.3.1.cmml"><mo id="S4.E1.m1.4.4.2.2.1.3.1.2.2" movablelimits="false" xref="S4.E1.m1.4.4.2.2.1.3.1.2.2.cmml">∑</mo><mrow id="S4.E1.m1.4.4.2.2.1.3.1.2.3" xref="S4.E1.m1.4.4.2.2.1.3.1.2.3.cmml"><mi id="S4.E1.m1.4.4.2.2.1.3.1.2.3.2" xref="S4.E1.m1.4.4.2.2.1.3.1.2.3.2.cmml">k</mi><mo id="S4.E1.m1.4.4.2.2.1.3.1.2.3.1" xref="S4.E1.m1.4.4.2.2.1.3.1.2.3.1.cmml">=</mo><mn id="S4.E1.m1.4.4.2.2.1.3.1.2.3.3" xref="S4.E1.m1.4.4.2.2.1.3.1.2.3.3.cmml">1</mn></mrow><mi id="S4.E1.m1.4.4.2.2.1.3.1.3" xref="S4.E1.m1.4.4.2.2.1.3.1.3.cmml">N</mi></munderover><mrow id="S4.E1.m1.4.4.2.2.1.3.2" xref="S4.E1.m1.4.4.2.2.1.3.2.cmml"><msub id="S4.E1.m1.4.4.2.2.1.3.2.2" xref="S4.E1.m1.4.4.2.2.1.3.2.2.cmml"><mi id="S4.E1.m1.4.4.2.2.1.3.2.2.2" xref="S4.E1.m1.4.4.2.2.1.3.2.2.2.cmml">p</mi><mi id="S4.E1.m1.4.4.2.2.1.3.2.2.3" xref="S4.E1.m1.4.4.2.2.1.3.2.2.3.cmml">k</mi></msub><mo id="S4.E1.m1.4.4.2.2.1.3.2.1" xref="S4.E1.m1.4.4.2.2.1.3.2.1.cmml">⁢</mo><msub id="S4.E1.m1.4.4.2.2.1.3.2.3" xref="S4.E1.m1.4.4.2.2.1.3.2.3.cmml"><mi id="S4.E1.m1.4.4.2.2.1.3.2.3.2" xref="S4.E1.m1.4.4.2.2.1.3.2.3.2.cmml">F</mi><mi id="S4.E1.m1.4.4.2.2.1.3.2.3.3" xref="S4.E1.m1.4.4.2.2.1.3.2.3.3.cmml">k</mi></msub><mo id="S4.E1.m1.4.4.2.2.1.3.2.1a" xref="S4.E1.m1.4.4.2.2.1.3.2.1.cmml">⁢</mo><mrow id="S4.E1.m1.4.4.2.2.1.3.2.4.2" xref="S4.E1.m1.4.4.2.2.1.3.2.cmml"><mo id="S4.E1.m1.4.4.2.2.1.3.2.4.2.1" stretchy="false" xref="S4.E1.m1.4.4.2.2.1.3.2.cmml">(</mo><mi id="S4.E1.m1.2.2" xref="S4.E1.m1.2.2.cmml">w</mi><mo id="S4.E1.m1.4.4.2.2.1.3.2.4.2.2" stretchy="false" xref="S4.E1.m1.4.4.2.2.1.3.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S4.E1.m1.4.4.2.2.3" xref="S4.E1.m1.4.4.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E1.m1.4b"><apply id="S4.E1.m1.4.4.3.cmml" xref="S4.E1.m1.4.4.2"><apply id="S4.E1.m1.3.3.1.1.cmml" xref="S4.E1.m1.3.3.1.1"><csymbol cd="ambiguous" id="S4.E1.m1.3.3.1.1.1.cmml" xref="S4.E1.m1.3.3.1.1">subscript</csymbol><min id="S4.E1.m1.3.3.1.1.2.cmml" xref="S4.E1.m1.3.3.1.1.2"></min><ci id="S4.E1.m1.3.3.1.1.3.cmml" xref="S4.E1.m1.3.3.1.1.3">𝑤</ci></apply><apply id="S4.E1.m1.4.4.2.2.1.cmml" xref="S4.E1.m1.4.4.2.2.1"><eq id="S4.E1.m1.4.4.2.2.1.1.cmml" xref="S4.E1.m1.4.4.2.2.1.1"></eq><apply id="S4.E1.m1.4.4.2.2.1.2.cmml" xref="S4.E1.m1.4.4.2.2.1.2"><times id="S4.E1.m1.4.4.2.2.1.2.1.cmml" xref="S4.E1.m1.4.4.2.2.1.2.1"></times><ci id="S4.E1.m1.4.4.2.2.1.2.2.cmml" xref="S4.E1.m1.4.4.2.2.1.2.2">𝐹</ci><ci id="S4.E1.m1.1.1.cmml" xref="S4.E1.m1.1.1">𝑤</ci></apply><apply id="S4.E1.m1.4.4.2.2.1.3.cmml" xref="S4.E1.m1.4.4.2.2.1.3"><apply id="S4.E1.m1.4.4.2.2.1.3.1.cmml" xref="S4.E1.m1.4.4.2.2.1.3.1"><csymbol cd="ambiguous" id="S4.E1.m1.4.4.2.2.1.3.1.1.cmml" xref="S4.E1.m1.4.4.2.2.1.3.1">superscript</csymbol><apply id="S4.E1.m1.4.4.2.2.1.3.1.2.cmml" xref="S4.E1.m1.4.4.2.2.1.3.1"><csymbol cd="ambiguous" id="S4.E1.m1.4.4.2.2.1.3.1.2.1.cmml" xref="S4.E1.m1.4.4.2.2.1.3.1">subscript</csymbol><sum id="S4.E1.m1.4.4.2.2.1.3.1.2.2.cmml" xref="S4.E1.m1.4.4.2.2.1.3.1.2.2"></sum><apply id="S4.E1.m1.4.4.2.2.1.3.1.2.3.cmml" xref="S4.E1.m1.4.4.2.2.1.3.1.2.3"><eq id="S4.E1.m1.4.4.2.2.1.3.1.2.3.1.cmml" xref="S4.E1.m1.4.4.2.2.1.3.1.2.3.1"></eq><ci 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id="S4.E1.m1.4.4.2.2.1.3.2.3.2.cmml" xref="S4.E1.m1.4.4.2.2.1.3.2.3.2">𝐹</ci><ci id="S4.E1.m1.4.4.2.2.1.3.2.3.3.cmml" xref="S4.E1.m1.4.4.2.2.1.3.2.3.3">𝑘</ci></apply><ci id="S4.E1.m1.2.2.cmml" xref="S4.E1.m1.2.2">𝑤</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E1.m1.4c">\min_{w}\left\{F(w)=\sum_{k=1}^{N}p_{k}F_{k}(w)\right\}</annotation><annotation encoding="application/x-llamapun" id="S4.E1.m1.4d">roman_min start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT { italic_F ( italic_w ) = ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w ) }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p2.9">Here, <math alttext="p_{k}" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.1"><semantics id="S4.SS1.p2.1.m1.1a"><msub id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml"><mi id="S4.SS1.p2.1.m1.1.1.2" xref="S4.SS1.p2.1.m1.1.1.2.cmml">p</mi><mi id="S4.SS1.p2.1.m1.1.1.3" xref="S4.SS1.p2.1.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.1b"><apply id="S4.SS1.p2.1.m1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.1.m1.1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1">subscript</csymbol><ci id="S4.SS1.p2.1.m1.1.1.2.cmml" xref="S4.SS1.p2.1.m1.1.1.2">𝑝</ci><ci id="S4.SS1.p2.1.m1.1.1.3.cmml" xref="S4.SS1.p2.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.1c">p_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.1d">italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> denotes the weights of the <math alttext="k" class="ltx_Math" display="inline" id="S4.SS1.p2.2.m2.1"><semantics id="S4.SS1.p2.2.m2.1a"><mi id="S4.SS1.p2.2.m2.1.1" xref="S4.SS1.p2.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.2.m2.1b"><ci id="S4.SS1.p2.2.m2.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.1d">italic_k</annotation></semantics></math>-th device (<math alttext="p_{k}\geq 0" class="ltx_Math" display="inline" id="S4.SS1.p2.3.m3.1"><semantics id="S4.SS1.p2.3.m3.1a"><mrow id="S4.SS1.p2.3.m3.1.1" xref="S4.SS1.p2.3.m3.1.1.cmml"><msub id="S4.SS1.p2.3.m3.1.1.2" xref="S4.SS1.p2.3.m3.1.1.2.cmml"><mi id="S4.SS1.p2.3.m3.1.1.2.2" xref="S4.SS1.p2.3.m3.1.1.2.2.cmml">p</mi><mi id="S4.SS1.p2.3.m3.1.1.2.3" xref="S4.SS1.p2.3.m3.1.1.2.3.cmml">k</mi></msub><mo id="S4.SS1.p2.3.m3.1.1.1" xref="S4.SS1.p2.3.m3.1.1.1.cmml">≥</mo><mn id="S4.SS1.p2.3.m3.1.1.3" xref="S4.SS1.p2.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.3.m3.1b"><apply id="S4.SS1.p2.3.m3.1.1.cmml" xref="S4.SS1.p2.3.m3.1.1"><geq id="S4.SS1.p2.3.m3.1.1.1.cmml" xref="S4.SS1.p2.3.m3.1.1.1"></geq><apply id="S4.SS1.p2.3.m3.1.1.2.cmml" xref="S4.SS1.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p2.3.m3.1.1.2.1.cmml" xref="S4.SS1.p2.3.m3.1.1.2">subscript</csymbol><ci id="S4.SS1.p2.3.m3.1.1.2.2.cmml" xref="S4.SS1.p2.3.m3.1.1.2.2">𝑝</ci><ci id="S4.SS1.p2.3.m3.1.1.2.3.cmml" xref="S4.SS1.p2.3.m3.1.1.2.3">𝑘</ci></apply><cn id="S4.SS1.p2.3.m3.1.1.3.cmml" type="integer" xref="S4.SS1.p2.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.3.m3.1c">p_{k}\geq 0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.3.m3.1d">italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ≥ 0</annotation></semantics></math>, <math alttext="\displaystyle\sum_{k=1}^{N}p_{k}=1" class="ltx_Math" display="inline" id="S4.SS1.p2.4.m4.1"><semantics id="S4.SS1.p2.4.m4.1a"><mrow id="S4.SS1.p2.4.m4.1.1" xref="S4.SS1.p2.4.m4.1.1.cmml"><mrow id="S4.SS1.p2.4.m4.1.1.2" xref="S4.SS1.p2.4.m4.1.1.2.cmml"><mstyle displaystyle="true" id="S4.SS1.p2.4.m4.1.1.2.1" xref="S4.SS1.p2.4.m4.1.1.2.1.cmml"><munderover id="S4.SS1.p2.4.m4.1.1.2.1a" xref="S4.SS1.p2.4.m4.1.1.2.1.cmml"><mo id="S4.SS1.p2.4.m4.1.1.2.1.2.2" movablelimits="false" xref="S4.SS1.p2.4.m4.1.1.2.1.2.2.cmml">∑</mo><mrow id="S4.SS1.p2.4.m4.1.1.2.1.2.3" xref="S4.SS1.p2.4.m4.1.1.2.1.2.3.cmml"><mi id="S4.SS1.p2.4.m4.1.1.2.1.2.3.2" xref="S4.SS1.p2.4.m4.1.1.2.1.2.3.2.cmml">k</mi><mo id="S4.SS1.p2.4.m4.1.1.2.1.2.3.1" xref="S4.SS1.p2.4.m4.1.1.2.1.2.3.1.cmml">=</mo><mn id="S4.SS1.p2.4.m4.1.1.2.1.2.3.3" xref="S4.SS1.p2.4.m4.1.1.2.1.2.3.3.cmml">1</mn></mrow><mi id="S4.SS1.p2.4.m4.1.1.2.1.3" xref="S4.SS1.p2.4.m4.1.1.2.1.3.cmml">N</mi></munderover></mstyle><msub id="S4.SS1.p2.4.m4.1.1.2.2" xref="S4.SS1.p2.4.m4.1.1.2.2.cmml"><mi id="S4.SS1.p2.4.m4.1.1.2.2.2" xref="S4.SS1.p2.4.m4.1.1.2.2.2.cmml">p</mi><mi id="S4.SS1.p2.4.m4.1.1.2.2.3" xref="S4.SS1.p2.4.m4.1.1.2.2.3.cmml">k</mi></msub></mrow><mo id="S4.SS1.p2.4.m4.1.1.1" xref="S4.SS1.p2.4.m4.1.1.1.cmml">=</mo><mn id="S4.SS1.p2.4.m4.1.1.3" xref="S4.SS1.p2.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.4.m4.1b"><apply id="S4.SS1.p2.4.m4.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1"><eq id="S4.SS1.p2.4.m4.1.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1.1"></eq><apply id="S4.SS1.p2.4.m4.1.1.2.cmml" xref="S4.SS1.p2.4.m4.1.1.2"><apply id="S4.SS1.p2.4.m4.1.1.2.1.cmml" xref="S4.SS1.p2.4.m4.1.1.2.1"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m4.1.1.2.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1.2.1">superscript</csymbol><apply id="S4.SS1.p2.4.m4.1.1.2.1.2.cmml" xref="S4.SS1.p2.4.m4.1.1.2.1"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m4.1.1.2.1.2.1.cmml" xref="S4.SS1.p2.4.m4.1.1.2.1">subscript</csymbol><sum id="S4.SS1.p2.4.m4.1.1.2.1.2.2.cmml" xref="S4.SS1.p2.4.m4.1.1.2.1.2.2"></sum><apply id="S4.SS1.p2.4.m4.1.1.2.1.2.3.cmml" xref="S4.SS1.p2.4.m4.1.1.2.1.2.3"><eq id="S4.SS1.p2.4.m4.1.1.2.1.2.3.1.cmml" xref="S4.SS1.p2.4.m4.1.1.2.1.2.3.1"></eq><ci id="S4.SS1.p2.4.m4.1.1.2.1.2.3.2.cmml" xref="S4.SS1.p2.4.m4.1.1.2.1.2.3.2">𝑘</ci><cn id="S4.SS1.p2.4.m4.1.1.2.1.2.3.3.cmml" type="integer" xref="S4.SS1.p2.4.m4.1.1.2.1.2.3.3">1</cn></apply></apply><ci id="S4.SS1.p2.4.m4.1.1.2.1.3.cmml" xref="S4.SS1.p2.4.m4.1.1.2.1.3">𝑁</ci></apply><apply id="S4.SS1.p2.4.m4.1.1.2.2.cmml" xref="S4.SS1.p2.4.m4.1.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m4.1.1.2.2.1.cmml" xref="S4.SS1.p2.4.m4.1.1.2.2">subscript</csymbol><ci id="S4.SS1.p2.4.m4.1.1.2.2.2.cmml" xref="S4.SS1.p2.4.m4.1.1.2.2.2">𝑝</ci><ci id="S4.SS1.p2.4.m4.1.1.2.2.3.cmml" xref="S4.SS1.p2.4.m4.1.1.2.2.3">𝑘</ci></apply></apply><cn id="S4.SS1.p2.4.m4.1.1.3.cmml" type="integer" xref="S4.SS1.p2.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.4.m4.1c">\displaystyle\sum_{k=1}^{N}p_{k}=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.4.m4.1d">∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = 1</annotation></semantics></math>), <math alttext="N" class="ltx_Math" display="inline" id="S4.SS1.p2.5.m5.1"><semantics id="S4.SS1.p2.5.m5.1a"><mi id="S4.SS1.p2.5.m5.1.1" xref="S4.SS1.p2.5.m5.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.5.m5.1b"><ci id="S4.SS1.p2.5.m5.1.1.cmml" xref="S4.SS1.p2.5.m5.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.5.m5.1c">N</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.5.m5.1d">italic_N</annotation></semantics></math> is the total number of devices, and <math alttext="w" class="ltx_Math" display="inline" id="S4.SS1.p2.6.m6.1"><semantics id="S4.SS1.p2.6.m6.1a"><mi id="S4.SS1.p2.6.m6.1.1" xref="S4.SS1.p2.6.m6.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.6.m6.1b"><ci id="S4.SS1.p2.6.m6.1.1.cmml" xref="S4.SS1.p2.6.m6.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.6.m6.1c">w</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.6.m6.1d">italic_w</annotation></semantics></math> represents the model parameters. The objective function for each device <math alttext="k" class="ltx_Math" display="inline" id="S4.SS1.p2.7.m7.1"><semantics id="S4.SS1.p2.7.m7.1a"><mi id="S4.SS1.p2.7.m7.1.1" xref="S4.SS1.p2.7.m7.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.7.m7.1b"><ci id="S4.SS1.p2.7.m7.1.1.cmml" xref="S4.SS1.p2.7.m7.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.7.m7.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.7.m7.1d">italic_k</annotation></semantics></math> (where <math alttext="1\leq k\leq N" class="ltx_Math" display="inline" id="S4.SS1.p2.8.m8.1"><semantics id="S4.SS1.p2.8.m8.1a"><mrow id="S4.SS1.p2.8.m8.1.1" xref="S4.SS1.p2.8.m8.1.1.cmml"><mn id="S4.SS1.p2.8.m8.1.1.2" xref="S4.SS1.p2.8.m8.1.1.2.cmml">1</mn><mo id="S4.SS1.p2.8.m8.1.1.3" xref="S4.SS1.p2.8.m8.1.1.3.cmml">≤</mo><mi id="S4.SS1.p2.8.m8.1.1.4" xref="S4.SS1.p2.8.m8.1.1.4.cmml">k</mi><mo id="S4.SS1.p2.8.m8.1.1.5" xref="S4.SS1.p2.8.m8.1.1.5.cmml">≤</mo><mi id="S4.SS1.p2.8.m8.1.1.6" xref="S4.SS1.p2.8.m8.1.1.6.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.8.m8.1b"><apply id="S4.SS1.p2.8.m8.1.1.cmml" xref="S4.SS1.p2.8.m8.1.1"><and id="S4.SS1.p2.8.m8.1.1a.cmml" xref="S4.SS1.p2.8.m8.1.1"></and><apply id="S4.SS1.p2.8.m8.1.1b.cmml" xref="S4.SS1.p2.8.m8.1.1"><leq id="S4.SS1.p2.8.m8.1.1.3.cmml" xref="S4.SS1.p2.8.m8.1.1.3"></leq><cn id="S4.SS1.p2.8.m8.1.1.2.cmml" type="integer" xref="S4.SS1.p2.8.m8.1.1.2">1</cn><ci id="S4.SS1.p2.8.m8.1.1.4.cmml" xref="S4.SS1.p2.8.m8.1.1.4">𝑘</ci></apply><apply id="S4.SS1.p2.8.m8.1.1c.cmml" xref="S4.SS1.p2.8.m8.1.1"><leq id="S4.SS1.p2.8.m8.1.1.5.cmml" xref="S4.SS1.p2.8.m8.1.1.5"></leq><share href="https://arxiv.org/html/2503.11828v1#S4.SS1.p2.8.m8.1.1.4.cmml" id="S4.SS1.p2.8.m8.1.1d.cmml" xref="S4.SS1.p2.8.m8.1.1"></share><ci id="S4.SS1.p2.8.m8.1.1.6.cmml" xref="S4.SS1.p2.8.m8.1.1.6">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.8.m8.1c">1\leq k\leq N</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.8.m8.1d">1 ≤ italic_k ≤ italic_N</annotation></semantics></math>) using the entire dataset <math alttext="\mathbf{x}_{k}=\{x_{k,1},\ldots,x_{k,n_{k}}\}" class="ltx_Math" display="inline" id="S4.SS1.p2.9.m9.7"><semantics id="S4.SS1.p2.9.m9.7a"><mrow id="S4.SS1.p2.9.m9.7.7" xref="S4.SS1.p2.9.m9.7.7.cmml"><msub id="S4.SS1.p2.9.m9.7.7.4" xref="S4.SS1.p2.9.m9.7.7.4.cmml"><mi id="S4.SS1.p2.9.m9.7.7.4.2" xref="S4.SS1.p2.9.m9.7.7.4.2.cmml">𝐱</mi><mi id="S4.SS1.p2.9.m9.7.7.4.3" xref="S4.SS1.p2.9.m9.7.7.4.3.cmml">k</mi></msub><mo id="S4.SS1.p2.9.m9.7.7.3" xref="S4.SS1.p2.9.m9.7.7.3.cmml">=</mo><mrow id="S4.SS1.p2.9.m9.7.7.2.2" xref="S4.SS1.p2.9.m9.7.7.2.3.cmml"><mo id="S4.SS1.p2.9.m9.7.7.2.2.3" stretchy="false" xref="S4.SS1.p2.9.m9.7.7.2.3.cmml">{</mo><msub id="S4.SS1.p2.9.m9.6.6.1.1.1" xref="S4.SS1.p2.9.m9.6.6.1.1.1.cmml"><mi id="S4.SS1.p2.9.m9.6.6.1.1.1.2" xref="S4.SS1.p2.9.m9.6.6.1.1.1.2.cmml">x</mi><mrow id="S4.SS1.p2.9.m9.2.2.2.4" xref="S4.SS1.p2.9.m9.2.2.2.3.cmml"><mi id="S4.SS1.p2.9.m9.1.1.1.1" xref="S4.SS1.p2.9.m9.1.1.1.1.cmml">k</mi><mo id="S4.SS1.p2.9.m9.2.2.2.4.1" xref="S4.SS1.p2.9.m9.2.2.2.3.cmml">,</mo><mn id="S4.SS1.p2.9.m9.2.2.2.2" xref="S4.SS1.p2.9.m9.2.2.2.2.cmml">1</mn></mrow></msub><mo id="S4.SS1.p2.9.m9.7.7.2.2.4" xref="S4.SS1.p2.9.m9.7.7.2.3.cmml">,</mo><mi id="S4.SS1.p2.9.m9.5.5" mathvariant="normal" xref="S4.SS1.p2.9.m9.5.5.cmml">…</mi><mo id="S4.SS1.p2.9.m9.7.7.2.2.5" xref="S4.SS1.p2.9.m9.7.7.2.3.cmml">,</mo><msub id="S4.SS1.p2.9.m9.7.7.2.2.2" xref="S4.SS1.p2.9.m9.7.7.2.2.2.cmml"><mi id="S4.SS1.p2.9.m9.7.7.2.2.2.2" xref="S4.SS1.p2.9.m9.7.7.2.2.2.2.cmml">x</mi><mrow id="S4.SS1.p2.9.m9.4.4.2.2" xref="S4.SS1.p2.9.m9.4.4.2.3.cmml"><mi id="S4.SS1.p2.9.m9.3.3.1.1" xref="S4.SS1.p2.9.m9.3.3.1.1.cmml">k</mi><mo id="S4.SS1.p2.9.m9.4.4.2.2.2" xref="S4.SS1.p2.9.m9.4.4.2.3.cmml">,</mo><msub id="S4.SS1.p2.9.m9.4.4.2.2.1" xref="S4.SS1.p2.9.m9.4.4.2.2.1.cmml"><mi id="S4.SS1.p2.9.m9.4.4.2.2.1.2" xref="S4.SS1.p2.9.m9.4.4.2.2.1.2.cmml">n</mi><mi id="S4.SS1.p2.9.m9.4.4.2.2.1.3" xref="S4.SS1.p2.9.m9.4.4.2.2.1.3.cmml">k</mi></msub></mrow></msub><mo id="S4.SS1.p2.9.m9.7.7.2.2.6" stretchy="false" xref="S4.SS1.p2.9.m9.7.7.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.9.m9.7b"><apply id="S4.SS1.p2.9.m9.7.7.cmml" xref="S4.SS1.p2.9.m9.7.7"><eq id="S4.SS1.p2.9.m9.7.7.3.cmml" xref="S4.SS1.p2.9.m9.7.7.3"></eq><apply id="S4.SS1.p2.9.m9.7.7.4.cmml" xref="S4.SS1.p2.9.m9.7.7.4"><csymbol cd="ambiguous" id="S4.SS1.p2.9.m9.7.7.4.1.cmml" xref="S4.SS1.p2.9.m9.7.7.4">subscript</csymbol><ci id="S4.SS1.p2.9.m9.7.7.4.2.cmml" xref="S4.SS1.p2.9.m9.7.7.4.2">𝐱</ci><ci id="S4.SS1.p2.9.m9.7.7.4.3.cmml" xref="S4.SS1.p2.9.m9.7.7.4.3">𝑘</ci></apply><set id="S4.SS1.p2.9.m9.7.7.2.3.cmml" xref="S4.SS1.p2.9.m9.7.7.2.2"><apply id="S4.SS1.p2.9.m9.6.6.1.1.1.cmml" xref="S4.SS1.p2.9.m9.6.6.1.1.1"><csymbol 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id="S4.SS1.p2.9.m9.4.4.2.2.1.1.cmml" xref="S4.SS1.p2.9.m9.4.4.2.2.1">subscript</csymbol><ci id="S4.SS1.p2.9.m9.4.4.2.2.1.2.cmml" xref="S4.SS1.p2.9.m9.4.4.2.2.1.2">𝑛</ci><ci id="S4.SS1.p2.9.m9.4.4.2.2.1.3.cmml" xref="S4.SS1.p2.9.m9.4.4.2.2.1.3">𝑘</ci></apply></list></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.9.m9.7c">\mathbf{x}_{k}=\{x_{k,1},\ldots,x_{k,n_{k}}\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.9.m9.7d">bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = { italic_x start_POSTSUBSCRIPT italic_k , 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_k , italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT }</annotation></semantics></math> is expressed as:</p> </div> <div class="ltx_para" id="S4.SS1.p3"> <table class="ltx_equation ltx_eqn_table" id="S4.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="F_{k}(w)=\frac{1}{n_{k}}\sum_{j=1}^{n_{k}}\mathcal{L}oss(w;x_{k,j})" class="ltx_Math" display="block" id="S4.E2.m1.5"><semantics id="S4.E2.m1.5a"><mrow id="S4.E2.m1.5.5" xref="S4.E2.m1.5.5.cmml"><mrow id="S4.E2.m1.5.5.3" xref="S4.E2.m1.5.5.3.cmml"><msub id="S4.E2.m1.5.5.3.2" xref="S4.E2.m1.5.5.3.2.cmml"><mi id="S4.E2.m1.5.5.3.2.2" xref="S4.E2.m1.5.5.3.2.2.cmml">F</mi><mi id="S4.E2.m1.5.5.3.2.3" xref="S4.E2.m1.5.5.3.2.3.cmml">k</mi></msub><mo id="S4.E2.m1.5.5.3.1" xref="S4.E2.m1.5.5.3.1.cmml">⁢</mo><mrow id="S4.E2.m1.5.5.3.3.2" xref="S4.E2.m1.5.5.3.cmml"><mo id="S4.E2.m1.5.5.3.3.2.1" stretchy="false" xref="S4.E2.m1.5.5.3.cmml">(</mo><mi id="S4.E2.m1.3.3" xref="S4.E2.m1.3.3.cmml">w</mi><mo id="S4.E2.m1.5.5.3.3.2.2" stretchy="false" xref="S4.E2.m1.5.5.3.cmml">)</mo></mrow></mrow><mo id="S4.E2.m1.5.5.2" xref="S4.E2.m1.5.5.2.cmml">=</mo><mrow id="S4.E2.m1.5.5.1" xref="S4.E2.m1.5.5.1.cmml"><mfrac id="S4.E2.m1.5.5.1.3" xref="S4.E2.m1.5.5.1.3.cmml"><mn id="S4.E2.m1.5.5.1.3.2" xref="S4.E2.m1.5.5.1.3.2.cmml">1</mn><msub id="S4.E2.m1.5.5.1.3.3" xref="S4.E2.m1.5.5.1.3.3.cmml"><mi id="S4.E2.m1.5.5.1.3.3.2" xref="S4.E2.m1.5.5.1.3.3.2.cmml">n</mi><mi id="S4.E2.m1.5.5.1.3.3.3" xref="S4.E2.m1.5.5.1.3.3.3.cmml">k</mi></msub></mfrac><mo id="S4.E2.m1.5.5.1.2" xref="S4.E2.m1.5.5.1.2.cmml">⁢</mo><mrow id="S4.E2.m1.5.5.1.1" xref="S4.E2.m1.5.5.1.1.cmml"><munderover id="S4.E2.m1.5.5.1.1.2" xref="S4.E2.m1.5.5.1.1.2.cmml"><mo id="S4.E2.m1.5.5.1.1.2.2.2" movablelimits="false" xref="S4.E2.m1.5.5.1.1.2.2.2.cmml">∑</mo><mrow id="S4.E2.m1.5.5.1.1.2.2.3" xref="S4.E2.m1.5.5.1.1.2.2.3.cmml"><mi id="S4.E2.m1.5.5.1.1.2.2.3.2" xref="S4.E2.m1.5.5.1.1.2.2.3.2.cmml">j</mi><mo id="S4.E2.m1.5.5.1.1.2.2.3.1" xref="S4.E2.m1.5.5.1.1.2.2.3.1.cmml">=</mo><mn id="S4.E2.m1.5.5.1.1.2.2.3.3" xref="S4.E2.m1.5.5.1.1.2.2.3.3.cmml">1</mn></mrow><msub id="S4.E2.m1.5.5.1.1.2.3" xref="S4.E2.m1.5.5.1.1.2.3.cmml"><mi 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xref="S4.E2.m1.4.4.cmml">w</mi><mo id="S4.E2.m1.5.5.1.1.1.1.1.3" xref="S4.E2.m1.5.5.1.1.1.1.2.cmml">;</mo><msub id="S4.E2.m1.5.5.1.1.1.1.1.1" xref="S4.E2.m1.5.5.1.1.1.1.1.1.cmml"><mi id="S4.E2.m1.5.5.1.1.1.1.1.1.2" xref="S4.E2.m1.5.5.1.1.1.1.1.1.2.cmml">x</mi><mrow id="S4.E2.m1.2.2.2.4" xref="S4.E2.m1.2.2.2.3.cmml"><mi id="S4.E2.m1.1.1.1.1" xref="S4.E2.m1.1.1.1.1.cmml">k</mi><mo id="S4.E2.m1.2.2.2.4.1" xref="S4.E2.m1.2.2.2.3.cmml">,</mo><mi id="S4.E2.m1.2.2.2.2" xref="S4.E2.m1.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S4.E2.m1.5.5.1.1.1.1.1.4" stretchy="false" xref="S4.E2.m1.5.5.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E2.m1.5b"><apply id="S4.E2.m1.5.5.cmml" xref="S4.E2.m1.5.5"><eq id="S4.E2.m1.5.5.2.cmml" xref="S4.E2.m1.5.5.2"></eq><apply id="S4.E2.m1.5.5.3.cmml" xref="S4.E2.m1.5.5.3"><times id="S4.E2.m1.5.5.3.1.cmml" xref="S4.E2.m1.5.5.3.1"></times><apply id="S4.E2.m1.5.5.3.2.cmml" xref="S4.E2.m1.5.5.3.2"><csymbol 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id="S4.E2.m1.5.5.1.1.2.3.3.cmml" xref="S4.E2.m1.5.5.1.1.2.3.3">𝑘</ci></apply></apply><apply id="S4.E2.m1.5.5.1.1.1.cmml" xref="S4.E2.m1.5.5.1.1.1"><times id="S4.E2.m1.5.5.1.1.1.2.cmml" xref="S4.E2.m1.5.5.1.1.1.2"></times><ci id="S4.E2.m1.5.5.1.1.1.3.cmml" xref="S4.E2.m1.5.5.1.1.1.3">ℒ</ci><ci id="S4.E2.m1.5.5.1.1.1.4.cmml" xref="S4.E2.m1.5.5.1.1.1.4">𝑜</ci><ci id="S4.E2.m1.5.5.1.1.1.5.cmml" xref="S4.E2.m1.5.5.1.1.1.5">𝑠</ci><ci id="S4.E2.m1.5.5.1.1.1.6.cmml" xref="S4.E2.m1.5.5.1.1.1.6">𝑠</ci><list id="S4.E2.m1.5.5.1.1.1.1.2.cmml" xref="S4.E2.m1.5.5.1.1.1.1.1"><ci id="S4.E2.m1.4.4.cmml" xref="S4.E2.m1.4.4">𝑤</ci><apply id="S4.E2.m1.5.5.1.1.1.1.1.1.cmml" xref="S4.E2.m1.5.5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E2.m1.5.5.1.1.1.1.1.1.1.cmml" xref="S4.E2.m1.5.5.1.1.1.1.1.1">subscript</csymbol><ci id="S4.E2.m1.5.5.1.1.1.1.1.1.2.cmml" xref="S4.E2.m1.5.5.1.1.1.1.1.1.2">𝑥</ci><list id="S4.E2.m1.2.2.2.3.cmml" xref="S4.E2.m1.2.2.2.4"><ci id="S4.E2.m1.1.1.1.1.cmml" xref="S4.E2.m1.1.1.1.1">𝑘</ci><ci id="S4.E2.m1.2.2.2.2.cmml" xref="S4.E2.m1.2.2.2.2">𝑗</ci></list></apply></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E2.m1.5c">F_{k}(w)=\frac{1}{n_{k}}\sum_{j=1}^{n_{k}}\mathcal{L}oss(w;x_{k,j})</annotation><annotation encoding="application/x-llamapun" id="S4.E2.m1.5d">italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w ) = divide start_ARG 1 end_ARG start_ARG italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUPERSCRIPT caligraphic_L italic_o italic_s italic_s ( italic_w ; italic_x start_POSTSUBSCRIPT italic_k , italic_j end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p3.6">where <math alttext="\mathcal{L}oss(\cdot;\cdot)" class="ltx_Math" display="inline" id="S4.SS1.p3.1.m1.2"><semantics id="S4.SS1.p3.1.m1.2a"><mrow id="S4.SS1.p3.1.m1.2.3" xref="S4.SS1.p3.1.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.1.m1.2.3.2" xref="S4.SS1.p3.1.m1.2.3.2.cmml">ℒ</mi><mo id="S4.SS1.p3.1.m1.2.3.1" xref="S4.SS1.p3.1.m1.2.3.1.cmml">⁢</mo><mi id="S4.SS1.p3.1.m1.2.3.3" xref="S4.SS1.p3.1.m1.2.3.3.cmml">o</mi><mo id="S4.SS1.p3.1.m1.2.3.1a" xref="S4.SS1.p3.1.m1.2.3.1.cmml">⁢</mo><mi id="S4.SS1.p3.1.m1.2.3.4" xref="S4.SS1.p3.1.m1.2.3.4.cmml">s</mi><mo id="S4.SS1.p3.1.m1.2.3.1b" xref="S4.SS1.p3.1.m1.2.3.1.cmml">⁢</mo><mi id="S4.SS1.p3.1.m1.2.3.5" xref="S4.SS1.p3.1.m1.2.3.5.cmml">s</mi><mo id="S4.SS1.p3.1.m1.2.3.1c" xref="S4.SS1.p3.1.m1.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.p3.1.m1.2.3.6.2" xref="S4.SS1.p3.1.m1.2.3.6.1.cmml"><mo id="S4.SS1.p3.1.m1.2.3.6.2.1" stretchy="false" xref="S4.SS1.p3.1.m1.2.3.6.1.cmml">(</mo><mo id="S4.SS1.p3.1.m1.1.1" lspace="0em" rspace="0em" xref="S4.SS1.p3.1.m1.1.1.cmml">⋅</mo><mo id="S4.SS1.p3.1.m1.2.3.6.2.2" rspace="0em" xref="S4.SS1.p3.1.m1.2.3.6.1.cmml">;</mo><mo id="S4.SS1.p3.1.m1.2.2" lspace="0em" rspace="0em" xref="S4.SS1.p3.1.m1.2.2.cmml">⋅</mo><mo id="S4.SS1.p3.1.m1.2.3.6.2.3" stretchy="false" xref="S4.SS1.p3.1.m1.2.3.6.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.1.m1.2b"><apply id="S4.SS1.p3.1.m1.2.3.cmml" xref="S4.SS1.p3.1.m1.2.3"><times id="S4.SS1.p3.1.m1.2.3.1.cmml" xref="S4.SS1.p3.1.m1.2.3.1"></times><ci id="S4.SS1.p3.1.m1.2.3.2.cmml" xref="S4.SS1.p3.1.m1.2.3.2">ℒ</ci><ci id="S4.SS1.p3.1.m1.2.3.3.cmml" xref="S4.SS1.p3.1.m1.2.3.3">𝑜</ci><ci id="S4.SS1.p3.1.m1.2.3.4.cmml" xref="S4.SS1.p3.1.m1.2.3.4">𝑠</ci><ci id="S4.SS1.p3.1.m1.2.3.5.cmml" xref="S4.SS1.p3.1.m1.2.3.5">𝑠</ci><list id="S4.SS1.p3.1.m1.2.3.6.1.cmml" xref="S4.SS1.p3.1.m1.2.3.6.2"><ci id="S4.SS1.p3.1.m1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1">⋅</ci><ci id="S4.SS1.p3.1.m1.2.2.cmml" xref="S4.SS1.p3.1.m1.2.2">⋅</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.1.m1.2c">\mathcal{L}oss(\cdot;\cdot)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.1.m1.2d">caligraphic_L italic_o italic_s italic_s ( ⋅ ; ⋅ )</annotation></semantics></math> represents a user-specified loss function and <math alttext="n_{k}" class="ltx_Math" display="inline" id="S4.SS1.p3.2.m2.1"><semantics id="S4.SS1.p3.2.m2.1a"><msub id="S4.SS1.p3.2.m2.1.1" xref="S4.SS1.p3.2.m2.1.1.cmml"><mi id="S4.SS1.p3.2.m2.1.1.2" xref="S4.SS1.p3.2.m2.1.1.2.cmml">n</mi><mi id="S4.SS1.p3.2.m2.1.1.3" xref="S4.SS1.p3.2.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.2.m2.1b"><apply id="S4.SS1.p3.2.m2.1.1.cmml" xref="S4.SS1.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.2.m2.1.1.1.cmml" xref="S4.SS1.p3.2.m2.1.1">subscript</csymbol><ci id="S4.SS1.p3.2.m2.1.1.2.cmml" xref="S4.SS1.p3.2.m2.1.1.2">𝑛</ci><ci id="S4.SS1.p3.2.m2.1.1.3.cmml" xref="S4.SS1.p3.2.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.2.m2.1c">n_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.2.m2.1d">italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> represents the number of samples associated with the <math alttext="k" class="ltx_Math" display="inline" id="S4.SS1.p3.3.m3.1"><semantics id="S4.SS1.p3.3.m3.1a"><mi id="S4.SS1.p3.3.m3.1.1" xref="S4.SS1.p3.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.3.m3.1b"><ci id="S4.SS1.p3.3.m3.1.1.cmml" xref="S4.SS1.p3.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.3.m3.1d">italic_k</annotation></semantics></math>-th device. If we uniformly select <math alttext="\xi_{k}" class="ltx_Math" display="inline" id="S4.SS1.p3.4.m4.1"><semantics id="S4.SS1.p3.4.m4.1a"><msub id="S4.SS1.p3.4.m4.1.1" xref="S4.SS1.p3.4.m4.1.1.cmml"><mi id="S4.SS1.p3.4.m4.1.1.2" xref="S4.SS1.p3.4.m4.1.1.2.cmml">ξ</mi><mi id="S4.SS1.p3.4.m4.1.1.3" xref="S4.SS1.p3.4.m4.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.4.m4.1b"><apply id="S4.SS1.p3.4.m4.1.1.cmml" xref="S4.SS1.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.4.m4.1.1.1.cmml" xref="S4.SS1.p3.4.m4.1.1">subscript</csymbol><ci id="S4.SS1.p3.4.m4.1.1.2.cmml" xref="S4.SS1.p3.4.m4.1.1.2">𝜉</ci><ci id="S4.SS1.p3.4.m4.1.1.3.cmml" xref="S4.SS1.p3.4.m4.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.4.m4.1c">\xi_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.4.m4.1d">italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> samples from device <math alttext="k" class="ltx_Math" display="inline" id="S4.SS1.p3.5.m5.1"><semantics id="S4.SS1.p3.5.m5.1a"><mi id="S4.SS1.p3.5.m5.1.1" xref="S4.SS1.p3.5.m5.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.5.m5.1b"><ci id="S4.SS1.p3.5.m5.1.1.cmml" xref="S4.SS1.p3.5.m5.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.5.m5.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.5.m5.1d">italic_k</annotation></semantics></math>’s dataset <math alttext="\mathbf{x}_{k}" class="ltx_Math" display="inline" id="S4.SS1.p3.6.m6.1"><semantics id="S4.SS1.p3.6.m6.1a"><msub id="S4.SS1.p3.6.m6.1.1" xref="S4.SS1.p3.6.m6.1.1.cmml"><mi id="S4.SS1.p3.6.m6.1.1.2" xref="S4.SS1.p3.6.m6.1.1.2.cmml">𝐱</mi><mi id="S4.SS1.p3.6.m6.1.1.3" xref="S4.SS1.p3.6.m6.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.6.m6.1b"><apply id="S4.SS1.p3.6.m6.1.1.cmml" xref="S4.SS1.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.6.m6.1.1.1.cmml" xref="S4.SS1.p3.6.m6.1.1">subscript</csymbol><ci id="S4.SS1.p3.6.m6.1.1.2.cmml" xref="S4.SS1.p3.6.m6.1.1.2">𝐱</ci><ci id="S4.SS1.p3.6.m6.1.1.3.cmml" xref="S4.SS1.p3.6.m6.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.6.m6.1c">\mathbf{x}_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.6.m6.1d">bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, the objective function becomes:</p> <table class="ltx_equation ltx_eqn_table" id="S4.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="F_{k}(w)=\frac{1}{n_{k}}\mathcal{L}oss(w;\xi_{k})" class="ltx_Math" display="block" id="S4.E3.m1.3"><semantics id="S4.E3.m1.3a"><mrow id="S4.E3.m1.3.3" xref="S4.E3.m1.3.3.cmml"><mrow id="S4.E3.m1.3.3.3" xref="S4.E3.m1.3.3.3.cmml"><msub id="S4.E3.m1.3.3.3.2" xref="S4.E3.m1.3.3.3.2.cmml"><mi id="S4.E3.m1.3.3.3.2.2" xref="S4.E3.m1.3.3.3.2.2.cmml">F</mi><mi id="S4.E3.m1.3.3.3.2.3" xref="S4.E3.m1.3.3.3.2.3.cmml">k</mi></msub><mo id="S4.E3.m1.3.3.3.1" xref="S4.E3.m1.3.3.3.1.cmml">⁢</mo><mrow id="S4.E3.m1.3.3.3.3.2" xref="S4.E3.m1.3.3.3.cmml"><mo id="S4.E3.m1.3.3.3.3.2.1" stretchy="false" xref="S4.E3.m1.3.3.3.cmml">(</mo><mi id="S4.E3.m1.1.1" xref="S4.E3.m1.1.1.cmml">w</mi><mo id="S4.E3.m1.3.3.3.3.2.2" stretchy="false" xref="S4.E3.m1.3.3.3.cmml">)</mo></mrow></mrow><mo id="S4.E3.m1.3.3.2" xref="S4.E3.m1.3.3.2.cmml">=</mo><mrow id="S4.E3.m1.3.3.1" xref="S4.E3.m1.3.3.1.cmml"><mfrac id="S4.E3.m1.3.3.1.3" xref="S4.E3.m1.3.3.1.3.cmml"><mn id="S4.E3.m1.3.3.1.3.2" xref="S4.E3.m1.3.3.1.3.2.cmml">1</mn><msub id="S4.E3.m1.3.3.1.3.3" xref="S4.E3.m1.3.3.1.3.3.cmml"><mi id="S4.E3.m1.3.3.1.3.3.2" xref="S4.E3.m1.3.3.1.3.3.2.cmml">n</mi><mi id="S4.E3.m1.3.3.1.3.3.3" xref="S4.E3.m1.3.3.1.3.3.3.cmml">k</mi></msub></mfrac><mo id="S4.E3.m1.3.3.1.2" xref="S4.E3.m1.3.3.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S4.E3.m1.3.3.1.4" xref="S4.E3.m1.3.3.1.4.cmml">ℒ</mi><mo id="S4.E3.m1.3.3.1.2a" xref="S4.E3.m1.3.3.1.2.cmml">⁢</mo><mi id="S4.E3.m1.3.3.1.5" xref="S4.E3.m1.3.3.1.5.cmml">o</mi><mo id="S4.E3.m1.3.3.1.2b" xref="S4.E3.m1.3.3.1.2.cmml">⁢</mo><mi id="S4.E3.m1.3.3.1.6" xref="S4.E3.m1.3.3.1.6.cmml">s</mi><mo id="S4.E3.m1.3.3.1.2c" xref="S4.E3.m1.3.3.1.2.cmml">⁢</mo><mi id="S4.E3.m1.3.3.1.7" xref="S4.E3.m1.3.3.1.7.cmml">s</mi><mo id="S4.E3.m1.3.3.1.2d" xref="S4.E3.m1.3.3.1.2.cmml">⁢</mo><mrow id="S4.E3.m1.3.3.1.1.1" xref="S4.E3.m1.3.3.1.1.2.cmml"><mo id="S4.E3.m1.3.3.1.1.1.2" stretchy="false" xref="S4.E3.m1.3.3.1.1.2.cmml">(</mo><mi id="S4.E3.m1.2.2" xref="S4.E3.m1.2.2.cmml">w</mi><mo id="S4.E3.m1.3.3.1.1.1.3" xref="S4.E3.m1.3.3.1.1.2.cmml">;</mo><msub id="S4.E3.m1.3.3.1.1.1.1" xref="S4.E3.m1.3.3.1.1.1.1.cmml"><mi id="S4.E3.m1.3.3.1.1.1.1.2" xref="S4.E3.m1.3.3.1.1.1.1.2.cmml">ξ</mi><mi id="S4.E3.m1.3.3.1.1.1.1.3" xref="S4.E3.m1.3.3.1.1.1.1.3.cmml">k</mi></msub><mo id="S4.E3.m1.3.3.1.1.1.4" stretchy="false" xref="S4.E3.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E3.m1.3b"><apply id="S4.E3.m1.3.3.cmml" xref="S4.E3.m1.3.3"><eq id="S4.E3.m1.3.3.2.cmml" xref="S4.E3.m1.3.3.2"></eq><apply id="S4.E3.m1.3.3.3.cmml" xref="S4.E3.m1.3.3.3"><times id="S4.E3.m1.3.3.3.1.cmml" xref="S4.E3.m1.3.3.3.1"></times><apply id="S4.E3.m1.3.3.3.2.cmml" xref="S4.E3.m1.3.3.3.2"><csymbol cd="ambiguous" id="S4.E3.m1.3.3.3.2.1.cmml" xref="S4.E3.m1.3.3.3.2">subscript</csymbol><ci id="S4.E3.m1.3.3.3.2.2.cmml" xref="S4.E3.m1.3.3.3.2.2">𝐹</ci><ci id="S4.E3.m1.3.3.3.2.3.cmml" xref="S4.E3.m1.3.3.3.2.3">𝑘</ci></apply><ci id="S4.E3.m1.1.1.cmml" xref="S4.E3.m1.1.1">𝑤</ci></apply><apply id="S4.E3.m1.3.3.1.cmml" xref="S4.E3.m1.3.3.1"><times id="S4.E3.m1.3.3.1.2.cmml" xref="S4.E3.m1.3.3.1.2"></times><apply id="S4.E3.m1.3.3.1.3.cmml" xref="S4.E3.m1.3.3.1.3"><divide id="S4.E3.m1.3.3.1.3.1.cmml" xref="S4.E3.m1.3.3.1.3"></divide><cn id="S4.E3.m1.3.3.1.3.2.cmml" type="integer" xref="S4.E3.m1.3.3.1.3.2">1</cn><apply id="S4.E3.m1.3.3.1.3.3.cmml" xref="S4.E3.m1.3.3.1.3.3"><csymbol cd="ambiguous" id="S4.E3.m1.3.3.1.3.3.1.cmml" xref="S4.E3.m1.3.3.1.3.3">subscript</csymbol><ci id="S4.E3.m1.3.3.1.3.3.2.cmml" xref="S4.E3.m1.3.3.1.3.3.2">𝑛</ci><ci id="S4.E3.m1.3.3.1.3.3.3.cmml" xref="S4.E3.m1.3.3.1.3.3.3">𝑘</ci></apply></apply><ci id="S4.E3.m1.3.3.1.4.cmml" xref="S4.E3.m1.3.3.1.4">ℒ</ci><ci id="S4.E3.m1.3.3.1.5.cmml" xref="S4.E3.m1.3.3.1.5">𝑜</ci><ci id="S4.E3.m1.3.3.1.6.cmml" xref="S4.E3.m1.3.3.1.6">𝑠</ci><ci id="S4.E3.m1.3.3.1.7.cmml" xref="S4.E3.m1.3.3.1.7">𝑠</ci><list id="S4.E3.m1.3.3.1.1.2.cmml" xref="S4.E3.m1.3.3.1.1.1"><ci id="S4.E3.m1.2.2.cmml" xref="S4.E3.m1.2.2">𝑤</ci><apply id="S4.E3.m1.3.3.1.1.1.1.cmml" xref="S4.E3.m1.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S4.E3.m1.3.3.1.1.1.1.1.cmml" xref="S4.E3.m1.3.3.1.1.1.1">subscript</csymbol><ci id="S4.E3.m1.3.3.1.1.1.1.2.cmml" xref="S4.E3.m1.3.3.1.1.1.1.2">𝜉</ci><ci id="S4.E3.m1.3.3.1.1.1.1.3.cmml" xref="S4.E3.m1.3.3.1.1.1.1.3">𝑘</ci></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E3.m1.3c">F_{k}(w)=\frac{1}{n_{k}}\mathcal{L}oss(w;\xi_{k})</annotation><annotation encoding="application/x-llamapun" id="S4.E3.m1.3d">italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w ) = divide start_ARG 1 end_ARG start_ARG italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG caligraphic_L italic_o italic_s italic_s ( italic_w ; italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS1.p4"> <p class="ltx_p" id="S4.SS1.p4.4">We assume that <math alttext="F_{1},\ldots,F_{N}" class="ltx_Math" display="inline" id="S4.SS1.p4.1.m1.3"><semantics id="S4.SS1.p4.1.m1.3a"><mrow id="S4.SS1.p4.1.m1.3.3.2" xref="S4.SS1.p4.1.m1.3.3.3.cmml"><msub id="S4.SS1.p4.1.m1.2.2.1.1" xref="S4.SS1.p4.1.m1.2.2.1.1.cmml"><mi id="S4.SS1.p4.1.m1.2.2.1.1.2" xref="S4.SS1.p4.1.m1.2.2.1.1.2.cmml">F</mi><mn id="S4.SS1.p4.1.m1.2.2.1.1.3" xref="S4.SS1.p4.1.m1.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.SS1.p4.1.m1.3.3.2.3" xref="S4.SS1.p4.1.m1.3.3.3.cmml">,</mo><mi id="S4.SS1.p4.1.m1.1.1" mathvariant="normal" xref="S4.SS1.p4.1.m1.1.1.cmml">…</mi><mo id="S4.SS1.p4.1.m1.3.3.2.4" xref="S4.SS1.p4.1.m1.3.3.3.cmml">,</mo><msub id="S4.SS1.p4.1.m1.3.3.2.2" xref="S4.SS1.p4.1.m1.3.3.2.2.cmml"><mi id="S4.SS1.p4.1.m1.3.3.2.2.2" xref="S4.SS1.p4.1.m1.3.3.2.2.2.cmml">F</mi><mi id="S4.SS1.p4.1.m1.3.3.2.2.3" xref="S4.SS1.p4.1.m1.3.3.2.2.3.cmml">N</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.1.m1.3b"><list id="S4.SS1.p4.1.m1.3.3.3.cmml" xref="S4.SS1.p4.1.m1.3.3.2"><apply id="S4.SS1.p4.1.m1.2.2.1.1.cmml" xref="S4.SS1.p4.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p4.1.m1.2.2.1.1.1.cmml" xref="S4.SS1.p4.1.m1.2.2.1.1">subscript</csymbol><ci id="S4.SS1.p4.1.m1.2.2.1.1.2.cmml" xref="S4.SS1.p4.1.m1.2.2.1.1.2">𝐹</ci><cn id="S4.SS1.p4.1.m1.2.2.1.1.3.cmml" type="integer" xref="S4.SS1.p4.1.m1.2.2.1.1.3">1</cn></apply><ci id="S4.SS1.p4.1.m1.1.1.cmml" xref="S4.SS1.p4.1.m1.1.1">…</ci><apply id="S4.SS1.p4.1.m1.3.3.2.2.cmml" xref="S4.SS1.p4.1.m1.3.3.2.2"><csymbol cd="ambiguous" id="S4.SS1.p4.1.m1.3.3.2.2.1.cmml" xref="S4.SS1.p4.1.m1.3.3.2.2">subscript</csymbol><ci id="S4.SS1.p4.1.m1.3.3.2.2.2.cmml" xref="S4.SS1.p4.1.m1.3.3.2.2.2">𝐹</ci><ci id="S4.SS1.p4.1.m1.3.3.2.2.3.cmml" xref="S4.SS1.p4.1.m1.3.3.2.2.3">𝑁</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.1.m1.3c">F_{1},\ldots,F_{N}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.1.m1.3d">italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_F start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT</annotation></semantics></math> are all L-smooth and <math alttext="\mu" class="ltx_Math" display="inline" id="S4.SS1.p4.2.m2.1"><semantics id="S4.SS1.p4.2.m2.1a"><mi id="S4.SS1.p4.2.m2.1.1" xref="S4.SS1.p4.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.2.m2.1b"><ci id="S4.SS1.p4.2.m2.1.1.cmml" xref="S4.SS1.p4.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.2.m2.1d">italic_μ</annotation></semantics></math>-strongly convex <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib7" title="">7</a>]</cite>, ensuring that the gradient does not change too rapidly and the value of the function between any two points is not only above the tangent line but also above the tangent line plus a positive term proportional to <math alttext="\|w-w^{\prime}\|^{2}" class="ltx_Math" display="inline" id="S4.SS1.p4.3.m3.1"><semantics id="S4.SS1.p4.3.m3.1a"><msup id="S4.SS1.p4.3.m3.1.1" xref="S4.SS1.p4.3.m3.1.1.cmml"><mrow id="S4.SS1.p4.3.m3.1.1.1.1" xref="S4.SS1.p4.3.m3.1.1.1.2.cmml"><mo id="S4.SS1.p4.3.m3.1.1.1.1.2" stretchy="false" xref="S4.SS1.p4.3.m3.1.1.1.2.1.cmml">‖</mo><mrow id="S4.SS1.p4.3.m3.1.1.1.1.1" xref="S4.SS1.p4.3.m3.1.1.1.1.1.cmml"><mi id="S4.SS1.p4.3.m3.1.1.1.1.1.2" xref="S4.SS1.p4.3.m3.1.1.1.1.1.2.cmml">w</mi><mo id="S4.SS1.p4.3.m3.1.1.1.1.1.1" xref="S4.SS1.p4.3.m3.1.1.1.1.1.1.cmml">−</mo><msup id="S4.SS1.p4.3.m3.1.1.1.1.1.3" xref="S4.SS1.p4.3.m3.1.1.1.1.1.3.cmml"><mi id="S4.SS1.p4.3.m3.1.1.1.1.1.3.2" xref="S4.SS1.p4.3.m3.1.1.1.1.1.3.2.cmml">w</mi><mo id="S4.SS1.p4.3.m3.1.1.1.1.1.3.3" xref="S4.SS1.p4.3.m3.1.1.1.1.1.3.3.cmml">′</mo></msup></mrow><mo id="S4.SS1.p4.3.m3.1.1.1.1.3" stretchy="false" xref="S4.SS1.p4.3.m3.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S4.SS1.p4.3.m3.1.1.3" xref="S4.SS1.p4.3.m3.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.3.m3.1b"><apply id="S4.SS1.p4.3.m3.1.1.cmml" xref="S4.SS1.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.p4.3.m3.1.1.2.cmml" xref="S4.SS1.p4.3.m3.1.1">superscript</csymbol><apply id="S4.SS1.p4.3.m3.1.1.1.2.cmml" xref="S4.SS1.p4.3.m3.1.1.1.1"><csymbol cd="latexml" id="S4.SS1.p4.3.m3.1.1.1.2.1.cmml" xref="S4.SS1.p4.3.m3.1.1.1.1.2">norm</csymbol><apply id="S4.SS1.p4.3.m3.1.1.1.1.1.cmml" xref="S4.SS1.p4.3.m3.1.1.1.1.1"><minus id="S4.SS1.p4.3.m3.1.1.1.1.1.1.cmml" xref="S4.SS1.p4.3.m3.1.1.1.1.1.1"></minus><ci id="S4.SS1.p4.3.m3.1.1.1.1.1.2.cmml" xref="S4.SS1.p4.3.m3.1.1.1.1.1.2">𝑤</ci><apply id="S4.SS1.p4.3.m3.1.1.1.1.1.3.cmml" xref="S4.SS1.p4.3.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p4.3.m3.1.1.1.1.1.3.1.cmml" xref="S4.SS1.p4.3.m3.1.1.1.1.1.3">superscript</csymbol><ci id="S4.SS1.p4.3.m3.1.1.1.1.1.3.2.cmml" xref="S4.SS1.p4.3.m3.1.1.1.1.1.3.2">𝑤</ci><ci id="S4.SS1.p4.3.m3.1.1.1.1.1.3.3.cmml" xref="S4.SS1.p4.3.m3.1.1.1.1.1.3.3">′</ci></apply></apply></apply><cn id="S4.SS1.p4.3.m3.1.1.3.cmml" type="integer" xref="S4.SS1.p4.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.3.m3.1c">\|w-w^{\prime}\|^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.3.m3.1d">∥ italic_w - italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. For <math alttext="k=1,\ldots,N" class="ltx_Math" display="inline" id="S4.SS1.p4.4.m4.3"><semantics id="S4.SS1.p4.4.m4.3a"><mrow id="S4.SS1.p4.4.m4.3.4" xref="S4.SS1.p4.4.m4.3.4.cmml"><mi id="S4.SS1.p4.4.m4.3.4.2" xref="S4.SS1.p4.4.m4.3.4.2.cmml">k</mi><mo id="S4.SS1.p4.4.m4.3.4.1" xref="S4.SS1.p4.4.m4.3.4.1.cmml">=</mo><mrow id="S4.SS1.p4.4.m4.3.4.3.2" xref="S4.SS1.p4.4.m4.3.4.3.1.cmml"><mn id="S4.SS1.p4.4.m4.1.1" xref="S4.SS1.p4.4.m4.1.1.cmml">1</mn><mo id="S4.SS1.p4.4.m4.3.4.3.2.1" xref="S4.SS1.p4.4.m4.3.4.3.1.cmml">,</mo><mi id="S4.SS1.p4.4.m4.2.2" mathvariant="normal" xref="S4.SS1.p4.4.m4.2.2.cmml">…</mi><mo id="S4.SS1.p4.4.m4.3.4.3.2.2" xref="S4.SS1.p4.4.m4.3.4.3.1.cmml">,</mo><mi id="S4.SS1.p4.4.m4.3.3" xref="S4.SS1.p4.4.m4.3.3.cmml">N</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.4.m4.3b"><apply id="S4.SS1.p4.4.m4.3.4.cmml" xref="S4.SS1.p4.4.m4.3.4"><eq id="S4.SS1.p4.4.m4.3.4.1.cmml" xref="S4.SS1.p4.4.m4.3.4.1"></eq><ci id="S4.SS1.p4.4.m4.3.4.2.cmml" xref="S4.SS1.p4.4.m4.3.4.2">𝑘</ci><list id="S4.SS1.p4.4.m4.3.4.3.1.cmml" xref="S4.SS1.p4.4.m4.3.4.3.2"><cn id="S4.SS1.p4.4.m4.1.1.cmml" type="integer" xref="S4.SS1.p4.4.m4.1.1">1</cn><ci id="S4.SS1.p4.4.m4.2.2.cmml" xref="S4.SS1.p4.4.m4.2.2">…</ci><ci id="S4.SS1.p4.4.m4.3.3.cmml" xref="S4.SS1.p4.4.m4.3.3">𝑁</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.4.m4.3c">k=1,\ldots,N</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.4.m4.3d">italic_k = 1 , … , italic_N</annotation></semantics></math>, we also assume:</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="E\|\nabla F_{k}(w_{k},\xi_{k})-\nabla F_{k}(w_{k})\|^{2}\leq\sigma_{k}^{2}" class="ltx_Math" display="block" id="S4.Ex1.m1.1"><semantics id="S4.Ex1.m1.1a"><mrow id="S4.Ex1.m1.1.1" xref="S4.Ex1.m1.1.1.cmml"><mrow id="S4.Ex1.m1.1.1.1" xref="S4.Ex1.m1.1.1.1.cmml"><mi id="S4.Ex1.m1.1.1.1.3" xref="S4.Ex1.m1.1.1.1.3.cmml">E</mi><mo id="S4.Ex1.m1.1.1.1.2" xref="S4.Ex1.m1.1.1.1.2.cmml">⁢</mo><msup id="S4.Ex1.m1.1.1.1.1" xref="S4.Ex1.m1.1.1.1.1.cmml"><mrow id="S4.Ex1.m1.1.1.1.1.1.1" xref="S4.Ex1.m1.1.1.1.1.1.2.cmml"><mo id="S4.Ex1.m1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex1.m1.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="S4.Ex1.m1.1.1.1.1.1.1.1" xref="S4.Ex1.m1.1.1.1.1.1.1.1.cmml"><mrow id="S4.Ex1.m1.1.1.1.1.1.1.1.2" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2.cmml"><mrow id="S4.Ex1.m1.1.1.1.1.1.1.1.2.4" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2.4.cmml"><mo id="S4.Ex1.m1.1.1.1.1.1.1.1.2.4.1" rspace="0.167em" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2.4.1.cmml">∇</mo><msub id="S4.Ex1.m1.1.1.1.1.1.1.1.2.4.2" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2.4.2.cmml"><mi id="S4.Ex1.m1.1.1.1.1.1.1.1.2.4.2.2" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2.4.2.2.cmml">F</mi><mi id="S4.Ex1.m1.1.1.1.1.1.1.1.2.4.2.3" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2.4.2.3.cmml">k</mi></msub></mrow><mo id="S4.Ex1.m1.1.1.1.1.1.1.1.2.3" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2.3.cmml">⁢</mo><mrow id="S4.Ex1.m1.1.1.1.1.1.1.1.2.2.2" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2.2.3.cmml"><mo id="S4.Ex1.m1.1.1.1.1.1.1.1.2.2.2.3" stretchy="false" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2.2.3.cmml">(</mo><msub id="S4.Ex1.m1.1.1.1.1.1.1.1.1.1.1.1" xref="S4.Ex1.m1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex1.m1.1.1.1.1.1.1.1.1.1.1.1.2" xref="S4.Ex1.m1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">w</mi><mi id="S4.Ex1.m1.1.1.1.1.1.1.1.1.1.1.1.3" xref="S4.Ex1.m1.1.1.1.1.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S4.Ex1.m1.1.1.1.1.1.1.1.2.2.2.4" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2.2.3.cmml">,</mo><msub id="S4.Ex1.m1.1.1.1.1.1.1.1.2.2.2.2" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2.2.2.2.cmml"><mi id="S4.Ex1.m1.1.1.1.1.1.1.1.2.2.2.2.2" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2.2.2.2.2.cmml">ξ</mi><mi id="S4.Ex1.m1.1.1.1.1.1.1.1.2.2.2.2.3" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2.2.2.2.3.cmml">k</mi></msub><mo id="S4.Ex1.m1.1.1.1.1.1.1.1.2.2.2.5" stretchy="false" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.Ex1.m1.1.1.1.1.1.1.1.4" xref="S4.Ex1.m1.1.1.1.1.1.1.1.4.cmml">−</mo><mrow id="S4.Ex1.m1.1.1.1.1.1.1.1.3" xref="S4.Ex1.m1.1.1.1.1.1.1.1.3.cmml"><mrow id="S4.Ex1.m1.1.1.1.1.1.1.1.3.3" xref="S4.Ex1.m1.1.1.1.1.1.1.1.3.3.cmml"><mo id="S4.Ex1.m1.1.1.1.1.1.1.1.3.3.1" rspace="0.167em" xref="S4.Ex1.m1.1.1.1.1.1.1.1.3.3.1.cmml">∇</mo><msub id="S4.Ex1.m1.1.1.1.1.1.1.1.3.3.2" xref="S4.Ex1.m1.1.1.1.1.1.1.1.3.3.2.cmml"><mi id="S4.Ex1.m1.1.1.1.1.1.1.1.3.3.2.2" xref="S4.Ex1.m1.1.1.1.1.1.1.1.3.3.2.2.cmml">F</mi><mi id="S4.Ex1.m1.1.1.1.1.1.1.1.3.3.2.3" xref="S4.Ex1.m1.1.1.1.1.1.1.1.3.3.2.3.cmml">k</mi></msub></mrow><mo id="S4.Ex1.m1.1.1.1.1.1.1.1.3.2" xref="S4.Ex1.m1.1.1.1.1.1.1.1.3.2.cmml">⁢</mo><mrow id="S4.Ex1.m1.1.1.1.1.1.1.1.3.1.1" xref="S4.Ex1.m1.1.1.1.1.1.1.1.3.1.1.1.cmml"><mo id="S4.Ex1.m1.1.1.1.1.1.1.1.3.1.1.2" stretchy="false" xref="S4.Ex1.m1.1.1.1.1.1.1.1.3.1.1.1.cmml">(</mo><msub id="S4.Ex1.m1.1.1.1.1.1.1.1.3.1.1.1" xref="S4.Ex1.m1.1.1.1.1.1.1.1.3.1.1.1.cmml"><mi id="S4.Ex1.m1.1.1.1.1.1.1.1.3.1.1.1.2" xref="S4.Ex1.m1.1.1.1.1.1.1.1.3.1.1.1.2.cmml">w</mi><mi id="S4.Ex1.m1.1.1.1.1.1.1.1.3.1.1.1.3" xref="S4.Ex1.m1.1.1.1.1.1.1.1.3.1.1.1.3.cmml">k</mi></msub><mo id="S4.Ex1.m1.1.1.1.1.1.1.1.3.1.1.3" stretchy="false" xref="S4.Ex1.m1.1.1.1.1.1.1.1.3.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex1.m1.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex1.m1.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S4.Ex1.m1.1.1.1.1.3" xref="S4.Ex1.m1.1.1.1.1.3.cmml">2</mn></msup></mrow><mo id="S4.Ex1.m1.1.1.2" xref="S4.Ex1.m1.1.1.2.cmml">≤</mo><msubsup id="S4.Ex1.m1.1.1.3" xref="S4.Ex1.m1.1.1.3.cmml"><mi id="S4.Ex1.m1.1.1.3.2.2" xref="S4.Ex1.m1.1.1.3.2.2.cmml">σ</mi><mi id="S4.Ex1.m1.1.1.3.2.3" xref="S4.Ex1.m1.1.1.3.2.3.cmml">k</mi><mn id="S4.Ex1.m1.1.1.3.3" xref="S4.Ex1.m1.1.1.3.3.cmml">2</mn></msubsup></mrow><annotation-xml 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end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p4.6">These assumptions indicate that the noise’s impact on gradient estimation is limited. We consider <math alttext="F_{k}^{*}" class="ltx_Math" display="inline" id="S4.SS1.p4.5.m1.1"><semantics id="S4.SS1.p4.5.m1.1a"><msubsup id="S4.SS1.p4.5.m1.1.1" xref="S4.SS1.p4.5.m1.1.1.cmml"><mi id="S4.SS1.p4.5.m1.1.1.2.2" xref="S4.SS1.p4.5.m1.1.1.2.2.cmml">F</mi><mi id="S4.SS1.p4.5.m1.1.1.2.3" xref="S4.SS1.p4.5.m1.1.1.2.3.cmml">k</mi><mo id="S4.SS1.p4.5.m1.1.1.3" xref="S4.SS1.p4.5.m1.1.1.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.5.m1.1b"><apply id="S4.SS1.p4.5.m1.1.1.cmml" xref="S4.SS1.p4.5.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p4.5.m1.1.1.1.cmml" xref="S4.SS1.p4.5.m1.1.1">superscript</csymbol><apply id="S4.SS1.p4.5.m1.1.1.2.cmml" xref="S4.SS1.p4.5.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p4.5.m1.1.1.2.1.cmml" xref="S4.SS1.p4.5.m1.1.1">subscript</csymbol><ci id="S4.SS1.p4.5.m1.1.1.2.2.cmml" xref="S4.SS1.p4.5.m1.1.1.2.2">𝐹</ci><ci id="S4.SS1.p4.5.m1.1.1.2.3.cmml" xref="S4.SS1.p4.5.m1.1.1.2.3">𝑘</ci></apply><times id="S4.SS1.p4.5.m1.1.1.3.cmml" xref="S4.SS1.p4.5.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.5.m1.1c">F_{k}^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.5.m1.1d">italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> to be the local optimal solution, i.e., it represents the minimum values of <math alttext="F_{k}" class="ltx_Math" display="inline" id="S4.SS1.p4.6.m2.1"><semantics id="S4.SS1.p4.6.m2.1a"><msub id="S4.SS1.p4.6.m2.1.1" xref="S4.SS1.p4.6.m2.1.1.cmml"><mi id="S4.SS1.p4.6.m2.1.1.2" xref="S4.SS1.p4.6.m2.1.1.2.cmml">F</mi><mi id="S4.SS1.p4.6.m2.1.1.3" xref="S4.SS1.p4.6.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.6.m2.1b"><apply id="S4.SS1.p4.6.m2.1.1.cmml" xref="S4.SS1.p4.6.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p4.6.m2.1.1.1.cmml" xref="S4.SS1.p4.6.m2.1.1">subscript</csymbol><ci id="S4.SS1.p4.6.m2.1.1.2.cmml" xref="S4.SS1.p4.6.m2.1.1.2">𝐹</ci><ci id="S4.SS1.p4.6.m2.1.1.3.cmml" xref="S4.SS1.p4.6.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.6.m2.1c">F_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.6.m2.1d">italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. We use the term</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="F^{\ast}=\frac{1}{N}\sum_{k=1}^{N}F_{k}^{*}" class="ltx_Math" display="block" id="S4.Ex3.m1.1"><semantics id="S4.Ex3.m1.1a"><mrow id="S4.Ex3.m1.1.1" xref="S4.Ex3.m1.1.1.cmml"><msup id="S4.Ex3.m1.1.1.2" xref="S4.Ex3.m1.1.1.2.cmml"><mi id="S4.Ex3.m1.1.1.2.2" xref="S4.Ex3.m1.1.1.2.2.cmml">F</mi><mo id="S4.Ex3.m1.1.1.2.3" xref="S4.Ex3.m1.1.1.2.3.cmml">∗</mo></msup><mo id="S4.Ex3.m1.1.1.1" xref="S4.Ex3.m1.1.1.1.cmml">=</mo><mrow id="S4.Ex3.m1.1.1.3" xref="S4.Ex3.m1.1.1.3.cmml"><mfrac id="S4.Ex3.m1.1.1.3.2" xref="S4.Ex3.m1.1.1.3.2.cmml"><mn id="S4.Ex3.m1.1.1.3.2.2" xref="S4.Ex3.m1.1.1.3.2.2.cmml">1</mn><mi id="S4.Ex3.m1.1.1.3.2.3" xref="S4.Ex3.m1.1.1.3.2.3.cmml">N</mi></mfrac><mo id="S4.Ex3.m1.1.1.3.1" xref="S4.Ex3.m1.1.1.3.1.cmml">⁢</mo><mrow 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id="S4.Ex3.m1.1.1.3.3.1.1.cmml" xref="S4.Ex3.m1.1.1.3.3.1">superscript</csymbol><apply id="S4.Ex3.m1.1.1.3.3.1.2.cmml" xref="S4.Ex3.m1.1.1.3.3.1"><csymbol cd="ambiguous" id="S4.Ex3.m1.1.1.3.3.1.2.1.cmml" xref="S4.Ex3.m1.1.1.3.3.1">subscript</csymbol><sum id="S4.Ex3.m1.1.1.3.3.1.2.2.cmml" xref="S4.Ex3.m1.1.1.3.3.1.2.2"></sum><apply id="S4.Ex3.m1.1.1.3.3.1.2.3.cmml" xref="S4.Ex3.m1.1.1.3.3.1.2.3"><eq id="S4.Ex3.m1.1.1.3.3.1.2.3.1.cmml" xref="S4.Ex3.m1.1.1.3.3.1.2.3.1"></eq><ci id="S4.Ex3.m1.1.1.3.3.1.2.3.2.cmml" xref="S4.Ex3.m1.1.1.3.3.1.2.3.2">𝑘</ci><cn id="S4.Ex3.m1.1.1.3.3.1.2.3.3.cmml" type="integer" xref="S4.Ex3.m1.1.1.3.3.1.2.3.3">1</cn></apply></apply><ci id="S4.Ex3.m1.1.1.3.3.1.3.cmml" xref="S4.Ex3.m1.1.1.3.3.1.3">𝑁</ci></apply><apply id="S4.Ex3.m1.1.1.3.3.2.cmml" xref="S4.Ex3.m1.1.1.3.3.2"><csymbol cd="ambiguous" id="S4.Ex3.m1.1.1.3.3.2.1.cmml" xref="S4.Ex3.m1.1.1.3.3.2">superscript</csymbol><apply id="S4.Ex3.m1.1.1.3.3.2.2.cmml" xref="S4.Ex3.m1.1.1.3.3.2"><csymbol cd="ambiguous" id="S4.Ex3.m1.1.1.3.3.2.2.1.cmml" xref="S4.Ex3.m1.1.1.3.3.2">subscript</csymbol><ci id="S4.Ex3.m1.1.1.3.3.2.2.2.cmml" xref="S4.Ex3.m1.1.1.3.3.2.2.2">𝐹</ci><ci id="S4.Ex3.m1.1.1.3.3.2.2.3.cmml" xref="S4.Ex3.m1.1.1.3.3.2.2.3">𝑘</ci></apply><times id="S4.Ex3.m1.1.1.3.3.2.3.cmml" xref="S4.Ex3.m1.1.1.3.3.2.3"></times></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex3.m1.1c">F^{\ast}=\frac{1}{N}\sum_{k=1}^{N}F_{k}^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex3.m1.1d">italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_N end_ARG ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p4.9">and</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="Z=F^{\ast}-F^{\ast}_{k}" class="ltx_Math" display="block" id="S4.Ex4.m1.1"><semantics id="S4.Ex4.m1.1a"><mrow id="S4.Ex4.m1.1.1" xref="S4.Ex4.m1.1.1.cmml"><mi id="S4.Ex4.m1.1.1.2" xref="S4.Ex4.m1.1.1.2.cmml">Z</mi><mo id="S4.Ex4.m1.1.1.1" xref="S4.Ex4.m1.1.1.1.cmml">=</mo><mrow id="S4.Ex4.m1.1.1.3" xref="S4.Ex4.m1.1.1.3.cmml"><msup id="S4.Ex4.m1.1.1.3.2" xref="S4.Ex4.m1.1.1.3.2.cmml"><mi id="S4.Ex4.m1.1.1.3.2.2" xref="S4.Ex4.m1.1.1.3.2.2.cmml">F</mi><mo id="S4.Ex4.m1.1.1.3.2.3" xref="S4.Ex4.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S4.Ex4.m1.1.1.3.1" xref="S4.Ex4.m1.1.1.3.1.cmml">−</mo><msubsup id="S4.Ex4.m1.1.1.3.3" xref="S4.Ex4.m1.1.1.3.3.cmml"><mi id="S4.Ex4.m1.1.1.3.3.2.2" xref="S4.Ex4.m1.1.1.3.3.2.2.cmml">F</mi><mi id="S4.Ex4.m1.1.1.3.3.3" xref="S4.Ex4.m1.1.1.3.3.3.cmml">k</mi><mo id="S4.Ex4.m1.1.1.3.3.2.3" xref="S4.Ex4.m1.1.1.3.3.2.3.cmml">∗</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex4.m1.1b"><apply id="S4.Ex4.m1.1.1.cmml" xref="S4.Ex4.m1.1.1"><eq id="S4.Ex4.m1.1.1.1.cmml" xref="S4.Ex4.m1.1.1.1"></eq><ci id="S4.Ex4.m1.1.1.2.cmml" xref="S4.Ex4.m1.1.1.2">𝑍</ci><apply id="S4.Ex4.m1.1.1.3.cmml" xref="S4.Ex4.m1.1.1.3"><minus id="S4.Ex4.m1.1.1.3.1.cmml" xref="S4.Ex4.m1.1.1.3.1"></minus><apply id="S4.Ex4.m1.1.1.3.2.cmml" xref="S4.Ex4.m1.1.1.3.2"><csymbol cd="ambiguous" id="S4.Ex4.m1.1.1.3.2.1.cmml" xref="S4.Ex4.m1.1.1.3.2">superscript</csymbol><ci id="S4.Ex4.m1.1.1.3.2.2.cmml" xref="S4.Ex4.m1.1.1.3.2.2">𝐹</ci><ci id="S4.Ex4.m1.1.1.3.2.3.cmml" xref="S4.Ex4.m1.1.1.3.2.3">∗</ci></apply><apply id="S4.Ex4.m1.1.1.3.3.cmml" xref="S4.Ex4.m1.1.1.3.3"><csymbol cd="ambiguous" id="S4.Ex4.m1.1.1.3.3.1.cmml" xref="S4.Ex4.m1.1.1.3.3">subscript</csymbol><apply id="S4.Ex4.m1.1.1.3.3.2.cmml" xref="S4.Ex4.m1.1.1.3.3"><csymbol cd="ambiguous" id="S4.Ex4.m1.1.1.3.3.2.1.cmml" xref="S4.Ex4.m1.1.1.3.3">superscript</csymbol><ci id="S4.Ex4.m1.1.1.3.3.2.2.cmml" xref="S4.Ex4.m1.1.1.3.3.2.2">𝐹</ci><ci id="S4.Ex4.m1.1.1.3.3.2.3.cmml" xref="S4.Ex4.m1.1.1.3.3.2.3">∗</ci></apply><ci id="S4.Ex4.m1.1.1.3.3.3.cmml" xref="S4.Ex4.m1.1.1.3.3.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex4.m1.1c">Z=F^{\ast}-F^{\ast}_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex4.m1.1d">italic_Z = italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p4.7">to quantify the degree of non-IID. If the data is IID, <math alttext="Z=0" class="ltx_Math" display="inline" id="S4.SS1.p4.7.m1.1"><semantics id="S4.SS1.p4.7.m1.1a"><mrow id="S4.SS1.p4.7.m1.1.1" xref="S4.SS1.p4.7.m1.1.1.cmml"><mi id="S4.SS1.p4.7.m1.1.1.2" xref="S4.SS1.p4.7.m1.1.1.2.cmml">Z</mi><mo id="S4.SS1.p4.7.m1.1.1.1" xref="S4.SS1.p4.7.m1.1.1.1.cmml">=</mo><mn id="S4.SS1.p4.7.m1.1.1.3" xref="S4.SS1.p4.7.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.7.m1.1b"><apply id="S4.SS1.p4.7.m1.1.1.cmml" xref="S4.SS1.p4.7.m1.1.1"><eq id="S4.SS1.p4.7.m1.1.1.1.cmml" xref="S4.SS1.p4.7.m1.1.1.1"></eq><ci id="S4.SS1.p4.7.m1.1.1.2.cmml" xref="S4.SS1.p4.7.m1.1.1.2">𝑍</ci><cn id="S4.SS1.p4.7.m1.1.1.3.cmml" type="integer" xref="S4.SS1.p4.7.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.7.m1.1c">Z=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.7.m1.1d">italic_Z = 0</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S4.SS2.4.1.1">IV-B</span> </span><span class="ltx_text ltx_font_italic" id="S4.SS2.5.2">Convergence Analysis</span> </h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.3">We show that six different DFL deployments converge to the global optimal for strongly convex functions on Non-IID data. This means we are interested in evaluating <math alttext="E\left(F\left(\bar{w}\right)\right)-F\left(w^{\ast}\right)" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.3"><semantics id="S4.SS2.p1.1.m1.3a"><mrow id="S4.SS2.p1.1.m1.3.3" xref="S4.SS2.p1.1.m1.3.3.cmml"><mrow id="S4.SS2.p1.1.m1.2.2.1" xref="S4.SS2.p1.1.m1.2.2.1.cmml"><mi id="S4.SS2.p1.1.m1.2.2.1.3" xref="S4.SS2.p1.1.m1.2.2.1.3.cmml">E</mi><mo id="S4.SS2.p1.1.m1.2.2.1.2" xref="S4.SS2.p1.1.m1.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS2.p1.1.m1.2.2.1.1.1" xref="S4.SS2.p1.1.m1.2.2.1.1.1.1.cmml"><mo id="S4.SS2.p1.1.m1.2.2.1.1.1.2" xref="S4.SS2.p1.1.m1.2.2.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.p1.1.m1.2.2.1.1.1.1" xref="S4.SS2.p1.1.m1.2.2.1.1.1.1.cmml"><mi id="S4.SS2.p1.1.m1.2.2.1.1.1.1.2" xref="S4.SS2.p1.1.m1.2.2.1.1.1.1.2.cmml">F</mi><mo id="S4.SS2.p1.1.m1.2.2.1.1.1.1.1" xref="S4.SS2.p1.1.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS2.p1.1.m1.2.2.1.1.1.1.3.2" xref="S4.SS2.p1.1.m1.1.1.cmml"><mo id="S4.SS2.p1.1.m1.2.2.1.1.1.1.3.2.1" xref="S4.SS2.p1.1.m1.1.1.cmml">(</mo><mover accent="true" id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml"><mi id="S4.SS2.p1.1.m1.1.1.2" xref="S4.SS2.p1.1.m1.1.1.2.cmml">w</mi><mo id="S4.SS2.p1.1.m1.1.1.1" xref="S4.SS2.p1.1.m1.1.1.1.cmml">¯</mo></mover><mo id="S4.SS2.p1.1.m1.2.2.1.1.1.1.3.2.2" xref="S4.SS2.p1.1.m1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p1.1.m1.2.2.1.1.1.3" xref="S4.SS2.p1.1.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p1.1.m1.3.3.3" xref="S4.SS2.p1.1.m1.3.3.3.cmml">−</mo><mrow id="S4.SS2.p1.1.m1.3.3.2" xref="S4.SS2.p1.1.m1.3.3.2.cmml"><mi id="S4.SS2.p1.1.m1.3.3.2.3" xref="S4.SS2.p1.1.m1.3.3.2.3.cmml">F</mi><mo id="S4.SS2.p1.1.m1.3.3.2.2" xref="S4.SS2.p1.1.m1.3.3.2.2.cmml">⁢</mo><mrow id="S4.SS2.p1.1.m1.3.3.2.1.1" xref="S4.SS2.p1.1.m1.3.3.2.1.1.1.cmml"><mo id="S4.SS2.p1.1.m1.3.3.2.1.1.2" xref="S4.SS2.p1.1.m1.3.3.2.1.1.1.cmml">(</mo><msup id="S4.SS2.p1.1.m1.3.3.2.1.1.1" xref="S4.SS2.p1.1.m1.3.3.2.1.1.1.cmml"><mi id="S4.SS2.p1.1.m1.3.3.2.1.1.1.2" xref="S4.SS2.p1.1.m1.3.3.2.1.1.1.2.cmml">w</mi><mo id="S4.SS2.p1.1.m1.3.3.2.1.1.1.3" xref="S4.SS2.p1.1.m1.3.3.2.1.1.1.3.cmml">∗</mo></msup><mo id="S4.SS2.p1.1.m1.3.3.2.1.1.3" xref="S4.SS2.p1.1.m1.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.3b"><apply id="S4.SS2.p1.1.m1.3.3.cmml" xref="S4.SS2.p1.1.m1.3.3"><minus id="S4.SS2.p1.1.m1.3.3.3.cmml" xref="S4.SS2.p1.1.m1.3.3.3"></minus><apply id="S4.SS2.p1.1.m1.2.2.1.cmml" xref="S4.SS2.p1.1.m1.2.2.1"><times id="S4.SS2.p1.1.m1.2.2.1.2.cmml" xref="S4.SS2.p1.1.m1.2.2.1.2"></times><ci id="S4.SS2.p1.1.m1.2.2.1.3.cmml" xref="S4.SS2.p1.1.m1.2.2.1.3">𝐸</ci><apply id="S4.SS2.p1.1.m1.2.2.1.1.1.1.cmml" xref="S4.SS2.p1.1.m1.2.2.1.1.1"><times id="S4.SS2.p1.1.m1.2.2.1.1.1.1.1.cmml" xref="S4.SS2.p1.1.m1.2.2.1.1.1.1.1"></times><ci id="S4.SS2.p1.1.m1.2.2.1.1.1.1.2.cmml" xref="S4.SS2.p1.1.m1.2.2.1.1.1.1.2">𝐹</ci><apply id="S4.SS2.p1.1.m1.1.1.cmml" xref="S4.SS2.p1.1.m1.2.2.1.1.1.1.3.2"><ci id="S4.SS2.p1.1.m1.1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1.1">¯</ci><ci id="S4.SS2.p1.1.m1.1.1.2.cmml" xref="S4.SS2.p1.1.m1.1.1.2">𝑤</ci></apply></apply></apply><apply id="S4.SS2.p1.1.m1.3.3.2.cmml" xref="S4.SS2.p1.1.m1.3.3.2"><times id="S4.SS2.p1.1.m1.3.3.2.2.cmml" xref="S4.SS2.p1.1.m1.3.3.2.2"></times><ci id="S4.SS2.p1.1.m1.3.3.2.3.cmml" xref="S4.SS2.p1.1.m1.3.3.2.3">𝐹</ci><apply id="S4.SS2.p1.1.m1.3.3.2.1.1.1.cmml" xref="S4.SS2.p1.1.m1.3.3.2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.1.m1.3.3.2.1.1.1.1.cmml" xref="S4.SS2.p1.1.m1.3.3.2.1.1">superscript</csymbol><ci id="S4.SS2.p1.1.m1.3.3.2.1.1.1.2.cmml" xref="S4.SS2.p1.1.m1.3.3.2.1.1.1.2">𝑤</ci><ci id="S4.SS2.p1.1.m1.3.3.2.1.1.1.3.cmml" xref="S4.SS2.p1.1.m1.3.3.2.1.1.1.3">∗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.3c">E\left(F\left(\bar{w}\right)\right)-F\left(w^{\ast}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.1.m1.3d">italic_E ( italic_F ( over¯ start_ARG italic_w end_ARG ) ) - italic_F ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math>. Given that <math alttext="F_{k}" class="ltx_Math" display="inline" id="S4.SS2.p1.2.m2.1"><semantics id="S4.SS2.p1.2.m2.1a"><msub id="S4.SS2.p1.2.m2.1.1" xref="S4.SS2.p1.2.m2.1.1.cmml"><mi id="S4.SS2.p1.2.m2.1.1.2" xref="S4.SS2.p1.2.m2.1.1.2.cmml">F</mi><mi id="S4.SS2.p1.2.m2.1.1.3" xref="S4.SS2.p1.2.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.2.m2.1b"><apply id="S4.SS2.p1.2.m2.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.2.m2.1.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.p1.2.m2.1.1.2.cmml" xref="S4.SS2.p1.2.m2.1.1.2">𝐹</ci><ci id="S4.SS2.p1.2.m2.1.1.3.cmml" xref="S4.SS2.p1.2.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.2.m2.1c">F_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.2.m2.1d">italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is <math alttext="L" class="ltx_Math" display="inline" id="S4.SS2.p1.3.m3.1"><semantics id="S4.SS2.p1.3.m3.1a"><mi id="S4.SS2.p1.3.m3.1.1" xref="S4.SS2.p1.3.m3.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.3.m3.1b"><ci id="S4.SS2.p1.3.m3.1.1.cmml" xref="S4.SS2.p1.3.m3.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.3.m3.1c">L</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.3.m3.1d">italic_L</annotation></semantics></math>-smooth, we can derive the following inequality:</p> </div> <div class="ltx_para" id="S4.SS2.p2"> <table class="ltx_equation ltx_eqn_table" id="S4.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="F(\bar{w})\leq F(w^{*})+\langle\nabla F(w^{*}),\bar{w}-w^{*}\rangle+\frac{L}{2% }\|\bar{w}-w^{*}\|^{2}" class="ltx_Math" display="block" id="S4.E4.m1.5"><semantics id="S4.E4.m1.5a"><mrow id="S4.E4.m1.5.5" xref="S4.E4.m1.5.5.cmml"><mrow id="S4.E4.m1.5.5.6" xref="S4.E4.m1.5.5.6.cmml"><mi id="S4.E4.m1.5.5.6.2" xref="S4.E4.m1.5.5.6.2.cmml">F</mi><mo id="S4.E4.m1.5.5.6.1" xref="S4.E4.m1.5.5.6.1.cmml">⁢</mo><mrow id="S4.E4.m1.5.5.6.3.2" xref="S4.E4.m1.1.1.cmml"><mo id="S4.E4.m1.5.5.6.3.2.1" stretchy="false" xref="S4.E4.m1.1.1.cmml">(</mo><mover accent="true" id="S4.E4.m1.1.1" xref="S4.E4.m1.1.1.cmml"><mi id="S4.E4.m1.1.1.2" xref="S4.E4.m1.1.1.2.cmml">w</mi><mo id="S4.E4.m1.1.1.1" xref="S4.E4.m1.1.1.1.cmml">¯</mo></mover><mo id="S4.E4.m1.5.5.6.3.2.2" stretchy="false" xref="S4.E4.m1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E4.m1.5.5.5" xref="S4.E4.m1.5.5.5.cmml">≤</mo><mrow id="S4.E4.m1.5.5.4" xref="S4.E4.m1.5.5.4.cmml"><mrow id="S4.E4.m1.2.2.1.1" xref="S4.E4.m1.2.2.1.1.cmml"><mi id="S4.E4.m1.2.2.1.1.3" xref="S4.E4.m1.2.2.1.1.3.cmml">F</mi><mo id="S4.E4.m1.2.2.1.1.2" xref="S4.E4.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="S4.E4.m1.2.2.1.1.1.1" xref="S4.E4.m1.2.2.1.1.1.1.1.cmml"><mo 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xref="S4.E4.m1.4.4.3.3.2.2.1.cmml">−</mo><msup id="S4.E4.m1.4.4.3.3.2.2.3" xref="S4.E4.m1.4.4.3.3.2.2.3.cmml"><mi id="S4.E4.m1.4.4.3.3.2.2.3.2" xref="S4.E4.m1.4.4.3.3.2.2.3.2.cmml">w</mi><mo id="S4.E4.m1.4.4.3.3.2.2.3.3" xref="S4.E4.m1.4.4.3.3.2.2.3.3.cmml">∗</mo></msup></mrow><mo id="S4.E4.m1.4.4.3.3.2.5" stretchy="false" xref="S4.E4.m1.4.4.3.3.3.cmml">⟩</mo></mrow><mo id="S4.E4.m1.5.5.4.5a" xref="S4.E4.m1.5.5.4.5.cmml">+</mo><mrow id="S4.E4.m1.5.5.4.4" xref="S4.E4.m1.5.5.4.4.cmml"><mfrac id="S4.E4.m1.5.5.4.4.3" xref="S4.E4.m1.5.5.4.4.3.cmml"><mi id="S4.E4.m1.5.5.4.4.3.2" xref="S4.E4.m1.5.5.4.4.3.2.cmml">L</mi><mn id="S4.E4.m1.5.5.4.4.3.3" xref="S4.E4.m1.5.5.4.4.3.3.cmml">2</mn></mfrac><mo id="S4.E4.m1.5.5.4.4.2" xref="S4.E4.m1.5.5.4.4.2.cmml">⁢</mo><msup id="S4.E4.m1.5.5.4.4.1" xref="S4.E4.m1.5.5.4.4.1.cmml"><mrow id="S4.E4.m1.5.5.4.4.1.1.1" xref="S4.E4.m1.5.5.4.4.1.1.2.cmml"><mo id="S4.E4.m1.5.5.4.4.1.1.1.2" stretchy="false" xref="S4.E4.m1.5.5.4.4.1.1.2.1.cmml">‖</mo><mrow 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id="S4.E4.m1.5.5.4.4.1.1.1.1.3.1.cmml" xref="S4.E4.m1.5.5.4.4.1.1.1.1.3">superscript</csymbol><ci id="S4.E4.m1.5.5.4.4.1.1.1.1.3.2.cmml" xref="S4.E4.m1.5.5.4.4.1.1.1.1.3.2">𝑤</ci><times id="S4.E4.m1.5.5.4.4.1.1.1.1.3.3.cmml" xref="S4.E4.m1.5.5.4.4.1.1.1.1.3.3"></times></apply></apply></apply><cn id="S4.E4.m1.5.5.4.4.1.3.cmml" type="integer" xref="S4.E4.m1.5.5.4.4.1.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E4.m1.5c">F(\bar{w})\leq F(w^{*})+\langle\nabla F(w^{*}),\bar{w}-w^{*}\rangle+\frac{L}{2% }\|\bar{w}-w^{*}\|^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.E4.m1.5d">italic_F ( over¯ start_ARG italic_w end_ARG ) ≤ italic_F ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) + ⟨ ∇ italic_F ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) , over¯ start_ARG italic_w end_ARG - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ⟩ + divide start_ARG italic_L end_ARG start_ARG 2 end_ARG ∥ over¯ start_ARG italic_w end_ARG - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.p3"> <p class="ltx_p" id="S4.SS2.p3.2">Due to the monotonicity of the expected value, we have:</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="E(F(\bar{w}))\leq E(F(w^{*}))+E\left(\langle\nabla F(w^{*}),\bar{w}-w^{*}% \rangle\right)+\frac{L}{2}E\|\bar{w}-w^{*}\|^{2}" class="ltx_Math" display="block" id="S4.Ex5.m1.5"><semantics id="S4.Ex5.m1.5a"><mrow id="S4.Ex5.m1.5.5" 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italic_E ( italic_F ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ) + italic_E ( ⟨ ∇ italic_F ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) , over¯ start_ARG italic_w end_ARG - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ⟩ ) + divide start_ARG italic_L end_ARG start_ARG 2 end_ARG italic_E ∥ over¯ start_ARG italic_w end_ARG - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p3.1">Subtracting <math alttext="F_{k}^{*}" class="ltx_Math" display="inline" id="S4.SS2.p3.1.m1.1"><semantics id="S4.SS2.p3.1.m1.1a"><msubsup id="S4.SS2.p3.1.m1.1.1" xref="S4.SS2.p3.1.m1.1.1.cmml"><mi id="S4.SS2.p3.1.m1.1.1.2.2" xref="S4.SS2.p3.1.m1.1.1.2.2.cmml">F</mi><mi id="S4.SS2.p3.1.m1.1.1.2.3" xref="S4.SS2.p3.1.m1.1.1.2.3.cmml">k</mi><mo id="S4.SS2.p3.1.m1.1.1.3" xref="S4.SS2.p3.1.m1.1.1.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.1.m1.1b"><apply id="S4.SS2.p3.1.m1.1.1.cmml" xref="S4.SS2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.1.m1.1.1.1.cmml" xref="S4.SS2.p3.1.m1.1.1">superscript</csymbol><apply id="S4.SS2.p3.1.m1.1.1.2.cmml" xref="S4.SS2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.1.m1.1.1.2.1.cmml" xref="S4.SS2.p3.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p3.1.m1.1.1.2.2.cmml" xref="S4.SS2.p3.1.m1.1.1.2.2">𝐹</ci><ci id="S4.SS2.p3.1.m1.1.1.2.3.cmml" xref="S4.SS2.p3.1.m1.1.1.2.3">𝑘</ci></apply><times id="S4.SS2.p3.1.m1.1.1.3.cmml" xref="S4.SS2.p3.1.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.1.m1.1c">F_{k}^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.1.m1.1d">italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> from both sides yields:</p> <table class="ltx_equation ltx_eqn_table" id="S4.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{split}E(F(\bar{w}))-F^{*}\leq E(F(w^{*}))-F^{*}+\\ E\left(\langle\nabla F(w^{*}),\bar{w}-w^{*}\rangle\right)+\frac{L}{2}E\|\bar{w% }-w^{*}\|^{2}\end{split}" class="ltx_Math" display="block" id="S4.E5.m1.57"><semantics id="S4.E5.m1.57a"><mtable displaystyle="true" id="S4.E5.m1.57.57.8" rowspacing="0pt" xref="S4.E5.m1.53.53.4.cmml"><mtr id="S4.E5.m1.57.57.8a" xref="S4.E5.m1.53.53.4.cmml"><mtd class="ltx_align_right" columnalign="right" id="S4.E5.m1.57.57.8b" xref="S4.E5.m1.53.53.4.cmml"><mrow id="S4.E5.m1.55.55.6.51.25.25" xref="S4.E5.m1.53.53.4.cmml"><mrow id="S4.E5.m1.54.54.5.50.24.24.24" xref="S4.E5.m1.53.53.4.cmml"><mrow id="S4.E5.m1.54.54.5.50.24.24.24.1" xref="S4.E5.m1.53.53.4.cmml"><mi id="S4.E5.m1.1.1.1.1.1.1" xref="S4.E5.m1.1.1.1.1.1.1.cmml">E</mi><mo id="S4.E5.m1.54.54.5.50.24.24.24.1.2" xref="S4.E5.m1.53.53.4.cmml">⁢</mo><mrow id="S4.E5.m1.54.54.5.50.24.24.24.1.1.1" xref="S4.E5.m1.53.53.4.cmml"><mo id="S4.E5.m1.2.2.2.2.2.2" stretchy="false" xref="S4.E5.m1.53.53.4.cmml">(</mo><mrow id="S4.E5.m1.54.54.5.50.24.24.24.1.1.1.1" xref="S4.E5.m1.53.53.4.cmml"><mi id="S4.E5.m1.3.3.3.3.3.3" xref="S4.E5.m1.3.3.3.3.3.3.cmml">F</mi><mo id="S4.E5.m1.54.54.5.50.24.24.24.1.1.1.1.1" xref="S4.E5.m1.53.53.4.cmml">⁢</mo><mrow id="S4.E5.m1.54.54.5.50.24.24.24.1.1.1.1.2" xref="S4.E5.m1.53.53.4.cmml"><mo id="S4.E5.m1.4.4.4.4.4.4" stretchy="false" xref="S4.E5.m1.53.53.4.cmml">(</mo><mover accent="true" id="S4.E5.m1.5.5.5.5.5.5" xref="S4.E5.m1.5.5.5.5.5.5.cmml"><mi id="S4.E5.m1.5.5.5.5.5.5.2" xref="S4.E5.m1.5.5.5.5.5.5.2.cmml">w</mi><mo id="S4.E5.m1.5.5.5.5.5.5.1" xref="S4.E5.m1.5.5.5.5.5.5.1.cmml">¯</mo></mover><mo id="S4.E5.m1.6.6.6.6.6.6" stretchy="false" xref="S4.E5.m1.53.53.4.cmml">)</mo></mrow></mrow><mo id="S4.E5.m1.7.7.7.7.7.7" stretchy="false" xref="S4.E5.m1.53.53.4.cmml">)</mo></mrow></mrow><mo id="S4.E5.m1.8.8.8.8.8.8" xref="S4.E5.m1.8.8.8.8.8.8.cmml">−</mo><msup id="S4.E5.m1.54.54.5.50.24.24.24.2" xref="S4.E5.m1.53.53.4.cmml"><mi id="S4.E5.m1.9.9.9.9.9.9" xref="S4.E5.m1.9.9.9.9.9.9.cmml">F</mi><mo id="S4.E5.m1.10.10.10.10.10.10.1" xref="S4.E5.m1.10.10.10.10.10.10.1.cmml">∗</mo></msup></mrow><mo id="S4.E5.m1.11.11.11.11.11.11" xref="S4.E5.m1.11.11.11.11.11.11.cmml">≤</mo><mrow id="S4.E5.m1.55.55.6.51.25.25.25" xref="S4.E5.m1.53.53.4.cmml"><mrow id="S4.E5.m1.55.55.6.51.25.25.25.1" xref="S4.E5.m1.53.53.4.cmml"><mi id="S4.E5.m1.12.12.12.12.12.12" xref="S4.E5.m1.12.12.12.12.12.12.cmml">E</mi><mo id="S4.E5.m1.55.55.6.51.25.25.25.1.2" xref="S4.E5.m1.53.53.4.cmml">⁢</mo><mrow id="S4.E5.m1.55.55.6.51.25.25.25.1.1.1" xref="S4.E5.m1.53.53.4.cmml"><mo id="S4.E5.m1.13.13.13.13.13.13" stretchy="false" xref="S4.E5.m1.53.53.4.cmml">(</mo><mrow id="S4.E5.m1.55.55.6.51.25.25.25.1.1.1.1" xref="S4.E5.m1.53.53.4.cmml"><mi id="S4.E5.m1.14.14.14.14.14.14" xref="S4.E5.m1.14.14.14.14.14.14.cmml">F</mi><mo id="S4.E5.m1.55.55.6.51.25.25.25.1.1.1.1.2" xref="S4.E5.m1.53.53.4.cmml">⁢</mo><mrow id="S4.E5.m1.55.55.6.51.25.25.25.1.1.1.1.1.1" xref="S4.E5.m1.53.53.4.cmml"><mo id="S4.E5.m1.15.15.15.15.15.15" stretchy="false" xref="S4.E5.m1.53.53.4.cmml">(</mo><msup id="S4.E5.m1.55.55.6.51.25.25.25.1.1.1.1.1.1.1" xref="S4.E5.m1.53.53.4.cmml"><mi id="S4.E5.m1.16.16.16.16.16.16" xref="S4.E5.m1.16.16.16.16.16.16.cmml">w</mi><mo id="S4.E5.m1.17.17.17.17.17.17.1" xref="S4.E5.m1.17.17.17.17.17.17.1.cmml">∗</mo></msup><mo id="S4.E5.m1.18.18.18.18.18.18" stretchy="false" xref="S4.E5.m1.53.53.4.cmml">)</mo></mrow></mrow><mo id="S4.E5.m1.19.19.19.19.19.19" stretchy="false" xref="S4.E5.m1.53.53.4.cmml">)</mo></mrow></mrow><mo id="S4.E5.m1.20.20.20.20.20.20" xref="S4.E5.m1.20.20.20.20.20.20.cmml">−</mo><mrow id="S4.E5.m1.55.55.6.51.25.25.25.2" xref="S4.E5.m1.53.53.4.cmml"><msup id="S4.E5.m1.55.55.6.51.25.25.25.2.2" xref="S4.E5.m1.53.53.4.cmml"><mi id="S4.E5.m1.21.21.21.21.21.21" xref="S4.E5.m1.21.21.21.21.21.21.cmml">F</mi><mo id="S4.E5.m1.22.22.22.22.22.22.1" xref="S4.E5.m1.22.22.22.22.22.22.1.cmml">∗</mo></msup><mo id="S4.E5.m1.23.23.23.23.23.23" xref="S4.E5.m1.23.23.23.23.23.23.cmml">+</mo></mrow></mrow></mrow></mtd></mtr><mtr id="S4.E5.m1.57.57.8c" xref="S4.E5.m1.53.53.4.cmml"><mtd class="ltx_align_right" columnalign="right" id="S4.E5.m1.57.57.8d" xref="S4.E5.m1.53.53.4.cmml"><mrow id="S4.E5.m1.57.57.8.53.28.28" xref="S4.E5.m1.53.53.4.cmml"><mrow id="S4.E5.m1.56.56.7.52.27.27.27" xref="S4.E5.m1.53.53.4.cmml"><mi id="S4.E5.m1.24.24.24.1.1.1" xref="S4.E5.m1.24.24.24.1.1.1.cmml">E</mi><mo id="S4.E5.m1.56.56.7.52.27.27.27.2" xref="S4.E5.m1.53.53.4.cmml">⁢</mo><mrow id="S4.E5.m1.56.56.7.52.27.27.27.1.1" xref="S4.E5.m1.53.53.4.cmml"><mo id="S4.E5.m1.25.25.25.2.2.2" xref="S4.E5.m1.53.53.4.cmml">(</mo><mrow id="S4.E5.m1.56.56.7.52.27.27.27.1.1.1" xref="S4.E5.m1.53.53.4.cmml"><mo id="S4.E5.m1.26.26.26.3.3.3" stretchy="false" xref="S4.E5.m1.53.53.4.cmml">⟨</mo><mrow id="S4.E5.m1.56.56.7.52.27.27.27.1.1.1.1.1" xref="S4.E5.m1.53.53.4.cmml"><mrow id="S4.E5.m1.56.56.7.52.27.27.27.1.1.1.1.1.3" xref="S4.E5.m1.53.53.4.cmml"><mo id="S4.E5.m1.27.27.27.4.4.4" rspace="0.167em" xref="S4.E5.m1.27.27.27.4.4.4.cmml">∇</mo><mi id="S4.E5.m1.28.28.28.5.5.5" xref="S4.E5.m1.28.28.28.5.5.5.cmml">F</mi></mrow><mo id="S4.E5.m1.56.56.7.52.27.27.27.1.1.1.1.1.2" xref="S4.E5.m1.53.53.4.cmml">⁢</mo><mrow id="S4.E5.m1.56.56.7.52.27.27.27.1.1.1.1.1.1.1" xref="S4.E5.m1.53.53.4.cmml"><mo id="S4.E5.m1.29.29.29.6.6.6" stretchy="false" xref="S4.E5.m1.53.53.4.cmml">(</mo><msup id="S4.E5.m1.56.56.7.52.27.27.27.1.1.1.1.1.1.1.1" xref="S4.E5.m1.53.53.4.cmml"><mi id="S4.E5.m1.30.30.30.7.7.7" xref="S4.E5.m1.30.30.30.7.7.7.cmml">w</mi><mo id="S4.E5.m1.31.31.31.8.8.8.1" xref="S4.E5.m1.31.31.31.8.8.8.1.cmml">∗</mo></msup><mo id="S4.E5.m1.32.32.32.9.9.9" stretchy="false" xref="S4.E5.m1.53.53.4.cmml">)</mo></mrow></mrow><mo id="S4.E5.m1.33.33.33.10.10.10" xref="S4.E5.m1.53.53.4.cmml">,</mo><mrow id="S4.E5.m1.56.56.7.52.27.27.27.1.1.1.2.2" xref="S4.E5.m1.53.53.4.cmml"><mover accent="true" id="S4.E5.m1.34.34.34.11.11.11" xref="S4.E5.m1.34.34.34.11.11.11.cmml"><mi id="S4.E5.m1.34.34.34.11.11.11.2" xref="S4.E5.m1.34.34.34.11.11.11.2.cmml">w</mi><mo id="S4.E5.m1.34.34.34.11.11.11.1" xref="S4.E5.m1.34.34.34.11.11.11.1.cmml">¯</mo></mover><mo id="S4.E5.m1.35.35.35.12.12.12" xref="S4.E5.m1.35.35.35.12.12.12.cmml">−</mo><msup id="S4.E5.m1.56.56.7.52.27.27.27.1.1.1.2.2.1" xref="S4.E5.m1.53.53.4.cmml"><mi id="S4.E5.m1.36.36.36.13.13.13" xref="S4.E5.m1.36.36.36.13.13.13.cmml">w</mi><mo id="S4.E5.m1.37.37.37.14.14.14.1" xref="S4.E5.m1.37.37.37.14.14.14.1.cmml">∗</mo></msup></mrow><mo id="S4.E5.m1.38.38.38.15.15.15" stretchy="false" xref="S4.E5.m1.53.53.4.cmml">⟩</mo></mrow><mo id="S4.E5.m1.39.39.39.16.16.16" xref="S4.E5.m1.53.53.4.cmml">)</mo></mrow></mrow><mo id="S4.E5.m1.40.40.40.17.17.17" xref="S4.E5.m1.53.53.4.cmml">+</mo><mrow id="S4.E5.m1.57.57.8.53.28.28.28" xref="S4.E5.m1.53.53.4.cmml"><mfrac id="S4.E5.m1.41.41.41.18.18.18" xref="S4.E5.m1.41.41.41.18.18.18.cmml"><mi id="S4.E5.m1.41.41.41.18.18.18.2" xref="S4.E5.m1.41.41.41.18.18.18.2.cmml">L</mi><mn id="S4.E5.m1.41.41.41.18.18.18.3" xref="S4.E5.m1.41.41.41.18.18.18.3.cmml">2</mn></mfrac><mo id="S4.E5.m1.57.57.8.53.28.28.28.2" xref="S4.E5.m1.53.53.4.cmml">⁢</mo><mi id="S4.E5.m1.42.42.42.19.19.19" xref="S4.E5.m1.42.42.42.19.19.19.cmml">E</mi><mo id="S4.E5.m1.57.57.8.53.28.28.28.2a" xref="S4.E5.m1.53.53.4.cmml">⁢</mo><msup id="S4.E5.m1.57.57.8.53.28.28.28.1" xref="S4.E5.m1.53.53.4.cmml"><mrow id="S4.E5.m1.57.57.8.53.28.28.28.1.1.1" xref="S4.E5.m1.53.53.4.cmml"><mo id="S4.E5.m1.43.43.43.20.20.20" stretchy="false" xref="S4.E5.m1.53.53.4.cmml">‖</mo><mrow id="S4.E5.m1.57.57.8.53.28.28.28.1.1.1.1" xref="S4.E5.m1.53.53.4.cmml"><mover accent="true" id="S4.E5.m1.44.44.44.21.21.21" 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id="S4.E5.m1.52.52.3.3.2.1.1.1.1.1.2.cmml" xref="S4.E5.m1.57.57.8"></times><apply id="S4.E5.m1.52.52.3.3.2.1.1.1.1.1.3.cmml" xref="S4.E5.m1.57.57.8"><ci id="S4.E5.m1.27.27.27.4.4.4.cmml" xref="S4.E5.m1.27.27.27.4.4.4">∇</ci><ci id="S4.E5.m1.28.28.28.5.5.5.cmml" xref="S4.E5.m1.28.28.28.5.5.5">𝐹</ci></apply><apply id="S4.E5.m1.52.52.3.3.2.1.1.1.1.1.1.1.1.cmml" xref="S4.E5.m1.57.57.8"><csymbol cd="ambiguous" id="S4.E5.m1.52.52.3.3.2.1.1.1.1.1.1.1.1.1.cmml" xref="S4.E5.m1.57.57.8">superscript</csymbol><ci id="S4.E5.m1.30.30.30.7.7.7.cmml" xref="S4.E5.m1.30.30.30.7.7.7">𝑤</ci><times id="S4.E5.m1.31.31.31.8.8.8.1.cmml" xref="S4.E5.m1.31.31.31.8.8.8.1"></times></apply></apply><apply id="S4.E5.m1.52.52.3.3.2.1.1.1.2.2.cmml" xref="S4.E5.m1.57.57.8"><minus id="S4.E5.m1.35.35.35.12.12.12.cmml" xref="S4.E5.m1.35.35.35.12.12.12"></minus><apply id="S4.E5.m1.34.34.34.11.11.11.cmml" xref="S4.E5.m1.34.34.34.11.11.11"><ci id="S4.E5.m1.34.34.34.11.11.11.1.cmml" xref="S4.E5.m1.34.34.34.11.11.11.1">¯</ci><ci id="S4.E5.m1.34.34.34.11.11.11.2.cmml" xref="S4.E5.m1.34.34.34.11.11.11.2">𝑤</ci></apply><apply id="S4.E5.m1.52.52.3.3.2.1.1.1.2.2.3.cmml" xref="S4.E5.m1.57.57.8"><csymbol cd="ambiguous" id="S4.E5.m1.52.52.3.3.2.1.1.1.2.2.3.1.cmml" xref="S4.E5.m1.57.57.8">superscript</csymbol><ci id="S4.E5.m1.36.36.36.13.13.13.cmml" xref="S4.E5.m1.36.36.36.13.13.13">𝑤</ci><times id="S4.E5.m1.37.37.37.14.14.14.1.cmml" xref="S4.E5.m1.37.37.37.14.14.14.1"></times></apply></apply></list></apply><apply id="S4.E5.m1.53.53.4.4.3.cmml" xref="S4.E5.m1.57.57.8"><times id="S4.E5.m1.53.53.4.4.3.2.cmml" xref="S4.E5.m1.57.57.8"></times><apply id="S4.E5.m1.41.41.41.18.18.18.cmml" xref="S4.E5.m1.41.41.41.18.18.18"><divide id="S4.E5.m1.41.41.41.18.18.18.1.cmml" xref="S4.E5.m1.41.41.41.18.18.18"></divide><ci id="S4.E5.m1.41.41.41.18.18.18.2.cmml" xref="S4.E5.m1.41.41.41.18.18.18.2">𝐿</ci><cn id="S4.E5.m1.41.41.41.18.18.18.3.cmml" type="integer" xref="S4.E5.m1.41.41.41.18.18.18.3">2</cn></apply><ci id="S4.E5.m1.42.42.42.19.19.19.cmml" xref="S4.E5.m1.42.42.42.19.19.19">𝐸</ci><apply id="S4.E5.m1.53.53.4.4.3.1.cmml" xref="S4.E5.m1.57.57.8"><csymbol cd="ambiguous" id="S4.E5.m1.53.53.4.4.3.1.2.cmml" xref="S4.E5.m1.57.57.8">superscript</csymbol><apply id="S4.E5.m1.53.53.4.4.3.1.1.2.cmml" xref="S4.E5.m1.57.57.8"><csymbol cd="latexml" id="S4.E5.m1.53.53.4.4.3.1.1.2.1.cmml" xref="S4.E5.m1.57.57.8">delimited-∥∥</csymbol><apply id="S4.E5.m1.53.53.4.4.3.1.1.1.1.cmml" xref="S4.E5.m1.57.57.8"><minus id="S4.E5.m1.45.45.45.22.22.22.cmml" xref="S4.E5.m1.45.45.45.22.22.22"></minus><apply id="S4.E5.m1.44.44.44.21.21.21.cmml" xref="S4.E5.m1.44.44.44.21.21.21"><ci id="S4.E5.m1.44.44.44.21.21.21.1.cmml" xref="S4.E5.m1.44.44.44.21.21.21.1">¯</ci><ci id="S4.E5.m1.44.44.44.21.21.21.2.cmml" xref="S4.E5.m1.44.44.44.21.21.21.2">𝑤</ci></apply><apply id="S4.E5.m1.53.53.4.4.3.1.1.1.1.3.cmml" xref="S4.E5.m1.57.57.8"><csymbol cd="ambiguous" id="S4.E5.m1.53.53.4.4.3.1.1.1.1.3.1.cmml" xref="S4.E5.m1.57.57.8">superscript</csymbol><ci id="S4.E5.m1.46.46.46.23.23.23.cmml" xref="S4.E5.m1.46.46.46.23.23.23">𝑤</ci><times id="S4.E5.m1.47.47.47.24.24.24.1.cmml" xref="S4.E5.m1.47.47.47.24.24.24.1"></times></apply></apply></apply><cn id="S4.E5.m1.49.49.49.26.26.26.1.cmml" type="integer" xref="S4.E5.m1.49.49.49.26.26.26.1">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E5.m1.57c">\begin{split}E(F(\bar{w}))-F^{*}\leq E(F(w^{*}))-F^{*}+\\ E\left(\langle\nabla F(w^{*}),\bar{w}-w^{*}\rangle\right)+\frac{L}{2}E\|\bar{w% }-w^{*}\|^{2}\end{split}</annotation><annotation encoding="application/x-llamapun" id="S4.E5.m1.57d">start_ROW start_CELL italic_E ( italic_F ( over¯ start_ARG italic_w end_ARG ) ) - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ≤ italic_E ( italic_F ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ) - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT + end_CELL end_ROW start_ROW start_CELL italic_E ( ⟨ ∇ italic_F ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) , over¯ start_ARG italic_w end_ARG - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ⟩ ) + divide start_ARG italic_L end_ARG start_ARG 2 end_ARG italic_E ∥ over¯ start_ARG italic_w end_ARG - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.p4"> <p class="ltx_p" id="S4.SS2.p4.5">We have the function: <math alttext="E\left(F\left(w^{\ast}\right)\right)-F^{\ast}=0" class="ltx_Math" display="inline" id="S4.SS2.p4.1.m1.1"><semantics id="S4.SS2.p4.1.m1.1a"><mrow id="S4.SS2.p4.1.m1.1.1" xref="S4.SS2.p4.1.m1.1.1.cmml"><mrow id="S4.SS2.p4.1.m1.1.1.1" xref="S4.SS2.p4.1.m1.1.1.1.cmml"><mrow id="S4.SS2.p4.1.m1.1.1.1.1" xref="S4.SS2.p4.1.m1.1.1.1.1.cmml"><mi id="S4.SS2.p4.1.m1.1.1.1.1.3" xref="S4.SS2.p4.1.m1.1.1.1.1.3.cmml">E</mi><mo id="S4.SS2.p4.1.m1.1.1.1.1.2" xref="S4.SS2.p4.1.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p4.1.m1.1.1.1.1.1.1" xref="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.p4.1.m1.1.1.1.1.1.1.2" xref="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.p4.1.m1.1.1.1.1.1.1.1" xref="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.cmml"><mi id="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.3" xref="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.3.cmml">F</mi><mo id="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.2" xref="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.1.1" xref="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.1.1.2" xref="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.1.1.1" xref="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.1.1.1.2" xref="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.1.1.1.2.cmml">w</mi><mo id="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.1.1.1.3" xref="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.1.1.3" xref="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p4.1.m1.1.1.1.1.1.1.3" xref="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p4.1.m1.1.1.1.2" xref="S4.SS2.p4.1.m1.1.1.1.2.cmml">−</mo><msup id="S4.SS2.p4.1.m1.1.1.1.3" xref="S4.SS2.p4.1.m1.1.1.1.3.cmml"><mi id="S4.SS2.p4.1.m1.1.1.1.3.2" xref="S4.SS2.p4.1.m1.1.1.1.3.2.cmml">F</mi><mo id="S4.SS2.p4.1.m1.1.1.1.3.3" xref="S4.SS2.p4.1.m1.1.1.1.3.3.cmml">∗</mo></msup></mrow><mo id="S4.SS2.p4.1.m1.1.1.2" xref="S4.SS2.p4.1.m1.1.1.2.cmml">=</mo><mn id="S4.SS2.p4.1.m1.1.1.3" xref="S4.SS2.p4.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.1.m1.1b"><apply id="S4.SS2.p4.1.m1.1.1.cmml" xref="S4.SS2.p4.1.m1.1.1"><eq id="S4.SS2.p4.1.m1.1.1.2.cmml" xref="S4.SS2.p4.1.m1.1.1.2"></eq><apply id="S4.SS2.p4.1.m1.1.1.1.cmml" xref="S4.SS2.p4.1.m1.1.1.1"><minus id="S4.SS2.p4.1.m1.1.1.1.2.cmml" xref="S4.SS2.p4.1.m1.1.1.1.2"></minus><apply id="S4.SS2.p4.1.m1.1.1.1.1.cmml" xref="S4.SS2.p4.1.m1.1.1.1.1"><times id="S4.SS2.p4.1.m1.1.1.1.1.2.cmml" xref="S4.SS2.p4.1.m1.1.1.1.1.2"></times><ci id="S4.SS2.p4.1.m1.1.1.1.1.3.cmml" xref="S4.SS2.p4.1.m1.1.1.1.1.3">𝐸</ci><apply id="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.p4.1.m1.1.1.1.1.1.1"><times id="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.2"></times><ci id="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.3">𝐹</ci><apply id="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.1.1">superscript</csymbol><ci id="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.1.1.1.2">𝑤</ci><ci id="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.p4.1.m1.1.1.1.1.1.1.1.1.1.1.3">∗</ci></apply></apply></apply><apply id="S4.SS2.p4.1.m1.1.1.1.3.cmml" xref="S4.SS2.p4.1.m1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p4.1.m1.1.1.1.3.1.cmml" xref="S4.SS2.p4.1.m1.1.1.1.3">superscript</csymbol><ci id="S4.SS2.p4.1.m1.1.1.1.3.2.cmml" xref="S4.SS2.p4.1.m1.1.1.1.3.2">𝐹</ci><ci id="S4.SS2.p4.1.m1.1.1.1.3.3.cmml" xref="S4.SS2.p4.1.m1.1.1.1.3.3">∗</ci></apply></apply><cn id="S4.SS2.p4.1.m1.1.1.3.cmml" type="integer" xref="S4.SS2.p4.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.1.m1.1c">E\left(F\left(w^{\ast}\right)\right)-F^{\ast}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.1.m1.1d">italic_E ( italic_F ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ) - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = 0</annotation></semantics></math>. Strongly convex functions possess the property that their gradient is zero at the optimal point <math alttext="w^{*}" class="ltx_Math" display="inline" id="S4.SS2.p4.2.m2.1"><semantics id="S4.SS2.p4.2.m2.1a"><msup id="S4.SS2.p4.2.m2.1.1" xref="S4.SS2.p4.2.m2.1.1.cmml"><mi id="S4.SS2.p4.2.m2.1.1.2" xref="S4.SS2.p4.2.m2.1.1.2.cmml">w</mi><mo id="S4.SS2.p4.2.m2.1.1.3" xref="S4.SS2.p4.2.m2.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.2.m2.1b"><apply id="S4.SS2.p4.2.m2.1.1.cmml" xref="S4.SS2.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.2.m2.1.1.1.cmml" xref="S4.SS2.p4.2.m2.1.1">superscript</csymbol><ci id="S4.SS2.p4.2.m2.1.1.2.cmml" xref="S4.SS2.p4.2.m2.1.1.2">𝑤</ci><times id="S4.SS2.p4.2.m2.1.1.3.cmml" xref="S4.SS2.p4.2.m2.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.2.m2.1c">w^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.2.m2.1d">italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>. This is because the gradient vector <math alttext="\nabla F(w)" class="ltx_Math" display="inline" id="S4.SS2.p4.3.m3.1"><semantics id="S4.SS2.p4.3.m3.1a"><mrow id="S4.SS2.p4.3.m3.1.2" xref="S4.SS2.p4.3.m3.1.2.cmml"><mrow id="S4.SS2.p4.3.m3.1.2.2" xref="S4.SS2.p4.3.m3.1.2.2.cmml"><mo id="S4.SS2.p4.3.m3.1.2.2.1" rspace="0.167em" xref="S4.SS2.p4.3.m3.1.2.2.1.cmml">∇</mo><mi id="S4.SS2.p4.3.m3.1.2.2.2" xref="S4.SS2.p4.3.m3.1.2.2.2.cmml">F</mi></mrow><mo id="S4.SS2.p4.3.m3.1.2.1" xref="S4.SS2.p4.3.m3.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p4.3.m3.1.2.3.2" xref="S4.SS2.p4.3.m3.1.2.cmml"><mo id="S4.SS2.p4.3.m3.1.2.3.2.1" stretchy="false" xref="S4.SS2.p4.3.m3.1.2.cmml">(</mo><mi id="S4.SS2.p4.3.m3.1.1" xref="S4.SS2.p4.3.m3.1.1.cmml">w</mi><mo id="S4.SS2.p4.3.m3.1.2.3.2.2" stretchy="false" xref="S4.SS2.p4.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.3.m3.1b"><apply id="S4.SS2.p4.3.m3.1.2.cmml" xref="S4.SS2.p4.3.m3.1.2"><times id="S4.SS2.p4.3.m3.1.2.1.cmml" xref="S4.SS2.p4.3.m3.1.2.1"></times><apply id="S4.SS2.p4.3.m3.1.2.2.cmml" xref="S4.SS2.p4.3.m3.1.2.2"><ci id="S4.SS2.p4.3.m3.1.2.2.1.cmml" xref="S4.SS2.p4.3.m3.1.2.2.1">∇</ci><ci id="S4.SS2.p4.3.m3.1.2.2.2.cmml" xref="S4.SS2.p4.3.m3.1.2.2.2">𝐹</ci></apply><ci id="S4.SS2.p4.3.m3.1.1.cmml" xref="S4.SS2.p4.3.m3.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.3.m3.1c">\nabla F(w)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.3.m3.1d">∇ italic_F ( italic_w )</annotation></semantics></math> points toward the fastest growth of the function <math alttext="F(w)" class="ltx_Math" display="inline" id="S4.SS2.p4.4.m4.1"><semantics id="S4.SS2.p4.4.m4.1a"><mrow id="S4.SS2.p4.4.m4.1.2" xref="S4.SS2.p4.4.m4.1.2.cmml"><mi id="S4.SS2.p4.4.m4.1.2.2" xref="S4.SS2.p4.4.m4.1.2.2.cmml">F</mi><mo id="S4.SS2.p4.4.m4.1.2.1" xref="S4.SS2.p4.4.m4.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p4.4.m4.1.2.3.2" xref="S4.SS2.p4.4.m4.1.2.cmml"><mo id="S4.SS2.p4.4.m4.1.2.3.2.1" stretchy="false" xref="S4.SS2.p4.4.m4.1.2.cmml">(</mo><mi id="S4.SS2.p4.4.m4.1.1" xref="S4.SS2.p4.4.m4.1.1.cmml">w</mi><mo id="S4.SS2.p4.4.m4.1.2.3.2.2" stretchy="false" xref="S4.SS2.p4.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.4.m4.1b"><apply id="S4.SS2.p4.4.m4.1.2.cmml" xref="S4.SS2.p4.4.m4.1.2"><times id="S4.SS2.p4.4.m4.1.2.1.cmml" xref="S4.SS2.p4.4.m4.1.2.1"></times><ci id="S4.SS2.p4.4.m4.1.2.2.cmml" xref="S4.SS2.p4.4.m4.1.2.2">𝐹</ci><ci id="S4.SS2.p4.4.m4.1.1.cmml" xref="S4.SS2.p4.4.m4.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.4.m4.1c">F(w)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.4.m4.1d">italic_F ( italic_w )</annotation></semantics></math>. At the optimal point (local or global), the function reaches its minimum value, and consequently, the gradient <math alttext="\nabla F(w^{*})" class="ltx_Math" display="inline" id="S4.SS2.p4.5.m5.1"><semantics id="S4.SS2.p4.5.m5.1a"><mrow id="S4.SS2.p4.5.m5.1.1" xref="S4.SS2.p4.5.m5.1.1.cmml"><mrow id="S4.SS2.p4.5.m5.1.1.3" xref="S4.SS2.p4.5.m5.1.1.3.cmml"><mo id="S4.SS2.p4.5.m5.1.1.3.1" rspace="0.167em" xref="S4.SS2.p4.5.m5.1.1.3.1.cmml">∇</mo><mi id="S4.SS2.p4.5.m5.1.1.3.2" xref="S4.SS2.p4.5.m5.1.1.3.2.cmml">F</mi></mrow><mo id="S4.SS2.p4.5.m5.1.1.2" xref="S4.SS2.p4.5.m5.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p4.5.m5.1.1.1.1" xref="S4.SS2.p4.5.m5.1.1.1.1.1.cmml"><mo id="S4.SS2.p4.5.m5.1.1.1.1.2" stretchy="false" xref="S4.SS2.p4.5.m5.1.1.1.1.1.cmml">(</mo><msup id="S4.SS2.p4.5.m5.1.1.1.1.1" xref="S4.SS2.p4.5.m5.1.1.1.1.1.cmml"><mi id="S4.SS2.p4.5.m5.1.1.1.1.1.2" xref="S4.SS2.p4.5.m5.1.1.1.1.1.2.cmml">w</mi><mo id="S4.SS2.p4.5.m5.1.1.1.1.1.3" xref="S4.SS2.p4.5.m5.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S4.SS2.p4.5.m5.1.1.1.1.3" stretchy="false" xref="S4.SS2.p4.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.5.m5.1b"><apply id="S4.SS2.p4.5.m5.1.1.cmml" xref="S4.SS2.p4.5.m5.1.1"><times id="S4.SS2.p4.5.m5.1.1.2.cmml" xref="S4.SS2.p4.5.m5.1.1.2"></times><apply id="S4.SS2.p4.5.m5.1.1.3.cmml" xref="S4.SS2.p4.5.m5.1.1.3"><ci id="S4.SS2.p4.5.m5.1.1.3.1.cmml" xref="S4.SS2.p4.5.m5.1.1.3.1">∇</ci><ci id="S4.SS2.p4.5.m5.1.1.3.2.cmml" xref="S4.SS2.p4.5.m5.1.1.3.2">𝐹</ci></apply><apply id="S4.SS2.p4.5.m5.1.1.1.1.1.cmml" xref="S4.SS2.p4.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.5.m5.1.1.1.1.1.1.cmml" xref="S4.SS2.p4.5.m5.1.1.1.1">superscript</csymbol><ci id="S4.SS2.p4.5.m5.1.1.1.1.1.2.cmml" xref="S4.SS2.p4.5.m5.1.1.1.1.1.2">𝑤</ci><times id="S4.SS2.p4.5.m5.1.1.1.1.1.3.cmml" xref="S4.SS2.p4.5.m5.1.1.1.1.1.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.5.m5.1c">\nabla F(w^{*})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.5.m5.1d">∇ italic_F ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math> is zero. This can be expressed mathematically as:</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\nabla F(w^{*})=0" class="ltx_Math" display="block" id="S4.Ex6.m1.1"><semantics id="S4.Ex6.m1.1a"><mrow id="S4.Ex6.m1.1.1" xref="S4.Ex6.m1.1.1.cmml"><mrow id="S4.Ex6.m1.1.1.1" xref="S4.Ex6.m1.1.1.1.cmml"><mrow id="S4.Ex6.m1.1.1.1.3" xref="S4.Ex6.m1.1.1.1.3.cmml"><mo id="S4.Ex6.m1.1.1.1.3.1" rspace="0.167em" xref="S4.Ex6.m1.1.1.1.3.1.cmml">∇</mo><mi id="S4.Ex6.m1.1.1.1.3.2" xref="S4.Ex6.m1.1.1.1.3.2.cmml">F</mi></mrow><mo id="S4.Ex6.m1.1.1.1.2" xref="S4.Ex6.m1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex6.m1.1.1.1.1.1" xref="S4.Ex6.m1.1.1.1.1.1.1.cmml"><mo id="S4.Ex6.m1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex6.m1.1.1.1.1.1.1.cmml">(</mo><msup id="S4.Ex6.m1.1.1.1.1.1.1" xref="S4.Ex6.m1.1.1.1.1.1.1.cmml"><mi id="S4.Ex6.m1.1.1.1.1.1.1.2" xref="S4.Ex6.m1.1.1.1.1.1.1.2.cmml">w</mi><mo id="S4.Ex6.m1.1.1.1.1.1.1.3" xref="S4.Ex6.m1.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S4.Ex6.m1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex6.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex6.m1.1.1.2" xref="S4.Ex6.m1.1.1.2.cmml">=</mo><mn id="S4.Ex6.m1.1.1.3" xref="S4.Ex6.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex6.m1.1b"><apply id="S4.Ex6.m1.1.1.cmml" xref="S4.Ex6.m1.1.1"><eq id="S4.Ex6.m1.1.1.2.cmml" xref="S4.Ex6.m1.1.1.2"></eq><apply id="S4.Ex6.m1.1.1.1.cmml" xref="S4.Ex6.m1.1.1.1"><times id="S4.Ex6.m1.1.1.1.2.cmml" xref="S4.Ex6.m1.1.1.1.2"></times><apply id="S4.Ex6.m1.1.1.1.3.cmml" xref="S4.Ex6.m1.1.1.1.3"><ci id="S4.Ex6.m1.1.1.1.3.1.cmml" xref="S4.Ex6.m1.1.1.1.3.1">∇</ci><ci id="S4.Ex6.m1.1.1.1.3.2.cmml" xref="S4.Ex6.m1.1.1.1.3.2">𝐹</ci></apply><apply id="S4.Ex6.m1.1.1.1.1.1.1.cmml" xref="S4.Ex6.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex6.m1.1.1.1.1.1.1.1.cmml" xref="S4.Ex6.m1.1.1.1.1.1">superscript</csymbol><ci id="S4.Ex6.m1.1.1.1.1.1.1.2.cmml" xref="S4.Ex6.m1.1.1.1.1.1.1.2">𝑤</ci><times id="S4.Ex6.m1.1.1.1.1.1.1.3.cmml" xref="S4.Ex6.m1.1.1.1.1.1.1.3"></times></apply></apply><cn id="S4.Ex6.m1.1.1.3.cmml" type="integer" xref="S4.Ex6.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex6.m1.1c">\nabla F(w^{*})=0</annotation><annotation encoding="application/x-llamapun" id="S4.Ex6.m1.1d">∇ italic_F ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) = 0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p4.7">As a result, for any vector <math alttext="w-w^{*}" class="ltx_Math" display="inline" id="S4.SS2.p4.6.m1.1"><semantics id="S4.SS2.p4.6.m1.1a"><mrow id="S4.SS2.p4.6.m1.1.1" xref="S4.SS2.p4.6.m1.1.1.cmml"><mi id="S4.SS2.p4.6.m1.1.1.2" xref="S4.SS2.p4.6.m1.1.1.2.cmml">w</mi><mo id="S4.SS2.p4.6.m1.1.1.1" xref="S4.SS2.p4.6.m1.1.1.1.cmml">−</mo><msup id="S4.SS2.p4.6.m1.1.1.3" xref="S4.SS2.p4.6.m1.1.1.3.cmml"><mi id="S4.SS2.p4.6.m1.1.1.3.2" xref="S4.SS2.p4.6.m1.1.1.3.2.cmml">w</mi><mo id="S4.SS2.p4.6.m1.1.1.3.3" xref="S4.SS2.p4.6.m1.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.6.m1.1b"><apply id="S4.SS2.p4.6.m1.1.1.cmml" xref="S4.SS2.p4.6.m1.1.1"><minus id="S4.SS2.p4.6.m1.1.1.1.cmml" xref="S4.SS2.p4.6.m1.1.1.1"></minus><ci id="S4.SS2.p4.6.m1.1.1.2.cmml" xref="S4.SS2.p4.6.m1.1.1.2">𝑤</ci><apply id="S4.SS2.p4.6.m1.1.1.3.cmml" xref="S4.SS2.p4.6.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p4.6.m1.1.1.3.1.cmml" xref="S4.SS2.p4.6.m1.1.1.3">superscript</csymbol><ci id="S4.SS2.p4.6.m1.1.1.3.2.cmml" xref="S4.SS2.p4.6.m1.1.1.3.2">𝑤</ci><times id="S4.SS2.p4.6.m1.1.1.3.3.cmml" xref="S4.SS2.p4.6.m1.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.6.m1.1c">w-w^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.6.m1.1d">italic_w - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, the inner product <math alttext="\langle\nabla F(w^{*}),w-w^{*}\rangle" class="ltx_Math" display="inline" id="S4.SS2.p4.7.m2.2"><semantics id="S4.SS2.p4.7.m2.2a"><mrow id="S4.SS2.p4.7.m2.2.2.2" xref="S4.SS2.p4.7.m2.2.2.3.cmml"><mo id="S4.SS2.p4.7.m2.2.2.2.3" stretchy="false" xref="S4.SS2.p4.7.m2.2.2.3.cmml">⟨</mo><mrow id="S4.SS2.p4.7.m2.1.1.1.1" xref="S4.SS2.p4.7.m2.1.1.1.1.cmml"><mrow id="S4.SS2.p4.7.m2.1.1.1.1.3" xref="S4.SS2.p4.7.m2.1.1.1.1.3.cmml"><mo id="S4.SS2.p4.7.m2.1.1.1.1.3.1" rspace="0.167em" xref="S4.SS2.p4.7.m2.1.1.1.1.3.1.cmml">∇</mo><mi id="S4.SS2.p4.7.m2.1.1.1.1.3.2" xref="S4.SS2.p4.7.m2.1.1.1.1.3.2.cmml">F</mi></mrow><mo id="S4.SS2.p4.7.m2.1.1.1.1.2" xref="S4.SS2.p4.7.m2.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p4.7.m2.1.1.1.1.1.1" xref="S4.SS2.p4.7.m2.1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.p4.7.m2.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.p4.7.m2.1.1.1.1.1.1.1.cmml">(</mo><msup id="S4.SS2.p4.7.m2.1.1.1.1.1.1.1" xref="S4.SS2.p4.7.m2.1.1.1.1.1.1.1.cmml"><mi id="S4.SS2.p4.7.m2.1.1.1.1.1.1.1.2" xref="S4.SS2.p4.7.m2.1.1.1.1.1.1.1.2.cmml">w</mi><mo id="S4.SS2.p4.7.m2.1.1.1.1.1.1.1.3" xref="S4.SS2.p4.7.m2.1.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S4.SS2.p4.7.m2.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.p4.7.m2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p4.7.m2.2.2.2.4" xref="S4.SS2.p4.7.m2.2.2.3.cmml">,</mo><mrow id="S4.SS2.p4.7.m2.2.2.2.2" xref="S4.SS2.p4.7.m2.2.2.2.2.cmml"><mi id="S4.SS2.p4.7.m2.2.2.2.2.2" xref="S4.SS2.p4.7.m2.2.2.2.2.2.cmml">w</mi><mo id="S4.SS2.p4.7.m2.2.2.2.2.1" xref="S4.SS2.p4.7.m2.2.2.2.2.1.cmml">−</mo><msup id="S4.SS2.p4.7.m2.2.2.2.2.3" xref="S4.SS2.p4.7.m2.2.2.2.2.3.cmml"><mi id="S4.SS2.p4.7.m2.2.2.2.2.3.2" xref="S4.SS2.p4.7.m2.2.2.2.2.3.2.cmml">w</mi><mo id="S4.SS2.p4.7.m2.2.2.2.2.3.3" xref="S4.SS2.p4.7.m2.2.2.2.2.3.3.cmml">∗</mo></msup></mrow><mo id="S4.SS2.p4.7.m2.2.2.2.5" stretchy="false" xref="S4.SS2.p4.7.m2.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.7.m2.2b"><list id="S4.SS2.p4.7.m2.2.2.3.cmml" xref="S4.SS2.p4.7.m2.2.2.2"><apply id="S4.SS2.p4.7.m2.1.1.1.1.cmml" xref="S4.SS2.p4.7.m2.1.1.1.1"><times id="S4.SS2.p4.7.m2.1.1.1.1.2.cmml" xref="S4.SS2.p4.7.m2.1.1.1.1.2"></times><apply id="S4.SS2.p4.7.m2.1.1.1.1.3.cmml" xref="S4.SS2.p4.7.m2.1.1.1.1.3"><ci id="S4.SS2.p4.7.m2.1.1.1.1.3.1.cmml" xref="S4.SS2.p4.7.m2.1.1.1.1.3.1">∇</ci><ci id="S4.SS2.p4.7.m2.1.1.1.1.3.2.cmml" xref="S4.SS2.p4.7.m2.1.1.1.1.3.2">𝐹</ci></apply><apply id="S4.SS2.p4.7.m2.1.1.1.1.1.1.1.cmml" xref="S4.SS2.p4.7.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.7.m2.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.p4.7.m2.1.1.1.1.1.1">superscript</csymbol><ci id="S4.SS2.p4.7.m2.1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.p4.7.m2.1.1.1.1.1.1.1.2">𝑤</ci><times id="S4.SS2.p4.7.m2.1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.p4.7.m2.1.1.1.1.1.1.1.3"></times></apply></apply><apply id="S4.SS2.p4.7.m2.2.2.2.2.cmml" xref="S4.SS2.p4.7.m2.2.2.2.2"><minus id="S4.SS2.p4.7.m2.2.2.2.2.1.cmml" xref="S4.SS2.p4.7.m2.2.2.2.2.1"></minus><ci id="S4.SS2.p4.7.m2.2.2.2.2.2.cmml" xref="S4.SS2.p4.7.m2.2.2.2.2.2">𝑤</ci><apply id="S4.SS2.p4.7.m2.2.2.2.2.3.cmml" xref="S4.SS2.p4.7.m2.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.SS2.p4.7.m2.2.2.2.2.3.1.cmml" xref="S4.SS2.p4.7.m2.2.2.2.2.3">superscript</csymbol><ci id="S4.SS2.p4.7.m2.2.2.2.2.3.2.cmml" xref="S4.SS2.p4.7.m2.2.2.2.2.3.2">𝑤</ci><times id="S4.SS2.p4.7.m2.2.2.2.2.3.3.cmml" xref="S4.SS2.p4.7.m2.2.2.2.2.3.3"></times></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.7.m2.2c">\langle\nabla F(w^{*}),w-w^{*}\rangle</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.7.m2.2d">⟨ ∇ italic_F ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) , italic_w - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ⟩</annotation></semantics></math> is zero:</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\langle\nabla F(w^{*}),w-w^{*}\rangle=0" class="ltx_Math" display="block" id="S4.Ex7.m1.2"><semantics id="S4.Ex7.m1.2a"><mrow id="S4.Ex7.m1.2.2" xref="S4.Ex7.m1.2.2.cmml"><mrow id="S4.Ex7.m1.2.2.2.2" xref="S4.Ex7.m1.2.2.2.3.cmml"><mo id="S4.Ex7.m1.2.2.2.2.3" stretchy="false" xref="S4.Ex7.m1.2.2.2.3.cmml">⟨</mo><mrow id="S4.Ex7.m1.1.1.1.1.1" xref="S4.Ex7.m1.1.1.1.1.1.cmml"><mrow id="S4.Ex7.m1.1.1.1.1.1.3" xref="S4.Ex7.m1.1.1.1.1.1.3.cmml"><mo id="S4.Ex7.m1.1.1.1.1.1.3.1" rspace="0.167em" xref="S4.Ex7.m1.1.1.1.1.1.3.1.cmml">∇</mo><mi id="S4.Ex7.m1.1.1.1.1.1.3.2" xref="S4.Ex7.m1.1.1.1.1.1.3.2.cmml">F</mi></mrow><mo id="S4.Ex7.m1.1.1.1.1.1.2" xref="S4.Ex7.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex7.m1.1.1.1.1.1.1.1" xref="S4.Ex7.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex7.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex7.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S4.Ex7.m1.1.1.1.1.1.1.1.1" xref="S4.Ex7.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex7.m1.1.1.1.1.1.1.1.1.2" xref="S4.Ex7.m1.1.1.1.1.1.1.1.1.2.cmml">w</mi><mo id="S4.Ex7.m1.1.1.1.1.1.1.1.1.3" xref="S4.Ex7.m1.1.1.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S4.Ex7.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex7.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex7.m1.2.2.2.2.4" xref="S4.Ex7.m1.2.2.2.3.cmml">,</mo><mrow id="S4.Ex7.m1.2.2.2.2.2" xref="S4.Ex7.m1.2.2.2.2.2.cmml"><mi id="S4.Ex7.m1.2.2.2.2.2.2" xref="S4.Ex7.m1.2.2.2.2.2.2.cmml">w</mi><mo id="S4.Ex7.m1.2.2.2.2.2.1" xref="S4.Ex7.m1.2.2.2.2.2.1.cmml">−</mo><msup id="S4.Ex7.m1.2.2.2.2.2.3" xref="S4.Ex7.m1.2.2.2.2.2.3.cmml"><mi id="S4.Ex7.m1.2.2.2.2.2.3.2" xref="S4.Ex7.m1.2.2.2.2.2.3.2.cmml">w</mi><mo id="S4.Ex7.m1.2.2.2.2.2.3.3" xref="S4.Ex7.m1.2.2.2.2.2.3.3.cmml">∗</mo></msup></mrow><mo id="S4.Ex7.m1.2.2.2.2.5" stretchy="false" xref="S4.Ex7.m1.2.2.2.3.cmml">⟩</mo></mrow><mo id="S4.Ex7.m1.2.2.3" xref="S4.Ex7.m1.2.2.3.cmml">=</mo><mn id="S4.Ex7.m1.2.2.4" xref="S4.Ex7.m1.2.2.4.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex7.m1.2b"><apply id="S4.Ex7.m1.2.2.cmml" xref="S4.Ex7.m1.2.2"><eq id="S4.Ex7.m1.2.2.3.cmml" xref="S4.Ex7.m1.2.2.3"></eq><list id="S4.Ex7.m1.2.2.2.3.cmml" xref="S4.Ex7.m1.2.2.2.2"><apply id="S4.Ex7.m1.1.1.1.1.1.cmml" xref="S4.Ex7.m1.1.1.1.1.1"><times id="S4.Ex7.m1.1.1.1.1.1.2.cmml" xref="S4.Ex7.m1.1.1.1.1.1.2"></times><apply id="S4.Ex7.m1.1.1.1.1.1.3.cmml" xref="S4.Ex7.m1.1.1.1.1.1.3"><ci id="S4.Ex7.m1.1.1.1.1.1.3.1.cmml" xref="S4.Ex7.m1.1.1.1.1.1.3.1">∇</ci><ci id="S4.Ex7.m1.1.1.1.1.1.3.2.cmml" xref="S4.Ex7.m1.1.1.1.1.1.3.2">𝐹</ci></apply><apply id="S4.Ex7.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex7.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex7.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex7.m1.1.1.1.1.1.1.1">superscript</csymbol><ci id="S4.Ex7.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.Ex7.m1.1.1.1.1.1.1.1.1.2">𝑤</ci><times id="S4.Ex7.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.Ex7.m1.1.1.1.1.1.1.1.1.3"></times></apply></apply><apply id="S4.Ex7.m1.2.2.2.2.2.cmml" xref="S4.Ex7.m1.2.2.2.2.2"><minus id="S4.Ex7.m1.2.2.2.2.2.1.cmml" xref="S4.Ex7.m1.2.2.2.2.2.1"></minus><ci id="S4.Ex7.m1.2.2.2.2.2.2.cmml" xref="S4.Ex7.m1.2.2.2.2.2.2">𝑤</ci><apply id="S4.Ex7.m1.2.2.2.2.2.3.cmml" xref="S4.Ex7.m1.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.Ex7.m1.2.2.2.2.2.3.1.cmml" xref="S4.Ex7.m1.2.2.2.2.2.3">superscript</csymbol><ci id="S4.Ex7.m1.2.2.2.2.2.3.2.cmml" xref="S4.Ex7.m1.2.2.2.2.2.3.2">𝑤</ci><times id="S4.Ex7.m1.2.2.2.2.2.3.3.cmml" xref="S4.Ex7.m1.2.2.2.2.2.3.3"></times></apply></apply></list><cn id="S4.Ex7.m1.2.2.4.cmml" type="integer" xref="S4.Ex7.m1.2.2.4">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex7.m1.2c">\langle\nabla F(w^{*}),w-w^{*}\rangle=0</annotation><annotation encoding="application/x-llamapun" id="S4.Ex7.m1.2d">⟨ ∇ italic_F ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) , italic_w - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ⟩ = 0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p4.8">Hence we obtain the following:</p> </div> <div class="ltx_para" id="S4.SS2.p5"> <table class="ltx_equation ltx_eqn_table" id="S4.E6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="E(F(\bar{w}))-F\left(w^{\ast}\right)\leq\frac{L}{2}E||\bar{w}-w^{\ast}||^{2}" class="ltx_Math" display="block" id="S4.E6.m1.4"><semantics id="S4.E6.m1.4a"><mrow id="S4.E6.m1.4.4" xref="S4.E6.m1.4.4.cmml"><mrow id="S4.E6.m1.3.3.2" xref="S4.E6.m1.3.3.2.cmml"><mrow id="S4.E6.m1.2.2.1.1" xref="S4.E6.m1.2.2.1.1.cmml"><mi id="S4.E6.m1.2.2.1.1.3" xref="S4.E6.m1.2.2.1.1.3.cmml">E</mi><mo id="S4.E6.m1.2.2.1.1.2" xref="S4.E6.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="S4.E6.m1.2.2.1.1.1.1" xref="S4.E6.m1.2.2.1.1.1.1.1.cmml"><mo id="S4.E6.m1.2.2.1.1.1.1.2" stretchy="false" xref="S4.E6.m1.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S4.E6.m1.2.2.1.1.1.1.1" xref="S4.E6.m1.2.2.1.1.1.1.1.cmml"><mi id="S4.E6.m1.2.2.1.1.1.1.1.2" xref="S4.E6.m1.2.2.1.1.1.1.1.2.cmml">F</mi><mo id="S4.E6.m1.2.2.1.1.1.1.1.1" xref="S4.E6.m1.2.2.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E6.m1.2.2.1.1.1.1.1.3.2" xref="S4.E6.m1.1.1.cmml"><mo id="S4.E6.m1.2.2.1.1.1.1.1.3.2.1" stretchy="false" xref="S4.E6.m1.1.1.cmml">(</mo><mover accent="true" id="S4.E6.m1.1.1" xref="S4.E6.m1.1.1.cmml"><mi id="S4.E6.m1.1.1.2" xref="S4.E6.m1.1.1.2.cmml">w</mi><mo id="S4.E6.m1.1.1.1" xref="S4.E6.m1.1.1.1.cmml">¯</mo></mover><mo id="S4.E6.m1.2.2.1.1.1.1.1.3.2.2" stretchy="false" xref="S4.E6.m1.1.1.cmml">)</mo></mrow></mrow><mo 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start_ARG italic_w end_ARG - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.p6"> <p class="ltx_p" id="S4.SS2.p6.5">The inequality in Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4.E6" title="In IV-B Convergence Analysis ‣ IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">6</span></a> indicates that in a DFL deployment, the expected difference in loss values between the obtained model parameters <math alttext="\bar{w}" class="ltx_Math" display="inline" id="S4.SS2.p6.1.m1.1"><semantics id="S4.SS2.p6.1.m1.1a"><mover accent="true" id="S4.SS2.p6.1.m1.1.1" xref="S4.SS2.p6.1.m1.1.1.cmml"><mi id="S4.SS2.p6.1.m1.1.1.2" xref="S4.SS2.p6.1.m1.1.1.2.cmml">w</mi><mo id="S4.SS2.p6.1.m1.1.1.1" xref="S4.SS2.p6.1.m1.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.1.m1.1b"><apply id="S4.SS2.p6.1.m1.1.1.cmml" xref="S4.SS2.p6.1.m1.1.1"><ci id="S4.SS2.p6.1.m1.1.1.1.cmml" xref="S4.SS2.p6.1.m1.1.1.1">¯</ci><ci id="S4.SS2.p6.1.m1.1.1.2.cmml" xref="S4.SS2.p6.1.m1.1.1.2">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.1.m1.1c">\bar{w}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p6.1.m1.1d">over¯ start_ARG italic_w end_ARG</annotation></semantics></math> and the optimal model parameters <math alttext="w^{*}" class="ltx_Math" display="inline" id="S4.SS2.p6.2.m2.1"><semantics id="S4.SS2.p6.2.m2.1a"><msup id="S4.SS2.p6.2.m2.1.1" xref="S4.SS2.p6.2.m2.1.1.cmml"><mi id="S4.SS2.p6.2.m2.1.1.2" 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encoding="application/x-tex" id="S4.SS2.p6.4.m4.1c">\frac{L}{2}E\|\bar{w}-w^{*}\|^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p6.4.m4.1d">divide start_ARG italic_L end_ARG start_ARG 2 end_ARG italic_E ∥ over¯ start_ARG italic_w end_ARG - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. In other words, this inequality shows that the smaller the distance between the average model parameters and the optimal parameters, the smaller the error in the loss function values. Our subsequent steps involve simplifying <math alttext="\|\bar{w}-w^{*}\|^{2}" class="ltx_Math" display="inline" id="S4.SS2.p6.5.m5.1"><semantics id="S4.SS2.p6.5.m5.1a"><msup id="S4.SS2.p6.5.m5.1.1" xref="S4.SS2.p6.5.m5.1.1.cmml"><mrow id="S4.SS2.p6.5.m5.1.1.1.1" xref="S4.SS2.p6.5.m5.1.1.1.2.cmml"><mo id="S4.SS2.p6.5.m5.1.1.1.1.2" stretchy="false" xref="S4.SS2.p6.5.m5.1.1.1.2.1.cmml">‖</mo><mrow id="S4.SS2.p6.5.m5.1.1.1.1.1" xref="S4.SS2.p6.5.m5.1.1.1.1.1.cmml"><mover accent="true" id="S4.SS2.p6.5.m5.1.1.1.1.1.2" xref="S4.SS2.p6.5.m5.1.1.1.1.1.2.cmml"><mi id="S4.SS2.p6.5.m5.1.1.1.1.1.2.2" xref="S4.SS2.p6.5.m5.1.1.1.1.1.2.2.cmml">w</mi><mo id="S4.SS2.p6.5.m5.1.1.1.1.1.2.1" xref="S4.SS2.p6.5.m5.1.1.1.1.1.2.1.cmml">¯</mo></mover><mo id="S4.SS2.p6.5.m5.1.1.1.1.1.1" xref="S4.SS2.p6.5.m5.1.1.1.1.1.1.cmml">−</mo><msup id="S4.SS2.p6.5.m5.1.1.1.1.1.3" xref="S4.SS2.p6.5.m5.1.1.1.1.1.3.cmml"><mi id="S4.SS2.p6.5.m5.1.1.1.1.1.3.2" xref="S4.SS2.p6.5.m5.1.1.1.1.1.3.2.cmml">w</mi><mo id="S4.SS2.p6.5.m5.1.1.1.1.1.3.3" xref="S4.SS2.p6.5.m5.1.1.1.1.1.3.3.cmml">∗</mo></msup></mrow><mo id="S4.SS2.p6.5.m5.1.1.1.1.3" stretchy="false" xref="S4.SS2.p6.5.m5.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S4.SS2.p6.5.m5.1.1.3" xref="S4.SS2.p6.5.m5.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.5.m5.1b"><apply id="S4.SS2.p6.5.m5.1.1.cmml" xref="S4.SS2.p6.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS2.p6.5.m5.1.1.2.cmml" xref="S4.SS2.p6.5.m5.1.1">superscript</csymbol><apply id="S4.SS2.p6.5.m5.1.1.1.2.cmml" xref="S4.SS2.p6.5.m5.1.1.1.1"><csymbol cd="latexml" id="S4.SS2.p6.5.m5.1.1.1.2.1.cmml" xref="S4.SS2.p6.5.m5.1.1.1.1.2">norm</csymbol><apply id="S4.SS2.p6.5.m5.1.1.1.1.1.cmml" xref="S4.SS2.p6.5.m5.1.1.1.1.1"><minus id="S4.SS2.p6.5.m5.1.1.1.1.1.1.cmml" xref="S4.SS2.p6.5.m5.1.1.1.1.1.1"></minus><apply id="S4.SS2.p6.5.m5.1.1.1.1.1.2.cmml" xref="S4.SS2.p6.5.m5.1.1.1.1.1.2"><ci id="S4.SS2.p6.5.m5.1.1.1.1.1.2.1.cmml" xref="S4.SS2.p6.5.m5.1.1.1.1.1.2.1">¯</ci><ci id="S4.SS2.p6.5.m5.1.1.1.1.1.2.2.cmml" xref="S4.SS2.p6.5.m5.1.1.1.1.1.2.2">𝑤</ci></apply><apply id="S4.SS2.p6.5.m5.1.1.1.1.1.3.cmml" xref="S4.SS2.p6.5.m5.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p6.5.m5.1.1.1.1.1.3.1.cmml" xref="S4.SS2.p6.5.m5.1.1.1.1.1.3">superscript</csymbol><ci id="S4.SS2.p6.5.m5.1.1.1.1.1.3.2.cmml" xref="S4.SS2.p6.5.m5.1.1.1.1.1.3.2">𝑤</ci><times id="S4.SS2.p6.5.m5.1.1.1.1.1.3.3.cmml" xref="S4.SS2.p6.5.m5.1.1.1.1.1.3.3"></times></apply></apply></apply><cn id="S4.SS2.p6.5.m5.1.1.3.cmml" type="integer" xref="S4.SS2.p6.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.5.m5.1c">\|\bar{w}-w^{*}\|^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p6.5.m5.1d">∥ over¯ start_ARG italic_w end_ARG - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> based on different DFL deployments. Thus, we will derive different convergence formulas for each topology.</p> </div> <section class="ltx_subsubsection" id="S4.SS2.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection"><span class="ltx_text" id="S4.SS2.SSS1.4.1.1">IV-B</span>1 </span>Continuous Linear</h4> <div class="ltx_para" id="S4.SS2.SSS1.p1"> <p class="ltx_p" id="S4.SS2.SSS1.p1.3">The Equation (<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4.E7" title="In IV-B1 Continuous Linear ‣ IV-B Convergence Analysis ‣ IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">7</span></a>) shows the transfer of parameters. In detail, the continuous linear DFL <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib5" title="">5</a>]</cite> is structured as a line, with each node (device) connected to the next. Each device <math alttext="k" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.1.m1.1"><semantics id="S4.SS2.SSS1.p1.1.m1.1a"><mi id="S4.SS2.SSS1.p1.1.m1.1.1" xref="S4.SS2.SSS1.p1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.1.m1.1b"><ci id="S4.SS2.SSS1.p1.1.m1.1.1.cmml" xref="S4.SS2.SSS1.p1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.1.m1.1d">italic_k</annotation></semantics></math> receives the model parameters from the previous device <math alttext="k-1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.2.m2.1"><semantics id="S4.SS2.SSS1.p1.2.m2.1a"><mrow id="S4.SS2.SSS1.p1.2.m2.1.1" xref="S4.SS2.SSS1.p1.2.m2.1.1.cmml"><mi id="S4.SS2.SSS1.p1.2.m2.1.1.2" xref="S4.SS2.SSS1.p1.2.m2.1.1.2.cmml">k</mi><mo id="S4.SS2.SSS1.p1.2.m2.1.1.1" xref="S4.SS2.SSS1.p1.2.m2.1.1.1.cmml">−</mo><mn id="S4.SS2.SSS1.p1.2.m2.1.1.3" xref="S4.SS2.SSS1.p1.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.2.m2.1b"><apply id="S4.SS2.SSS1.p1.2.m2.1.1.cmml" xref="S4.SS2.SSS1.p1.2.m2.1.1"><minus id="S4.SS2.SSS1.p1.2.m2.1.1.1.cmml" xref="S4.SS2.SSS1.p1.2.m2.1.1.1"></minus><ci id="S4.SS2.SSS1.p1.2.m2.1.1.2.cmml" xref="S4.SS2.SSS1.p1.2.m2.1.1.2">𝑘</ci><cn id="S4.SS2.SSS1.p1.2.m2.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.2.m2.1c">k-1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.2.m2.1d">italic_k - 1</annotation></semantics></math> and updates the model using its local dataset <math alttext="x_{k}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.3.m3.1"><semantics id="S4.SS2.SSS1.p1.3.m3.1a"><msub id="S4.SS2.SSS1.p1.3.m3.1.1" xref="S4.SS2.SSS1.p1.3.m3.1.1.cmml"><mi id="S4.SS2.SSS1.p1.3.m3.1.1.2" xref="S4.SS2.SSS1.p1.3.m3.1.1.2.cmml">x</mi><mi id="S4.SS2.SSS1.p1.3.m3.1.1.3" xref="S4.SS2.SSS1.p1.3.m3.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.3.m3.1b"><apply id="S4.SS2.SSS1.p1.3.m3.1.1.cmml" xref="S4.SS2.SSS1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p1.3.m3.1.1.1.cmml" xref="S4.SS2.SSS1.p1.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p1.3.m3.1.1.2.cmml" xref="S4.SS2.SSS1.p1.3.m3.1.1.2">𝑥</ci><ci id="S4.SS2.SSS1.p1.3.m3.1.1.3.cmml" xref="S4.SS2.SSS1.p1.3.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.3.m3.1c">x_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.3.m3.1d">italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. Subsequently, it forwards the trained parameters along the line to the next one. This sequential and iterative training continues along the line, fostering collaboration among the clients to enhance the learning and performance of the entire system collectively. This process can be formulated as follows:</p> </div> <div class="ltx_para" id="S4.SS2.SSS1.p2"> <table class="ltx_equation ltx_eqn_table" id="S4.E7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="w_{k}=w_{k-1}-\eta_{k}\nabla F_{k}\left(w_{k-1},\xi_{k}\right)(1\leq k\leq N)" class="ltx_Math" display="block" id="S4.E7.m1.3"><semantics id="S4.E7.m1.3a"><mrow id="S4.E7.m1.3.3" xref="S4.E7.m1.3.3.cmml"><msub id="S4.E7.m1.3.3.5" xref="S4.E7.m1.3.3.5.cmml"><mi id="S4.E7.m1.3.3.5.2" xref="S4.E7.m1.3.3.5.2.cmml">w</mi><mi id="S4.E7.m1.3.3.5.3" xref="S4.E7.m1.3.3.5.3.cmml">k</mi></msub><mo id="S4.E7.m1.3.3.4" xref="S4.E7.m1.3.3.4.cmml">=</mo><mrow id="S4.E7.m1.3.3.3" xref="S4.E7.m1.3.3.3.cmml"><msub id="S4.E7.m1.3.3.3.5" xref="S4.E7.m1.3.3.3.5.cmml"><mi id="S4.E7.m1.3.3.3.5.2" xref="S4.E7.m1.3.3.3.5.2.cmml">w</mi><mrow id="S4.E7.m1.3.3.3.5.3" 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id="S4.E7.m1.3c">w_{k}=w_{k-1}-\eta_{k}\nabla F_{k}\left(w_{k-1},\xi_{k}\right)(1\leq k\leq N)</annotation><annotation encoding="application/x-llamapun" id="S4.E7.m1.3d">italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∇ italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT , italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ( 1 ≤ italic_k ≤ italic_N )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS1.p2.9">Here, <math alttext="w_{k}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p2.1.m1.1"><semantics id="S4.SS2.SSS1.p2.1.m1.1a"><msub id="S4.SS2.SSS1.p2.1.m1.1.1" xref="S4.SS2.SSS1.p2.1.m1.1.1.cmml"><mi id="S4.SS2.SSS1.p2.1.m1.1.1.2" xref="S4.SS2.SSS1.p2.1.m1.1.1.2.cmml">w</mi><mi id="S4.SS2.SSS1.p2.1.m1.1.1.3" xref="S4.SS2.SSS1.p2.1.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p2.1.m1.1b"><apply id="S4.SS2.SSS1.p2.1.m1.1.1.cmml" xref="S4.SS2.SSS1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p2.1.m1.1.1.1.cmml" xref="S4.SS2.SSS1.p2.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p2.1.m1.1.1.2.cmml" xref="S4.SS2.SSS1.p2.1.m1.1.1.2">𝑤</ci><ci id="S4.SS2.SSS1.p2.1.m1.1.1.3.cmml" xref="S4.SS2.SSS1.p2.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p2.1.m1.1c">w_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p2.1.m1.1d">italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> denotes the updated parameter for device <math alttext="k" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p2.2.m2.1"><semantics id="S4.SS2.SSS1.p2.2.m2.1a"><mi id="S4.SS2.SSS1.p2.2.m2.1.1" xref="S4.SS2.SSS1.p2.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p2.2.m2.1b"><ci id="S4.SS2.SSS1.p2.2.m2.1.1.cmml" xref="S4.SS2.SSS1.p2.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p2.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p2.2.m2.1d">italic_k</annotation></semantics></math>, <math alttext="\eta_{k}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p2.3.m3.1"><semantics id="S4.SS2.SSS1.p2.3.m3.1a"><msub id="S4.SS2.SSS1.p2.3.m3.1.1" xref="S4.SS2.SSS1.p2.3.m3.1.1.cmml"><mi id="S4.SS2.SSS1.p2.3.m3.1.1.2" xref="S4.SS2.SSS1.p2.3.m3.1.1.2.cmml">η</mi><mi id="S4.SS2.SSS1.p2.3.m3.1.1.3" xref="S4.SS2.SSS1.p2.3.m3.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p2.3.m3.1b"><apply id="S4.SS2.SSS1.p2.3.m3.1.1.cmml" xref="S4.SS2.SSS1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p2.3.m3.1.1.1.cmml" xref="S4.SS2.SSS1.p2.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p2.3.m3.1.1.2.cmml" xref="S4.SS2.SSS1.p2.3.m3.1.1.2">𝜂</ci><ci id="S4.SS2.SSS1.p2.3.m3.1.1.3.cmml" xref="S4.SS2.SSS1.p2.3.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p2.3.m3.1c">\eta_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p2.3.m3.1d">italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is the learning rate of device <math alttext="k" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p2.4.m4.1"><semantics id="S4.SS2.SSS1.p2.4.m4.1a"><mi id="S4.SS2.SSS1.p2.4.m4.1.1" xref="S4.SS2.SSS1.p2.4.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p2.4.m4.1b"><ci id="S4.SS2.SSS1.p2.4.m4.1.1.cmml" xref="S4.SS2.SSS1.p2.4.m4.1.1">𝑘</ci></annotation-xml><annotation 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xref="S4.SS2.SSS1.p2.5.m5.2.2.2.2.2.3.cmml">k</mi></msub><mo id="S4.SS2.SSS1.p2.5.m5.2.2.2.2.5" xref="S4.SS2.SSS1.p2.5.m5.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p2.5.m5.2b"><apply id="S4.SS2.SSS1.p2.5.m5.2.2.cmml" xref="S4.SS2.SSS1.p2.5.m5.2.2"><times id="S4.SS2.SSS1.p2.5.m5.2.2.3.cmml" xref="S4.SS2.SSS1.p2.5.m5.2.2.3"></times><apply id="S4.SS2.SSS1.p2.5.m5.2.2.4.cmml" xref="S4.SS2.SSS1.p2.5.m5.2.2.4"><ci id="S4.SS2.SSS1.p2.5.m5.2.2.4.1.cmml" xref="S4.SS2.SSS1.p2.5.m5.2.2.4.1">∇</ci><apply id="S4.SS2.SSS1.p2.5.m5.2.2.4.2.cmml" xref="S4.SS2.SSS1.p2.5.m5.2.2.4.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p2.5.m5.2.2.4.2.1.cmml" xref="S4.SS2.SSS1.p2.5.m5.2.2.4.2">subscript</csymbol><ci id="S4.SS2.SSS1.p2.5.m5.2.2.4.2.2.cmml" xref="S4.SS2.SSS1.p2.5.m5.2.2.4.2.2">𝐹</ci><ci id="S4.SS2.SSS1.p2.5.m5.2.2.4.2.3.cmml" xref="S4.SS2.SSS1.p2.5.m5.2.2.4.2.3">𝑘</ci></apply></apply><interval closure="open" id="S4.SS2.SSS1.p2.5.m5.2.2.2.3.cmml" xref="S4.SS2.SSS1.p2.5.m5.2.2.2.2"><apply id="S4.SS2.SSS1.p2.5.m5.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p2.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p2.5.m5.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p2.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p2.5.m5.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.p2.5.m5.1.1.1.1.1.2">𝑤</ci><apply id="S4.SS2.SSS1.p2.5.m5.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.p2.5.m5.1.1.1.1.1.3"><minus id="S4.SS2.SSS1.p2.5.m5.1.1.1.1.1.3.1.cmml" xref="S4.SS2.SSS1.p2.5.m5.1.1.1.1.1.3.1"></minus><ci id="S4.SS2.SSS1.p2.5.m5.1.1.1.1.1.3.2.cmml" xref="S4.SS2.SSS1.p2.5.m5.1.1.1.1.1.3.2">𝑘</ci><cn id="S4.SS2.SSS1.p2.5.m5.1.1.1.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS1.p2.5.m5.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="S4.SS2.SSS1.p2.5.m5.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.p2.5.m5.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p2.5.m5.2.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.p2.5.m5.2.2.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS1.p2.5.m5.2.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.p2.5.m5.2.2.2.2.2.2">𝜉</ci><ci id="S4.SS2.SSS1.p2.5.m5.2.2.2.2.2.3.cmml" xref="S4.SS2.SSS1.p2.5.m5.2.2.2.2.2.3">𝑘</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p2.5.m5.2c">\nabla F_{k}\left(w_{k-1},\xi_{k}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p2.5.m5.2d">∇ italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT , italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> represents the gradient of the local objective function <math alttext="F_{k}\left(w_{k-1},\xi_{k}\right)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p2.6.m6.2"><semantics id="S4.SS2.SSS1.p2.6.m6.2a"><mrow id="S4.SS2.SSS1.p2.6.m6.2.2" xref="S4.SS2.SSS1.p2.6.m6.2.2.cmml"><msub id="S4.SS2.SSS1.p2.6.m6.2.2.4" xref="S4.SS2.SSS1.p2.6.m6.2.2.4.cmml"><mi id="S4.SS2.SSS1.p2.6.m6.2.2.4.2" xref="S4.SS2.SSS1.p2.6.m6.2.2.4.2.cmml">F</mi><mi id="S4.SS2.SSS1.p2.6.m6.2.2.4.3" xref="S4.SS2.SSS1.p2.6.m6.2.2.4.3.cmml">k</mi></msub><mo id="S4.SS2.SSS1.p2.6.m6.2.2.3" xref="S4.SS2.SSS1.p2.6.m6.2.2.3.cmml">⁢</mo><mrow id="S4.SS2.SSS1.p2.6.m6.2.2.2.2" xref="S4.SS2.SSS1.p2.6.m6.2.2.2.3.cmml"><mo id="S4.SS2.SSS1.p2.6.m6.2.2.2.2.3" xref="S4.SS2.SSS1.p2.6.m6.2.2.2.3.cmml">(</mo><msub id="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1" xref="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.2" xref="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.2.cmml">w</mi><mrow id="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.3" xref="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.3.cmml"><mi id="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.3.2" xref="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.3.2.cmml">k</mi><mo id="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.3.1" xref="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.3.1.cmml">−</mo><mn id="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.3.3" xref="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S4.SS2.SSS1.p2.6.m6.2.2.2.2.4" xref="S4.SS2.SSS1.p2.6.m6.2.2.2.3.cmml">,</mo><msub id="S4.SS2.SSS1.p2.6.m6.2.2.2.2.2" xref="S4.SS2.SSS1.p2.6.m6.2.2.2.2.2.cmml"><mi id="S4.SS2.SSS1.p2.6.m6.2.2.2.2.2.2" xref="S4.SS2.SSS1.p2.6.m6.2.2.2.2.2.2.cmml">ξ</mi><mi id="S4.SS2.SSS1.p2.6.m6.2.2.2.2.2.3" xref="S4.SS2.SSS1.p2.6.m6.2.2.2.2.2.3.cmml">k</mi></msub><mo id="S4.SS2.SSS1.p2.6.m6.2.2.2.2.5" xref="S4.SS2.SSS1.p2.6.m6.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p2.6.m6.2b"><apply id="S4.SS2.SSS1.p2.6.m6.2.2.cmml" xref="S4.SS2.SSS1.p2.6.m6.2.2"><times id="S4.SS2.SSS1.p2.6.m6.2.2.3.cmml" xref="S4.SS2.SSS1.p2.6.m6.2.2.3"></times><apply id="S4.SS2.SSS1.p2.6.m6.2.2.4.cmml" xref="S4.SS2.SSS1.p2.6.m6.2.2.4"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p2.6.m6.2.2.4.1.cmml" xref="S4.SS2.SSS1.p2.6.m6.2.2.4">subscript</csymbol><ci id="S4.SS2.SSS1.p2.6.m6.2.2.4.2.cmml" xref="S4.SS2.SSS1.p2.6.m6.2.2.4.2">𝐹</ci><ci id="S4.SS2.SSS1.p2.6.m6.2.2.4.3.cmml" xref="S4.SS2.SSS1.p2.6.m6.2.2.4.3">𝑘</ci></apply><interval closure="open" id="S4.SS2.SSS1.p2.6.m6.2.2.2.3.cmml" xref="S4.SS2.SSS1.p2.6.m6.2.2.2.2"><apply id="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.2">𝑤</ci><apply id="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.3"><minus id="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.3.1.cmml" xref="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.3.1"></minus><ci id="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.3.2.cmml" xref="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.3.2">𝑘</ci><cn id="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS1.p2.6.m6.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="S4.SS2.SSS1.p2.6.m6.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.p2.6.m6.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p2.6.m6.2.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.p2.6.m6.2.2.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS1.p2.6.m6.2.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.p2.6.m6.2.2.2.2.2.2">𝜉</ci><ci id="S4.SS2.SSS1.p2.6.m6.2.2.2.2.2.3.cmml" xref="S4.SS2.SSS1.p2.6.m6.2.2.2.2.2.3">𝑘</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p2.6.m6.2c">F_{k}\left(w_{k-1},\xi_{k}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p2.6.m6.2d">italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT , italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> for device <math alttext="k" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p2.7.m7.1"><semantics id="S4.SS2.SSS1.p2.7.m7.1a"><mi id="S4.SS2.SSS1.p2.7.m7.1.1" xref="S4.SS2.SSS1.p2.7.m7.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p2.7.m7.1b"><ci id="S4.SS2.SSS1.p2.7.m7.1.1.cmml" xref="S4.SS2.SSS1.p2.7.m7.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p2.7.m7.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p2.7.m7.1d">italic_k</annotation></semantics></math> based on the previous device’s weight <math alttext="w_{k-1}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p2.8.m8.1"><semantics id="S4.SS2.SSS1.p2.8.m8.1a"><msub id="S4.SS2.SSS1.p2.8.m8.1.1" xref="S4.SS2.SSS1.p2.8.m8.1.1.cmml"><mi id="S4.SS2.SSS1.p2.8.m8.1.1.2" xref="S4.SS2.SSS1.p2.8.m8.1.1.2.cmml">w</mi><mrow id="S4.SS2.SSS1.p2.8.m8.1.1.3" xref="S4.SS2.SSS1.p2.8.m8.1.1.3.cmml"><mi id="S4.SS2.SSS1.p2.8.m8.1.1.3.2" xref="S4.SS2.SSS1.p2.8.m8.1.1.3.2.cmml">k</mi><mo id="S4.SS2.SSS1.p2.8.m8.1.1.3.1" xref="S4.SS2.SSS1.p2.8.m8.1.1.3.1.cmml">−</mo><mn id="S4.SS2.SSS1.p2.8.m8.1.1.3.3" xref="S4.SS2.SSS1.p2.8.m8.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p2.8.m8.1b"><apply id="S4.SS2.SSS1.p2.8.m8.1.1.cmml" xref="S4.SS2.SSS1.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p2.8.m8.1.1.1.cmml" xref="S4.SS2.SSS1.p2.8.m8.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p2.8.m8.1.1.2.cmml" xref="S4.SS2.SSS1.p2.8.m8.1.1.2">𝑤</ci><apply id="S4.SS2.SSS1.p2.8.m8.1.1.3.cmml" xref="S4.SS2.SSS1.p2.8.m8.1.1.3"><minus id="S4.SS2.SSS1.p2.8.m8.1.1.3.1.cmml" xref="S4.SS2.SSS1.p2.8.m8.1.1.3.1"></minus><ci id="S4.SS2.SSS1.p2.8.m8.1.1.3.2.cmml" xref="S4.SS2.SSS1.p2.8.m8.1.1.3.2">𝑘</ci><cn id="S4.SS2.SSS1.p2.8.m8.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS1.p2.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p2.8.m8.1c">w_{k-1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p2.8.m8.1d">italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT</annotation></semantics></math> and its local data set <math alttext="x_{k}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p2.9.m9.1"><semantics id="S4.SS2.SSS1.p2.9.m9.1a"><msub id="S4.SS2.SSS1.p2.9.m9.1.1" xref="S4.SS2.SSS1.p2.9.m9.1.1.cmml"><mi id="S4.SS2.SSS1.p2.9.m9.1.1.2" xref="S4.SS2.SSS1.p2.9.m9.1.1.2.cmml">x</mi><mi id="S4.SS2.SSS1.p2.9.m9.1.1.3" xref="S4.SS2.SSS1.p2.9.m9.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p2.9.m9.1b"><apply id="S4.SS2.SSS1.p2.9.m9.1.1.cmml" xref="S4.SS2.SSS1.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p2.9.m9.1.1.1.cmml" xref="S4.SS2.SSS1.p2.9.m9.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p2.9.m9.1.1.2.cmml" xref="S4.SS2.SSS1.p2.9.m9.1.1.2">𝑥</ci><ci id="S4.SS2.SSS1.p2.9.m9.1.1.3.cmml" xref="S4.SS2.SSS1.p2.9.m9.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p2.9.m9.1c">x_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p2.9.m9.1d">italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.SSS1.p3"> <p class="ltx_p" id="S4.SS2.SSS1.p3.5">We set <math alttext="w_{k}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.1.m1.1"><semantics id="S4.SS2.SSS1.p3.1.m1.1a"><msub id="S4.SS2.SSS1.p3.1.m1.1.1" xref="S4.SS2.SSS1.p3.1.m1.1.1.cmml"><mi id="S4.SS2.SSS1.p3.1.m1.1.1.2" xref="S4.SS2.SSS1.p3.1.m1.1.1.2.cmml">w</mi><mi id="S4.SS2.SSS1.p3.1.m1.1.1.3" xref="S4.SS2.SSS1.p3.1.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.1.m1.1b"><apply id="S4.SS2.SSS1.p3.1.m1.1.1.cmml" xref="S4.SS2.SSS1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.1.m1.1.1.1.cmml" xref="S4.SS2.SSS1.p3.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p3.1.m1.1.1.2.cmml" xref="S4.SS2.SSS1.p3.1.m1.1.1.2">𝑤</ci><ci id="S4.SS2.SSS1.p3.1.m1.1.1.3.cmml" xref="S4.SS2.SSS1.p3.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.1.m1.1c">w_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.1.m1.1d">italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> as the parameter trained on device <math alttext="k" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.2.m2.1"><semantics id="S4.SS2.SSS1.p3.2.m2.1a"><mi id="S4.SS2.SSS1.p3.2.m2.1.1" xref="S4.SS2.SSS1.p3.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.2.m2.1b"><ci id="S4.SS2.SSS1.p3.2.m2.1.1.cmml" xref="S4.SS2.SSS1.p3.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.2.m2.1d">italic_k</annotation></semantics></math>. Since the local model on each device is the same, it can be seen as a big model, hence we suppose <math alttext="A_{k}=\nabla F_{k}\left(w_{k-1},\xi_{k}\right),\bar{A_{k}}=\nabla F_{k}\left(w% _{k-1}\right)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.3.m3.2"><semantics id="S4.SS2.SSS1.p3.3.m3.2a"><mrow id="S4.SS2.SSS1.p3.3.m3.2.2.2" xref="S4.SS2.SSS1.p3.3.m3.2.2.3.cmml"><mrow id="S4.SS2.SSS1.p3.3.m3.1.1.1.1" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.cmml"><msub id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.4" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.4.cmml"><mi id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.4.2" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.4.2.cmml">A</mi><mi id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.4.3" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.4.3.cmml">k</mi></msub><mo id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.3" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.3.cmml">=</mo><mrow id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.cmml"><mrow id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.4" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.4.cmml"><mo id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.4.1" rspace="0.167em" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.4.1.cmml">∇</mo><msub id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.4.2" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.4.2.cmml"><mi id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.4.2.2" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.4.2.2.cmml">F</mi><mi id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.4.2.3" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.4.2.3.cmml">k</mi></msub></mrow><mo id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.3" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.3.cmml">⁢</mo><mrow id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.2.2" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.2.3.cmml"><mo id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.2.2.3" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.2.3.cmml">(</mo><msub id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.2" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.2.cmml">w</mi><mrow id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.3" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.3.cmml"><mi id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.3.2" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.3.2.cmml">k</mi><mo id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.3.1" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.3.1.cmml">−</mo><mn id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.3.3" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.2.2.4" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.2.3.cmml">,</mo><msub id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.2.2.2" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.2.2.2.cmml"><mi id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.2.2.2.2" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.2.2.2.2.cmml">ξ</mi><mi id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.2.2.2.3" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.2.2.2.3.cmml">k</mi></msub><mo id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.2.2.5" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.SS2.SSS1.p3.3.m3.2.2.2.3" xref="S4.SS2.SSS1.p3.3.m3.2.2.3a.cmml">,</mo><mrow id="S4.SS2.SSS1.p3.3.m3.2.2.2.2" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.cmml"><mover accent="true" id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.3" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.3.cmml"><msub id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.3.2" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.3.2.cmml"><mi id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.3.2.2" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.3.2.2.cmml">A</mi><mi id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.3.2.3" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.3.2.3.cmml">k</mi></msub><mo id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.3.1" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.3.1.cmml">¯</mo></mover><mo id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.2" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.2.cmml">=</mo><mrow id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.cmml"><mrow id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.3" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.3.cmml"><mo id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.3.1" rspace="0.167em" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.3.1.cmml">∇</mo><msub id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.3.2" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.3.2.cmml"><mi id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.3.2.2" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.3.2.2.cmml">F</mi><mi id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.3.2.3" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.3.2.3.cmml">k</mi></msub></mrow><mo id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.2" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.cmml"><mo id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.2" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.cmml">(</mo><msub id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.2" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.2.cmml">w</mi><mrow id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.3" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.3.cmml"><mi id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.3.2" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.3.2.cmml">k</mi><mo id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.3.1" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.3.1.cmml">−</mo><mn id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.3.3" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.3" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.3.m3.2b"><apply id="S4.SS2.SSS1.p3.3.m3.2.2.3.cmml" xref="S4.SS2.SSS1.p3.3.m3.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.3.m3.2.2.3a.cmml" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.3">formulae-sequence</csymbol><apply id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.cmml" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1"><eq id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.3"></eq><apply id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.4.cmml" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.4"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.4.1.cmml" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.4">subscript</csymbol><ci id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.4.2.cmml" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.4.2">𝐴</ci><ci id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.4.3.cmml" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.4.3">𝑘</ci></apply><apply id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2"><times id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.3.cmml" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.3"></times><apply id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.4.cmml" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.4"><ci id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.4.1.cmml" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.4.1">∇</ci><apply id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.4.2.cmml" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.4.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.4.2.1.cmml" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.4.2">subscript</csymbol><ci id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.4.2.2.cmml" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.4.2.2">𝐹</ci><ci id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.4.2.3.cmml" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.4.2.3">𝑘</ci></apply></apply><interval closure="open" id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.2.3.cmml" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.2.2"><apply id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.2">𝑤</ci><apply id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.3"><minus id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.3.1.cmml" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.3.1"></minus><ci id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.3.2.cmml" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.3.2">𝑘</ci><cn id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.2.2.2.cmml" xref="S4.SS2.SSS1.p3.3.m3.1.1.1.1.2.2.2.2"><csymbol cd="ambiguous" 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xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.3.2.2">𝐴</ci><ci id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.3.2.3.cmml" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.3.2.3">𝑘</ci></apply></apply><apply id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1"><times id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.2.cmml" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.2"></times><apply id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.3.cmml" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.3"><ci id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.3.1.cmml" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.3.1">∇</ci><apply id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.3.2.cmml" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.3.2.1.cmml" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.3.2">subscript</csymbol><ci id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.3.2.2.cmml" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.3.2.2">𝐹</ci><ci id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.3.2.3.cmml" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.3.2.3">𝑘</ci></apply></apply><apply id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.2">𝑤</ci><apply id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.3"><minus id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.3.1.cmml" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.3.1"></minus><ci id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.3.2.cmml" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.3.2">𝑘</ci><cn id="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS1.p3.3.m3.2.2.2.2.1.1.1.1.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.3.m3.2c">A_{k}=\nabla F_{k}\left(w_{k-1},\xi_{k}\right),\bar{A_{k}}=\nabla F_{k}\left(w% _{k-1}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.3.m3.2d">italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = ∇ italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT , italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) , over¯ start_ARG italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG = ∇ italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT )</annotation></semantics></math>; therefore, <math alttext="E(A_{k})=\bar{A_{k}}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.4.m4.1"><semantics id="S4.SS2.SSS1.p3.4.m4.1a"><mrow id="S4.SS2.SSS1.p3.4.m4.1.1" xref="S4.SS2.SSS1.p3.4.m4.1.1.cmml"><mrow id="S4.SS2.SSS1.p3.4.m4.1.1.1" xref="S4.SS2.SSS1.p3.4.m4.1.1.1.cmml"><mi id="S4.SS2.SSS1.p3.4.m4.1.1.1.3" xref="S4.SS2.SSS1.p3.4.m4.1.1.1.3.cmml">E</mi><mo id="S4.SS2.SSS1.p3.4.m4.1.1.1.2" xref="S4.SS2.SSS1.p3.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS1.p3.4.m4.1.1.1.1.1" xref="S4.SS2.SSS1.p3.4.m4.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS1.p3.4.m4.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS1.p3.4.m4.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.SSS1.p3.4.m4.1.1.1.1.1.1" xref="S4.SS2.SSS1.p3.4.m4.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.p3.4.m4.1.1.1.1.1.1.2" xref="S4.SS2.SSS1.p3.4.m4.1.1.1.1.1.1.2.cmml">A</mi><mi id="S4.SS2.SSS1.p3.4.m4.1.1.1.1.1.1.3" xref="S4.SS2.SSS1.p3.4.m4.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S4.SS2.SSS1.p3.4.m4.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS1.p3.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS1.p3.4.m4.1.1.2" xref="S4.SS2.SSS1.p3.4.m4.1.1.2.cmml">=</mo><mover accent="true" id="S4.SS2.SSS1.p3.4.m4.1.1.3" xref="S4.SS2.SSS1.p3.4.m4.1.1.3.cmml"><msub id="S4.SS2.SSS1.p3.4.m4.1.1.3.2" xref="S4.SS2.SSS1.p3.4.m4.1.1.3.2.cmml"><mi id="S4.SS2.SSS1.p3.4.m4.1.1.3.2.2" xref="S4.SS2.SSS1.p3.4.m4.1.1.3.2.2.cmml">A</mi><mi id="S4.SS2.SSS1.p3.4.m4.1.1.3.2.3" xref="S4.SS2.SSS1.p3.4.m4.1.1.3.2.3.cmml">k</mi></msub><mo id="S4.SS2.SSS1.p3.4.m4.1.1.3.1" xref="S4.SS2.SSS1.p3.4.m4.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.4.m4.1b"><apply id="S4.SS2.SSS1.p3.4.m4.1.1.cmml" xref="S4.SS2.SSS1.p3.4.m4.1.1"><eq id="S4.SS2.SSS1.p3.4.m4.1.1.2.cmml" xref="S4.SS2.SSS1.p3.4.m4.1.1.2"></eq><apply id="S4.SS2.SSS1.p3.4.m4.1.1.1.cmml" xref="S4.SS2.SSS1.p3.4.m4.1.1.1"><times id="S4.SS2.SSS1.p3.4.m4.1.1.1.2.cmml" xref="S4.SS2.SSS1.p3.4.m4.1.1.1.2"></times><ci id="S4.SS2.SSS1.p3.4.m4.1.1.1.3.cmml" xref="S4.SS2.SSS1.p3.4.m4.1.1.1.3">𝐸</ci><apply id="S4.SS2.SSS1.p3.4.m4.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p3.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.4.m4.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p3.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p3.4.m4.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.p3.4.m4.1.1.1.1.1.1.2">𝐴</ci><ci id="S4.SS2.SSS1.p3.4.m4.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.p3.4.m4.1.1.1.1.1.1.3">𝑘</ci></apply></apply><apply id="S4.SS2.SSS1.p3.4.m4.1.1.3.cmml" xref="S4.SS2.SSS1.p3.4.m4.1.1.3"><ci id="S4.SS2.SSS1.p3.4.m4.1.1.3.1.cmml" xref="S4.SS2.SSS1.p3.4.m4.1.1.3.1">¯</ci><apply id="S4.SS2.SSS1.p3.4.m4.1.1.3.2.cmml" xref="S4.SS2.SSS1.p3.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.4.m4.1.1.3.2.1.cmml" xref="S4.SS2.SSS1.p3.4.m4.1.1.3.2">subscript</csymbol><ci id="S4.SS2.SSS1.p3.4.m4.1.1.3.2.2.cmml" xref="S4.SS2.SSS1.p3.4.m4.1.1.3.2.2">𝐴</ci><ci id="S4.SS2.SSS1.p3.4.m4.1.1.3.2.3.cmml" xref="S4.SS2.SSS1.p3.4.m4.1.1.3.2.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.4.m4.1c">E(A_{k})=\bar{A_{k}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.4.m4.1d">italic_E ( italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = over¯ start_ARG italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG</annotation></semantics></math>, <math alttext="w_{k}=w_{k-1}-\eta_{k}A_{k}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.5.m5.1"><semantics id="S4.SS2.SSS1.p3.5.m5.1a"><mrow id="S4.SS2.SSS1.p3.5.m5.1.1" xref="S4.SS2.SSS1.p3.5.m5.1.1.cmml"><msub id="S4.SS2.SSS1.p3.5.m5.1.1.2" xref="S4.SS2.SSS1.p3.5.m5.1.1.2.cmml"><mi id="S4.SS2.SSS1.p3.5.m5.1.1.2.2" xref="S4.SS2.SSS1.p3.5.m5.1.1.2.2.cmml">w</mi><mi id="S4.SS2.SSS1.p3.5.m5.1.1.2.3" xref="S4.SS2.SSS1.p3.5.m5.1.1.2.3.cmml">k</mi></msub><mo id="S4.SS2.SSS1.p3.5.m5.1.1.1" xref="S4.SS2.SSS1.p3.5.m5.1.1.1.cmml">=</mo><mrow id="S4.SS2.SSS1.p3.5.m5.1.1.3" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.cmml"><msub id="S4.SS2.SSS1.p3.5.m5.1.1.3.2" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.2.cmml"><mi id="S4.SS2.SSS1.p3.5.m5.1.1.3.2.2" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.2.2.cmml">w</mi><mrow id="S4.SS2.SSS1.p3.5.m5.1.1.3.2.3" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.2.3.cmml"><mi id="S4.SS2.SSS1.p3.5.m5.1.1.3.2.3.2" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.2.3.2.cmml">k</mi><mo id="S4.SS2.SSS1.p3.5.m5.1.1.3.2.3.1" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.2.3.1.cmml">−</mo><mn id="S4.SS2.SSS1.p3.5.m5.1.1.3.2.3.3" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.2.3.3.cmml">1</mn></mrow></msub><mo id="S4.SS2.SSS1.p3.5.m5.1.1.3.1" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.1.cmml">−</mo><mrow id="S4.SS2.SSS1.p3.5.m5.1.1.3.3" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.3.cmml"><msub id="S4.SS2.SSS1.p3.5.m5.1.1.3.3.2" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.3.2.cmml"><mi id="S4.SS2.SSS1.p3.5.m5.1.1.3.3.2.2" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.3.2.2.cmml">η</mi><mi id="S4.SS2.SSS1.p3.5.m5.1.1.3.3.2.3" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.3.2.3.cmml">k</mi></msub><mo id="S4.SS2.SSS1.p3.5.m5.1.1.3.3.1" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.3.1.cmml">⁢</mo><msub id="S4.SS2.SSS1.p3.5.m5.1.1.3.3.3" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.3.3.cmml"><mi id="S4.SS2.SSS1.p3.5.m5.1.1.3.3.3.2" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.3.3.2.cmml">A</mi><mi id="S4.SS2.SSS1.p3.5.m5.1.1.3.3.3.3" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.3.3.3.cmml">k</mi></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.5.m5.1b"><apply id="S4.SS2.SSS1.p3.5.m5.1.1.cmml" xref="S4.SS2.SSS1.p3.5.m5.1.1"><eq id="S4.SS2.SSS1.p3.5.m5.1.1.1.cmml" xref="S4.SS2.SSS1.p3.5.m5.1.1.1"></eq><apply id="S4.SS2.SSS1.p3.5.m5.1.1.2.cmml" xref="S4.SS2.SSS1.p3.5.m5.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.5.m5.1.1.2.1.cmml" xref="S4.SS2.SSS1.p3.5.m5.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS1.p3.5.m5.1.1.2.2.cmml" xref="S4.SS2.SSS1.p3.5.m5.1.1.2.2">𝑤</ci><ci id="S4.SS2.SSS1.p3.5.m5.1.1.2.3.cmml" xref="S4.SS2.SSS1.p3.5.m5.1.1.2.3">𝑘</ci></apply><apply id="S4.SS2.SSS1.p3.5.m5.1.1.3.cmml" xref="S4.SS2.SSS1.p3.5.m5.1.1.3"><minus id="S4.SS2.SSS1.p3.5.m5.1.1.3.1.cmml" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.1"></minus><apply id="S4.SS2.SSS1.p3.5.m5.1.1.3.2.cmml" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.5.m5.1.1.3.2.1.cmml" 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xref="S4.SS2.SSS1.p3.5.m5.1.1.3.3.2.2">𝜂</ci><ci id="S4.SS2.SSS1.p3.5.m5.1.1.3.3.2.3.cmml" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.3.2.3">𝑘</ci></apply><apply id="S4.SS2.SSS1.p3.5.m5.1.1.3.3.3.cmml" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.3.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.5.m5.1.1.3.3.3.1.cmml" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.3.3">subscript</csymbol><ci id="S4.SS2.SSS1.p3.5.m5.1.1.3.3.3.2.cmml" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.3.3.2">𝐴</ci><ci id="S4.SS2.SSS1.p3.5.m5.1.1.3.3.3.3.cmml" xref="S4.SS2.SSS1.p3.5.m5.1.1.3.3.3.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.5.m5.1c">w_{k}=w_{k-1}-\eta_{k}A_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.5.m5.1d">italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. Thus, Equation( <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4.E6" title="In IV-B Convergence Analysis ‣ IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">6</span></a>) is equivalent to:</p> </div> <div class="ltx_para" id="S4.SS2.SSS1.p4"> <table class="ltx_equation ltx_eqn_table" id="S4.E8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{split}\begin{gathered}||w_{k}-w^{\ast}||^{2}=||w_{k-1}-\eta_{k}A_{k}-w^% {\ast}-\eta_{k}\bar{A}_{k}+\eta_{k}\bar{A}_{k}||^{2}\\ =||w_{k-1}-\eta_{k}\bar{A}_{k}-w^{\ast}||^{2}+2\eta_{k}<w_{k-1}-w^{\ast}-\eta_% {k}\bar{A}_{k},\\ \bar{A}_{k}-A_{k}>+\eta^{2}_{k}||A_{k}-\bar{A}_{k}||^{2}\end{gathered}\end{split}" class="ltx_Math" display="block" id="S4.E8.m1.89"><semantics id="S4.E8.m1.89a"><mtable displaystyle="true" 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xref="S4.E8.m1.33.33.33.33.33.33.33.33.33.33.1.cmml">k</mi></msub></mrow></mrow><mo id="S4.E8.m1.34.34.34.34.34.34.34.34.34.34b" stretchy="false" xref="S4.E8.m1.89.89.89.89.89.89.2.3.cmml">‖</mo></mrow><mn id="S4.E8.m1.36.36.36.36.36.36.36.36.36.36.1" xref="S4.E8.m1.36.36.36.36.36.36.36.36.36.36.1.cmml">2</mn></msup></mrow></mtd></mtr><mtr id="S4.E8.m1.89.89.89.89.89.89.6c"><mtd id="S4.E8.m1.89.89.89.89.89.89.6d"><mrow id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33"><mrow id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33.1"><mi id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33.1.3" xref="S4.E8.m1.89.89.89.89.89.89.2.3.cmml"></mi><mo id="S4.E8.m1.37.37.37.37.37.37.37.1.1.1" xref="S4.E8.m1.37.37.37.37.37.37.37.1.1.1.cmml">=</mo><mrow id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33.1.1"><msup id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33.1.1.1"><mrow id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33.1.1.1.1.1"><mo id="S4.E8.m1.38.38.38.38.38.38.38.2.2.2b" stretchy="false" xref="S4.E8.m1.89.89.89.89.89.89.2.3.cmml">‖</mo><mrow id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33.1.1.1.1.1.1"><msub id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33.1.1.1.1.1.1.1"><mi id="S4.E8.m1.40.40.40.40.40.40.40.4.4.4" xref="S4.E8.m1.40.40.40.40.40.40.40.4.4.4.cmml">w</mi><mrow id="S4.E8.m1.41.41.41.41.41.41.41.5.5.5.1" xref="S4.E8.m1.41.41.41.41.41.41.41.5.5.5.1.cmml"><mi id="S4.E8.m1.41.41.41.41.41.41.41.5.5.5.1.2" xref="S4.E8.m1.41.41.41.41.41.41.41.5.5.5.1.2.cmml">k</mi><mo id="S4.E8.m1.41.41.41.41.41.41.41.5.5.5.1.1" xref="S4.E8.m1.41.41.41.41.41.41.41.5.5.5.1.1.cmml">−</mo><mn id="S4.E8.m1.41.41.41.41.41.41.41.5.5.5.1.3" xref="S4.E8.m1.41.41.41.41.41.41.41.5.5.5.1.3.cmml">1</mn></mrow></msub><mo id="S4.E8.m1.42.42.42.42.42.42.42.6.6.6" xref="S4.E8.m1.42.42.42.42.42.42.42.6.6.6.cmml">−</mo><mrow id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33.1.1.1.1.1.1.2"><msub id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33.1.1.1.1.1.1.2.2"><mi id="S4.E8.m1.43.43.43.43.43.43.43.7.7.7" xref="S4.E8.m1.43.43.43.43.43.43.43.7.7.7.cmml">η</mi><mi id="S4.E8.m1.44.44.44.44.44.44.44.8.8.8.1" xref="S4.E8.m1.44.44.44.44.44.44.44.8.8.8.1.cmml">k</mi></msub><mo id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33.1.1.1.1.1.1.2.1" xref="S4.E8.m1.89.89.89.89.89.89.2.3.cmml">⁢</mo><msub id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33.1.1.1.1.1.1.2.3"><mover accent="true" id="S4.E8.m1.45.45.45.45.45.45.45.9.9.9" xref="S4.E8.m1.45.45.45.45.45.45.45.9.9.9.cmml"><mi id="S4.E8.m1.45.45.45.45.45.45.45.9.9.9.2" xref="S4.E8.m1.45.45.45.45.45.45.45.9.9.9.2.cmml">A</mi><mo id="S4.E8.m1.45.45.45.45.45.45.45.9.9.9.1" xref="S4.E8.m1.45.45.45.45.45.45.45.9.9.9.1.cmml">¯</mo></mover><mi id="S4.E8.m1.46.46.46.46.46.46.46.10.10.10.1" xref="S4.E8.m1.46.46.46.46.46.46.46.10.10.10.1.cmml">k</mi></msub></mrow><mo id="S4.E8.m1.42.42.42.42.42.42.42.6.6.6a" xref="S4.E8.m1.42.42.42.42.42.42.42.6.6.6.cmml">−</mo><msup id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33.1.1.1.1.1.1.3"><mi id="S4.E8.m1.48.48.48.48.48.48.48.12.12.12" xref="S4.E8.m1.48.48.48.48.48.48.48.12.12.12.cmml">w</mi><mo id="S4.E8.m1.49.49.49.49.49.49.49.13.13.13.1" xref="S4.E8.m1.49.49.49.49.49.49.49.13.13.13.1.cmml">∗</mo></msup></mrow><mo id="S4.E8.m1.50.50.50.50.50.50.50.14.14.14b" stretchy="false" xref="S4.E8.m1.89.89.89.89.89.89.2.3.cmml">‖</mo></mrow><mn id="S4.E8.m1.52.52.52.52.52.52.52.16.16.16.1" xref="S4.E8.m1.52.52.52.52.52.52.52.16.16.16.1.cmml">2</mn></msup><mo id="S4.E8.m1.53.53.53.53.53.53.53.17.17.17" xref="S4.E8.m1.53.53.53.53.53.53.53.17.17.17.cmml">+</mo><mrow id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33.1.1.2"><mn id="S4.E8.m1.54.54.54.54.54.54.54.18.18.18" xref="S4.E8.m1.54.54.54.54.54.54.54.18.18.18.cmml">2</mn><mo id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33.1.1.2.1" xref="S4.E8.m1.89.89.89.89.89.89.2.3.cmml">⁢</mo><msub id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33.1.1.2.2"><mi id="S4.E8.m1.55.55.55.55.55.55.55.19.19.19" xref="S4.E8.m1.55.55.55.55.55.55.55.19.19.19.cmml">η</mi><mi id="S4.E8.m1.56.56.56.56.56.56.56.20.20.20.1" xref="S4.E8.m1.56.56.56.56.56.56.56.20.20.20.1.cmml">k</mi></msub></mrow></mrow><mo id="S4.E8.m1.57.57.57.57.57.57.57.21.21.21" xref="S4.E8.m1.57.57.57.57.57.57.57.21.21.21.cmml"><</mo><mrow id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33.1.4"><msub id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33.1.4.1"><mi id="S4.E8.m1.58.58.58.58.58.58.58.22.22.22" xref="S4.E8.m1.58.58.58.58.58.58.58.22.22.22.cmml">w</mi><mrow id="S4.E8.m1.59.59.59.59.59.59.59.23.23.23.1" xref="S4.E8.m1.59.59.59.59.59.59.59.23.23.23.1.cmml"><mi id="S4.E8.m1.59.59.59.59.59.59.59.23.23.23.1.2" xref="S4.E8.m1.59.59.59.59.59.59.59.23.23.23.1.2.cmml">k</mi><mo id="S4.E8.m1.59.59.59.59.59.59.59.23.23.23.1.1" xref="S4.E8.m1.59.59.59.59.59.59.59.23.23.23.1.1.cmml">−</mo><mn id="S4.E8.m1.59.59.59.59.59.59.59.23.23.23.1.3" xref="S4.E8.m1.59.59.59.59.59.59.59.23.23.23.1.3.cmml">1</mn></mrow></msub><mo id="S4.E8.m1.60.60.60.60.60.60.60.24.24.24" xref="S4.E8.m1.60.60.60.60.60.60.60.24.24.24.cmml">−</mo><msup id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33.1.4.2"><mi id="S4.E8.m1.61.61.61.61.61.61.61.25.25.25" xref="S4.E8.m1.61.61.61.61.61.61.61.25.25.25.cmml">w</mi><mo id="S4.E8.m1.62.62.62.62.62.62.62.26.26.26.1" xref="S4.E8.m1.62.62.62.62.62.62.62.26.26.26.1.cmml">∗</mo></msup><mo id="S4.E8.m1.60.60.60.60.60.60.60.24.24.24a" xref="S4.E8.m1.60.60.60.60.60.60.60.24.24.24.cmml">−</mo><mrow id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33.1.4.3"><msub id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33.1.4.3.2"><mi id="S4.E8.m1.64.64.64.64.64.64.64.28.28.28" xref="S4.E8.m1.64.64.64.64.64.64.64.28.28.28.cmml">η</mi><mi id="S4.E8.m1.65.65.65.65.65.65.65.29.29.29.1" xref="S4.E8.m1.65.65.65.65.65.65.65.29.29.29.1.cmml">k</mi></msub><mo id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33.1.4.3.1" xref="S4.E8.m1.89.89.89.89.89.89.2.3.cmml">⁢</mo><msub id="S4.E8.m1.89.89.89.89.89.89.5.91.33.33.33.1.4.3.3"><mover accent="true" id="S4.E8.m1.66.66.66.66.66.66.66.30.30.30" xref="S4.E8.m1.66.66.66.66.66.66.66.30.30.30.cmml"><mi id="S4.E8.m1.66.66.66.66.66.66.66.30.30.30.2" xref="S4.E8.m1.66.66.66.66.66.66.66.30.30.30.2.cmml">A</mi><mo id="S4.E8.m1.66.66.66.66.66.66.66.30.30.30.1" xref="S4.E8.m1.66.66.66.66.66.66.66.30.30.30.1.cmml">¯</mo></mover><mi id="S4.E8.m1.67.67.67.67.67.67.67.31.31.31.1" xref="S4.E8.m1.67.67.67.67.67.67.67.31.31.31.1.cmml">k</mi></msub></mrow></mrow></mrow><mo id="S4.E8.m1.68.68.68.68.68.68.68.32.32.32" xref="S4.E8.m1.89.89.89.89.89.89.2.3.cmml">,</mo></mrow></mtd></mtr><mtr id="S4.E8.m1.89.89.89.89.89.89.6e"><mtd id="S4.E8.m1.89.89.89.89.89.89.6f"><mrow id="S4.E8.m1.89.89.89.89.89.89.6.92.21.21"><mrow id="S4.E8.m1.89.89.89.89.89.89.6.92.21.21.22"><msub id="S4.E8.m1.89.89.89.89.89.89.6.92.21.21.22.1"><mover accent="true" id="S4.E8.m1.69.69.69.69.69.69.69.1.1.1" xref="S4.E8.m1.69.69.69.69.69.69.69.1.1.1.cmml"><mi id="S4.E8.m1.69.69.69.69.69.69.69.1.1.1.2" xref="S4.E8.m1.69.69.69.69.69.69.69.1.1.1.2.cmml">A</mi><mo id="S4.E8.m1.69.69.69.69.69.69.69.1.1.1.1" xref="S4.E8.m1.69.69.69.69.69.69.69.1.1.1.1.cmml">¯</mo></mover><mi id="S4.E8.m1.70.70.70.70.70.70.70.2.2.2.1" xref="S4.E8.m1.70.70.70.70.70.70.70.2.2.2.1.cmml">k</mi></msub><mo id="S4.E8.m1.71.71.71.71.71.71.71.3.3.3" xref="S4.E8.m1.71.71.71.71.71.71.71.3.3.3.cmml">−</mo><msub id="S4.E8.m1.89.89.89.89.89.89.6.92.21.21.22.2"><mi id="S4.E8.m1.72.72.72.72.72.72.72.4.4.4" xref="S4.E8.m1.72.72.72.72.72.72.72.4.4.4.cmml">A</mi><mi id="S4.E8.m1.73.73.73.73.73.73.73.5.5.5.1" xref="S4.E8.m1.73.73.73.73.73.73.73.5.5.5.1.cmml">k</mi></msub></mrow><mo id="S4.E8.m1.74.74.74.74.74.74.74.6.6.6" xref="S4.E8.m1.74.74.74.74.74.74.74.6.6.6.cmml">></mo><mrow id="S4.E8.m1.89.89.89.89.89.89.6.92.21.21.21"><mo id="S4.E8.m1.89.89.89.89.89.89.6.92.21.21.21a" xref="S4.E8.m1.89.89.89.89.89.89.2.3.cmml">+</mo><mrow id="S4.E8.m1.89.89.89.89.89.89.6.92.21.21.21.1"><msubsup 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start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = | | italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL = | | italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT < italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT > + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | | italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(8)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS1.p4.7">There are three parts in Equation (<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4.E8" title="In IV-B1 Continuous Linear ‣ IV-B Convergence Analysis ‣ IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">8</span></a>): <math alttext="||w_{k-1}-\eta_{k}\bar{A}_{k}-w^{\ast}||^{2}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p4.1.m1.1"><semantics id="S4.SS2.SSS1.p4.1.m1.1a"><msup id="S4.SS2.SSS1.p4.1.m1.1.1" xref="S4.SS2.SSS1.p4.1.m1.1.1.cmml"><mrow id="S4.SS2.SSS1.p4.1.m1.1.1.1.1" xref="S4.SS2.SSS1.p4.1.m1.1.1.1.2.cmml"><mo id="S4.SS2.SSS1.p4.1.m1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS1.p4.1.m1.1.1.1.2.1.cmml">‖</mo><mrow id="S4.SS2.SSS1.p4.1.m1.1.1.1.1.1" 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end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="2\eta_{k}<w_{k-1}-w^{\ast}-\eta_{k}\bar{A}_{k},\bar{A}_{k}-A_{k}>" class="ltx_math_unparsed" display="inline" id="S4.SS2.SSS1.p4.2.m2.1"><semantics id="S4.SS2.SSS1.p4.2.m2.1a"><mrow id="S4.SS2.SSS1.p4.2.m2.1b"><mn id="S4.SS2.SSS1.p4.2.m2.1.1">2</mn><msub id="S4.SS2.SSS1.p4.2.m2.1.2"><mi id="S4.SS2.SSS1.p4.2.m2.1.2.2">η</mi><mi id="S4.SS2.SSS1.p4.2.m2.1.2.3">k</mi></msub><mo id="S4.SS2.SSS1.p4.2.m2.1.3"><</mo><msub id="S4.SS2.SSS1.p4.2.m2.1.4"><mi id="S4.SS2.SSS1.p4.2.m2.1.4.2">w</mi><mrow id="S4.SS2.SSS1.p4.2.m2.1.4.3"><mi id="S4.SS2.SSS1.p4.2.m2.1.4.3.2">k</mi><mo id="S4.SS2.SSS1.p4.2.m2.1.4.3.1">−</mo><mn id="S4.SS2.SSS1.p4.2.m2.1.4.3.3">1</mn></mrow></msub><mo id="S4.SS2.SSS1.p4.2.m2.1.5">−</mo><msup id="S4.SS2.SSS1.p4.2.m2.1.6"><mi id="S4.SS2.SSS1.p4.2.m2.1.6.2">w</mi><mo 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id="S4.SS2.SSS1.p4.2.m2.1c">2\eta_{k}<w_{k-1}-w^{\ast}-\eta_{k}\bar{A}_{k},\bar{A}_{k}-A_{k}></annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p4.2.m2.1d">2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT < italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ></annotation></semantics></math>, and <math alttext="\eta^{2}_{k}||A_{k}-\bar{A}_{k}||^{2}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p4.3.m3.1"><semantics id="S4.SS2.SSS1.p4.3.m3.1a"><mrow id="S4.SS2.SSS1.p4.3.m3.1.1" xref="S4.SS2.SSS1.p4.3.m3.1.1.cmml"><msubsup id="S4.SS2.SSS1.p4.3.m3.1.1.3" xref="S4.SS2.SSS1.p4.3.m3.1.1.3.cmml"><mi 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id="S4.SS2.SSS1.p4.3.m3.1.1.1.1.1.1.3.2.cmml" xref="S4.SS2.SSS1.p4.3.m3.1.1.1.1.1.1.3.2"><ci id="S4.SS2.SSS1.p4.3.m3.1.1.1.1.1.1.3.2.1.cmml" xref="S4.SS2.SSS1.p4.3.m3.1.1.1.1.1.1.3.2.1">¯</ci><ci id="S4.SS2.SSS1.p4.3.m3.1.1.1.1.1.1.3.2.2.cmml" xref="S4.SS2.SSS1.p4.3.m3.1.1.1.1.1.1.3.2.2">𝐴</ci></apply><ci id="S4.SS2.SSS1.p4.3.m3.1.1.1.1.1.1.3.3.cmml" xref="S4.SS2.SSS1.p4.3.m3.1.1.1.1.1.1.3.3">𝑘</ci></apply></apply></apply><cn id="S4.SS2.SSS1.p4.3.m3.1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p4.3.m3.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p4.3.m3.1c">\eta^{2}_{k}||A_{k}-\bar{A}_{k}||^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p4.3.m3.1d">italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | | italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. We can see that <math alttext="2\eta_{k}<w_{k-1}-w^{\ast}-\eta_{k}\bar{A}_{k},\bar{A}_{k}-A_{k}>=0" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p4.4.m4.2"><semantics id="S4.SS2.SSS1.p4.4.m4.2a"><mrow id="S4.SS2.SSS1.p4.4.m4.2.2.2" xref="S4.SS2.SSS1.p4.4.m4.2.2.3.cmml"><mrow id="S4.SS2.SSS1.p4.4.m4.1.1.1.1" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.cmml"><mrow id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.cmml"><mn id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.2" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.2.cmml">2</mn><mo id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.1" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.1.cmml">⁢</mo><msub id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.3" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.3.cmml"><mi id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.3.2" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.3.2.cmml">η</mi><mi id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.3.3" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.3.3.cmml">k</mi></msub></mrow><mo id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.1" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.1.cmml"><</mo><mrow id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.cmml"><msub id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.cmml"><mi id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.2" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.2.cmml">w</mi><mrow id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.3" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.3.cmml"><mi id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.3.2" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.3.2.cmml">k</mi><mo id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.3.1" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.3.1.cmml">−</mo><mn id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.3.3" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.3.3.cmml">1</mn></mrow></msub><mo id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.1" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.1.cmml">−</mo><msup id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.3" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.3.cmml"><mi id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.3.2" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.3.2.cmml">w</mi><mo id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.3.3" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.3.3.cmml">∗</mo></msup><mo id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.1a" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.1.cmml">−</mo><mrow id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.cmml"><msub id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.2" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.2.cmml"><mi id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.2.2" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.2.2.cmml">η</mi><mi id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.2.3" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.2.3.cmml">k</mi></msub><mo id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.1" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.1.cmml">⁢</mo><msub id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.3" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.3.cmml"><mover accent="true" id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.3.2" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.3.2.cmml"><mi id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.3.2.2" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.3.2.2.cmml">A</mi><mo id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.3.2.1" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.3.2.1.cmml">¯</mo></mover><mi id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.3.3" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.3.3.cmml">k</mi></msub></mrow></mrow></mrow><mo id="S4.SS2.SSS1.p4.4.m4.2.2.2.3" xref="S4.SS2.SSS1.p4.4.m4.2.2.3a.cmml">,</mo><mrow id="S4.SS2.SSS1.p4.4.m4.2.2.2.2" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.cmml"><mrow id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.cmml"><msub id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.2" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.2.cmml"><mover accent="true" id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.2.2" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.2.2.cmml"><mi id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.2.2.2" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.2.2.2.cmml">A</mi><mo id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.2.2.1" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.2.2.1.cmml">¯</mo></mover><mi id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.2.3" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.2.3.cmml">k</mi></msub><mo id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.1" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.1.cmml">−</mo><msub id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.3" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.3.cmml"><mi id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.3.2" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.3.2.cmml">A</mi><mi id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.3.3" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.3.3.cmml">k</mi></msub></mrow><mo id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.1" lspace="0.278em" rspace="0.278em" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.1.cmml">>=</mo><mn id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.3" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.3.cmml">0</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p4.4.m4.2b"><apply id="S4.SS2.SSS1.p4.4.m4.2.2.3.cmml" xref="S4.SS2.SSS1.p4.4.m4.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p4.4.m4.2.2.3a.cmml" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.3">formulae-sequence</csymbol><apply id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1"><lt id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.1"></lt><apply id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2"><times id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.1.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.1"></times><cn id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.2.cmml" type="integer" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.2">2</cn><apply id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.3.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.3.1.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.3">subscript</csymbol><ci id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.3.2.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.3.2">𝜂</ci><ci id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.3.3.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.2.3.3">𝑘</ci></apply></apply><apply id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3"><minus id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.1.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.1"></minus><apply id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.1.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2">subscript</csymbol><ci id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.2.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.2">𝑤</ci><apply id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.3.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.3"><minus id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.3.1.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.3.1"></minus><ci id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.3.2.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.3.2">𝑘</ci><cn id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.3.3.cmml" type="integer" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.2.3.3">1</cn></apply></apply><apply id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.3.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.3.1.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.3">superscript</csymbol><ci id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.3.2.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.3.2">𝑤</ci><ci id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.3.3.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.3.3">∗</ci></apply><apply id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4"><times id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.1.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.1"></times><apply id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.2.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.2.1.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.2">subscript</csymbol><ci id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.2.2.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.2.2">𝜂</ci><ci id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.2.3.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.2.3">𝑘</ci></apply><apply id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.3.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.3.1.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.3">subscript</csymbol><apply id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.3.2.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.3.2"><ci id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.3.2.1.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.3.2.1">¯</ci><ci id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.3.2.2.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.3.2.2">𝐴</ci></apply><ci id="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.3.3.cmml" xref="S4.SS2.SSS1.p4.4.m4.1.1.1.1.3.4.3.3">𝑘</ci></apply></apply></apply></apply><apply id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.cmml" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2"><geq id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.1"></geq><apply id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2"><minus id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.1"></minus><apply id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.2">subscript</csymbol><apply id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.2.2"><ci id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.2.2.1">¯</ci><ci id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.2.2.2">𝐴</ci></apply><ci id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.2.3.cmml" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.2.3">𝑘</ci></apply><apply id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.3.cmml" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.3.1.cmml" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.3">subscript</csymbol><ci id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.3.2.cmml" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.3.2">𝐴</ci><ci id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.3.3.cmml" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.2.3.3">𝑘</ci></apply></apply><cn id="S4.SS2.SSS1.p4.4.m4.2.2.2.2.3.cmml" type="integer" xref="S4.SS2.SSS1.p4.4.m4.2.2.2.2.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p4.4.m4.2c">2\eta_{k}<w_{k-1}-w^{\ast}-\eta_{k}\bar{A}_{k},\bar{A}_{k}-A_{k}>=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p4.4.m4.2d">2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT < italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT > = 0</annotation></semantics></math> because when we want to get the expectation for <math alttext="2\eta_{k}<w_{k-1}-w^{\ast}-\eta_{k}\bar{A}_{k},\bar{A}_{k}-A_{k}>" class="ltx_math_unparsed" display="inline" id="S4.SS2.SSS1.p4.5.m5.1"><semantics id="S4.SS2.SSS1.p4.5.m5.1a"><mrow id="S4.SS2.SSS1.p4.5.m5.1b"><mn id="S4.SS2.SSS1.p4.5.m5.1.1">2</mn><msub id="S4.SS2.SSS1.p4.5.m5.1.2"><mi id="S4.SS2.SSS1.p4.5.m5.1.2.2">η</mi><mi id="S4.SS2.SSS1.p4.5.m5.1.2.3">k</mi></msub><mo id="S4.SS2.SSS1.p4.5.m5.1.3"><</mo><msub id="S4.SS2.SSS1.p4.5.m5.1.4"><mi id="S4.SS2.SSS1.p4.5.m5.1.4.2">w</mi><mrow id="S4.SS2.SSS1.p4.5.m5.1.4.3"><mi id="S4.SS2.SSS1.p4.5.m5.1.4.3.2">k</mi><mo id="S4.SS2.SSS1.p4.5.m5.1.4.3.1">−</mo><mn id="S4.SS2.SSS1.p4.5.m5.1.4.3.3">1</mn></mrow></msub><mo id="S4.SS2.SSS1.p4.5.m5.1.5">−</mo><msup id="S4.SS2.SSS1.p4.5.m5.1.6"><mi id="S4.SS2.SSS1.p4.5.m5.1.6.2">w</mi><mo id="S4.SS2.SSS1.p4.5.m5.1.6.3">∗</mo></msup><mo id="S4.SS2.SSS1.p4.5.m5.1.7">−</mo><msub id="S4.SS2.SSS1.p4.5.m5.1.8"><mi id="S4.SS2.SSS1.p4.5.m5.1.8.2">η</mi><mi id="S4.SS2.SSS1.p4.5.m5.1.8.3">k</mi></msub><msub id="S4.SS2.SSS1.p4.5.m5.1.9"><mover accent="true" id="S4.SS2.SSS1.p4.5.m5.1.9.2"><mi id="S4.SS2.SSS1.p4.5.m5.1.9.2.2">A</mi><mo id="S4.SS2.SSS1.p4.5.m5.1.9.2.1">¯</mo></mover><mi id="S4.SS2.SSS1.p4.5.m5.1.9.3">k</mi></msub><mo id="S4.SS2.SSS1.p4.5.m5.1.10">,</mo><msub id="S4.SS2.SSS1.p4.5.m5.1.11"><mover accent="true" id="S4.SS2.SSS1.p4.5.m5.1.11.2"><mi id="S4.SS2.SSS1.p4.5.m5.1.11.2.2">A</mi><mo id="S4.SS2.SSS1.p4.5.m5.1.11.2.1">¯</mo></mover><mi id="S4.SS2.SSS1.p4.5.m5.1.11.3">k</mi></msub><mo id="S4.SS2.SSS1.p4.5.m5.1.12">−</mo><msub id="S4.SS2.SSS1.p4.5.m5.1.13"><mi id="S4.SS2.SSS1.p4.5.m5.1.13.2">A</mi><mi id="S4.SS2.SSS1.p4.5.m5.1.13.3">k</mi></msub><mo id="S4.SS2.SSS1.p4.5.m5.1.14">></mo></mrow><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p4.5.m5.1c">2\eta_{k}<w_{k-1}-w^{\ast}-\eta_{k}\bar{A}_{k},\bar{A}_{k}-A_{k}></annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p4.5.m5.1d">2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT < italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ></annotation></semantics></math>, the <math alttext="E(A_{k})=\bar{A}_{k}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p4.6.m6.1"><semantics id="S4.SS2.SSS1.p4.6.m6.1a"><mrow id="S4.SS2.SSS1.p4.6.m6.1.1" xref="S4.SS2.SSS1.p4.6.m6.1.1.cmml"><mrow id="S4.SS2.SSS1.p4.6.m6.1.1.1" xref="S4.SS2.SSS1.p4.6.m6.1.1.1.cmml"><mi id="S4.SS2.SSS1.p4.6.m6.1.1.1.3" xref="S4.SS2.SSS1.p4.6.m6.1.1.1.3.cmml">E</mi><mo id="S4.SS2.SSS1.p4.6.m6.1.1.1.2" xref="S4.SS2.SSS1.p4.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS1.p4.6.m6.1.1.1.1.1" xref="S4.SS2.SSS1.p4.6.m6.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS1.p4.6.m6.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS1.p4.6.m6.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.SSS1.p4.6.m6.1.1.1.1.1.1" xref="S4.SS2.SSS1.p4.6.m6.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.p4.6.m6.1.1.1.1.1.1.2" xref="S4.SS2.SSS1.p4.6.m6.1.1.1.1.1.1.2.cmml">A</mi><mi id="S4.SS2.SSS1.p4.6.m6.1.1.1.1.1.1.3" xref="S4.SS2.SSS1.p4.6.m6.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S4.SS2.SSS1.p4.6.m6.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS1.p4.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS1.p4.6.m6.1.1.2" xref="S4.SS2.SSS1.p4.6.m6.1.1.2.cmml">=</mo><msub id="S4.SS2.SSS1.p4.6.m6.1.1.3" xref="S4.SS2.SSS1.p4.6.m6.1.1.3.cmml"><mover accent="true" id="S4.SS2.SSS1.p4.6.m6.1.1.3.2" xref="S4.SS2.SSS1.p4.6.m6.1.1.3.2.cmml"><mi id="S4.SS2.SSS1.p4.6.m6.1.1.3.2.2" xref="S4.SS2.SSS1.p4.6.m6.1.1.3.2.2.cmml">A</mi><mo id="S4.SS2.SSS1.p4.6.m6.1.1.3.2.1" xref="S4.SS2.SSS1.p4.6.m6.1.1.3.2.1.cmml">¯</mo></mover><mi id="S4.SS2.SSS1.p4.6.m6.1.1.3.3" xref="S4.SS2.SSS1.p4.6.m6.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p4.6.m6.1b"><apply id="S4.SS2.SSS1.p4.6.m6.1.1.cmml" xref="S4.SS2.SSS1.p4.6.m6.1.1"><eq id="S4.SS2.SSS1.p4.6.m6.1.1.2.cmml" xref="S4.SS2.SSS1.p4.6.m6.1.1.2"></eq><apply id="S4.SS2.SSS1.p4.6.m6.1.1.1.cmml" xref="S4.SS2.SSS1.p4.6.m6.1.1.1"><times id="S4.SS2.SSS1.p4.6.m6.1.1.1.2.cmml" xref="S4.SS2.SSS1.p4.6.m6.1.1.1.2"></times><ci id="S4.SS2.SSS1.p4.6.m6.1.1.1.3.cmml" xref="S4.SS2.SSS1.p4.6.m6.1.1.1.3">𝐸</ci><apply id="S4.SS2.SSS1.p4.6.m6.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p4.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p4.6.m6.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p4.6.m6.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p4.6.m6.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.p4.6.m6.1.1.1.1.1.1.2">𝐴</ci><ci id="S4.SS2.SSS1.p4.6.m6.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.p4.6.m6.1.1.1.1.1.1.3">𝑘</ci></apply></apply><apply id="S4.SS2.SSS1.p4.6.m6.1.1.3.cmml" xref="S4.SS2.SSS1.p4.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p4.6.m6.1.1.3.1.cmml" xref="S4.SS2.SSS1.p4.6.m6.1.1.3">subscript</csymbol><apply id="S4.SS2.SSS1.p4.6.m6.1.1.3.2.cmml" xref="S4.SS2.SSS1.p4.6.m6.1.1.3.2"><ci id="S4.SS2.SSS1.p4.6.m6.1.1.3.2.1.cmml" xref="S4.SS2.SSS1.p4.6.m6.1.1.3.2.1">¯</ci><ci id="S4.SS2.SSS1.p4.6.m6.1.1.3.2.2.cmml" xref="S4.SS2.SSS1.p4.6.m6.1.1.3.2.2">𝐴</ci></apply><ci id="S4.SS2.SSS1.p4.6.m6.1.1.3.3.cmml" xref="S4.SS2.SSS1.p4.6.m6.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p4.6.m6.1c">E(A_{k})=\bar{A}_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p4.6.m6.1d">italic_E ( italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, which means <math alttext="\bar{A}_{k}-A_{k}=0" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p4.7.m7.1"><semantics id="S4.SS2.SSS1.p4.7.m7.1a"><mrow id="S4.SS2.SSS1.p4.7.m7.1.1" xref="S4.SS2.SSS1.p4.7.m7.1.1.cmml"><mrow id="S4.SS2.SSS1.p4.7.m7.1.1.2" xref="S4.SS2.SSS1.p4.7.m7.1.1.2.cmml"><msub id="S4.SS2.SSS1.p4.7.m7.1.1.2.2" xref="S4.SS2.SSS1.p4.7.m7.1.1.2.2.cmml"><mover accent="true" id="S4.SS2.SSS1.p4.7.m7.1.1.2.2.2" xref="S4.SS2.SSS1.p4.7.m7.1.1.2.2.2.cmml"><mi id="S4.SS2.SSS1.p4.7.m7.1.1.2.2.2.2" xref="S4.SS2.SSS1.p4.7.m7.1.1.2.2.2.2.cmml">A</mi><mo id="S4.SS2.SSS1.p4.7.m7.1.1.2.2.2.1" xref="S4.SS2.SSS1.p4.7.m7.1.1.2.2.2.1.cmml">¯</mo></mover><mi id="S4.SS2.SSS1.p4.7.m7.1.1.2.2.3" xref="S4.SS2.SSS1.p4.7.m7.1.1.2.2.3.cmml">k</mi></msub><mo id="S4.SS2.SSS1.p4.7.m7.1.1.2.1" xref="S4.SS2.SSS1.p4.7.m7.1.1.2.1.cmml">−</mo><msub id="S4.SS2.SSS1.p4.7.m7.1.1.2.3" xref="S4.SS2.SSS1.p4.7.m7.1.1.2.3.cmml"><mi id="S4.SS2.SSS1.p4.7.m7.1.1.2.3.2" xref="S4.SS2.SSS1.p4.7.m7.1.1.2.3.2.cmml">A</mi><mi id="S4.SS2.SSS1.p4.7.m7.1.1.2.3.3" xref="S4.SS2.SSS1.p4.7.m7.1.1.2.3.3.cmml">k</mi></msub></mrow><mo id="S4.SS2.SSS1.p4.7.m7.1.1.1" xref="S4.SS2.SSS1.p4.7.m7.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS1.p4.7.m7.1.1.3" xref="S4.SS2.SSS1.p4.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p4.7.m7.1b"><apply id="S4.SS2.SSS1.p4.7.m7.1.1.cmml" xref="S4.SS2.SSS1.p4.7.m7.1.1"><eq id="S4.SS2.SSS1.p4.7.m7.1.1.1.cmml" xref="S4.SS2.SSS1.p4.7.m7.1.1.1"></eq><apply id="S4.SS2.SSS1.p4.7.m7.1.1.2.cmml" xref="S4.SS2.SSS1.p4.7.m7.1.1.2"><minus id="S4.SS2.SSS1.p4.7.m7.1.1.2.1.cmml" xref="S4.SS2.SSS1.p4.7.m7.1.1.2.1"></minus><apply id="S4.SS2.SSS1.p4.7.m7.1.1.2.2.cmml" xref="S4.SS2.SSS1.p4.7.m7.1.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p4.7.m7.1.1.2.2.1.cmml" xref="S4.SS2.SSS1.p4.7.m7.1.1.2.2">subscript</csymbol><apply id="S4.SS2.SSS1.p4.7.m7.1.1.2.2.2.cmml" xref="S4.SS2.SSS1.p4.7.m7.1.1.2.2.2"><ci id="S4.SS2.SSS1.p4.7.m7.1.1.2.2.2.1.cmml" xref="S4.SS2.SSS1.p4.7.m7.1.1.2.2.2.1">¯</ci><ci id="S4.SS2.SSS1.p4.7.m7.1.1.2.2.2.2.cmml" xref="S4.SS2.SSS1.p4.7.m7.1.1.2.2.2.2">𝐴</ci></apply><ci id="S4.SS2.SSS1.p4.7.m7.1.1.2.2.3.cmml" xref="S4.SS2.SSS1.p4.7.m7.1.1.2.2.3">𝑘</ci></apply><apply id="S4.SS2.SSS1.p4.7.m7.1.1.2.3.cmml" xref="S4.SS2.SSS1.p4.7.m7.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p4.7.m7.1.1.2.3.1.cmml" xref="S4.SS2.SSS1.p4.7.m7.1.1.2.3">subscript</csymbol><ci id="S4.SS2.SSS1.p4.7.m7.1.1.2.3.2.cmml" xref="S4.SS2.SSS1.p4.7.m7.1.1.2.3.2">𝐴</ci><ci id="S4.SS2.SSS1.p4.7.m7.1.1.2.3.3.cmml" xref="S4.SS2.SSS1.p4.7.m7.1.1.2.3.3">𝑘</ci></apply></apply><cn id="S4.SS2.SSS1.p4.7.m7.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p4.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p4.7.m7.1c">\bar{A}_{k}-A_{k}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p4.7.m7.1d">over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = 0</annotation></semantics></math>. Therefore the dot product is also zero.</p> </div> <div class="ltx_para" id="S4.SS2.SSS1.p5"> <table class="ltx_equation ltx_eqn_table" id="S4.E9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{split}||w_{k-1}-\eta_{k}\bar{A}_{k}-w^{\ast}||^{2}=||w_{k-1}-w^{\ast}||% ^{2}\\ -2\eta_{k}<w_{k-1}-w^{\ast},\bar{A}_{k}>+\eta^{2}_{k}||\bar{A}_{k}||^{2}\end{split}" class="ltx_Math" display="block" id="S4.E9.m1.57"><semantics id="S4.E9.m1.57a"><mtable displaystyle="true" id="S4.E9.m1.57.57.6" rowspacing="0pt"><mtr id="S4.E9.m1.57.57.6a"><mtd class="ltx_align_right" columnalign="right" id="S4.E9.m1.57.57.6b"><mrow id="S4.E9.m1.55.55.4.53.28.28"><msup id="S4.E9.m1.54.54.3.52.27.27.27"><mrow id="S4.E9.m1.54.54.3.52.27.27.27.1.1"><mo id="S4.E9.m1.1.1.1.1.1.1b" stretchy="false" xref="S4.E9.m1.53.53.2.3.cmml">‖</mo><mrow id="S4.E9.m1.54.54.3.52.27.27.27.1.1.1"><msub 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italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = | | italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL - 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT < italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT > + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | | over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(9)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS1.p5.2">We can see there are three part in the equation (<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4.E9" title="In IV-B1 Continuous Linear ‣ IV-B Convergence Analysis ‣ IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">9</span></a>), since <math alttext="\eta^{2}_{k}||\bar{A}_{k}||^{2}=\eta^{2}_{k}||\nabla F_{k}\left(w_{k}\right)||% ^{2}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.1.m1.2"><semantics id="S4.SS2.SSS1.p5.1.m1.2a"><mrow id="S4.SS2.SSS1.p5.1.m1.2.2" xref="S4.SS2.SSS1.p5.1.m1.2.2.cmml"><mrow id="S4.SS2.SSS1.p5.1.m1.1.1.1" xref="S4.SS2.SSS1.p5.1.m1.1.1.1.cmml"><msubsup id="S4.SS2.SSS1.p5.1.m1.1.1.1.3" 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type="integer" xref="S4.SS2.SSS1.p5.1.m1.2.2.2.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.1.m1.2c">\eta^{2}_{k}||\bar{A}_{k}||^{2}=\eta^{2}_{k}||\nabla F_{k}\left(w_{k}\right)||% ^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.1.m1.2d">italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | | over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | | ∇ italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> and because all the local models are L-smooth, we can bound <math alttext="||\bar{A}_{k}||^{2}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p5.2.m2.1"><semantics id="S4.SS2.SSS1.p5.2.m2.1a"><msup id="S4.SS2.SSS1.p5.2.m2.1.1" xref="S4.SS2.SSS1.p5.2.m2.1.1.cmml"><mrow id="S4.SS2.SSS1.p5.2.m2.1.1.1.1" xref="S4.SS2.SSS1.p5.2.m2.1.1.1.2.cmml"><mo id="S4.SS2.SSS1.p5.2.m2.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS1.p5.2.m2.1.1.1.2.1.cmml">‖</mo><msub id="S4.SS2.SSS1.p5.2.m2.1.1.1.1.1" xref="S4.SS2.SSS1.p5.2.m2.1.1.1.1.1.cmml"><mover accent="true" id="S4.SS2.SSS1.p5.2.m2.1.1.1.1.1.2" xref="S4.SS2.SSS1.p5.2.m2.1.1.1.1.1.2.cmml"><mi id="S4.SS2.SSS1.p5.2.m2.1.1.1.1.1.2.2" xref="S4.SS2.SSS1.p5.2.m2.1.1.1.1.1.2.2.cmml">A</mi><mo id="S4.SS2.SSS1.p5.2.m2.1.1.1.1.1.2.1" xref="S4.SS2.SSS1.p5.2.m2.1.1.1.1.1.2.1.cmml">¯</mo></mover><mi id="S4.SS2.SSS1.p5.2.m2.1.1.1.1.1.3" xref="S4.SS2.SSS1.p5.2.m2.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S4.SS2.SSS1.p5.2.m2.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS1.p5.2.m2.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S4.SS2.SSS1.p5.2.m2.1.1.3" xref="S4.SS2.SSS1.p5.2.m2.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p5.2.m2.1b"><apply id="S4.SS2.SSS1.p5.2.m2.1.1.cmml" xref="S4.SS2.SSS1.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.2.m2.1.1.2.cmml" xref="S4.SS2.SSS1.p5.2.m2.1.1">superscript</csymbol><apply id="S4.SS2.SSS1.p5.2.m2.1.1.1.2.cmml" xref="S4.SS2.SSS1.p5.2.m2.1.1.1.1"><csymbol cd="latexml" id="S4.SS2.SSS1.p5.2.m2.1.1.1.2.1.cmml" xref="S4.SS2.SSS1.p5.2.m2.1.1.1.1.2">norm</csymbol><apply id="S4.SS2.SSS1.p5.2.m2.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p5.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p5.2.m2.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p5.2.m2.1.1.1.1.1">subscript</csymbol><apply id="S4.SS2.SSS1.p5.2.m2.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.p5.2.m2.1.1.1.1.1.2"><ci id="S4.SS2.SSS1.p5.2.m2.1.1.1.1.1.2.1.cmml" xref="S4.SS2.SSS1.p5.2.m2.1.1.1.1.1.2.1">¯</ci><ci id="S4.SS2.SSS1.p5.2.m2.1.1.1.1.1.2.2.cmml" xref="S4.SS2.SSS1.p5.2.m2.1.1.1.1.1.2.2">𝐴</ci></apply><ci id="S4.SS2.SSS1.p5.2.m2.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.p5.2.m2.1.1.1.1.1.3">𝑘</ci></apply></apply><cn id="S4.SS2.SSS1.p5.2.m2.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p5.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.2.m2.1c">||\bar{A}_{k}||^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.2.m2.1d">| | over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> as:</p> <table class="ltx_equation ltx_eqn_table" id="S4.E10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\eta^{2}_{k}||\nabla F_{k}\left(w_{k}\right)||^{2}\leq 2\eta^{2}L\left(F_{k}% \left(w_{k}\right)-F^{\ast}_{k}\right)" class="ltx_Math" display="block" id="S4.E10.m1.2"><semantics 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xref="S4.E10.m1.1.1.1.1.3.cmml">2</mn></msup></mrow><mo id="S4.E10.m1.2.2.3" xref="S4.E10.m1.2.2.3.cmml">≤</mo><mrow id="S4.E10.m1.2.2.2" xref="S4.E10.m1.2.2.2.cmml"><mn id="S4.E10.m1.2.2.2.3" xref="S4.E10.m1.2.2.2.3.cmml">2</mn><mo id="S4.E10.m1.2.2.2.2" xref="S4.E10.m1.2.2.2.2.cmml">⁢</mo><msup id="S4.E10.m1.2.2.2.4" xref="S4.E10.m1.2.2.2.4.cmml"><mi id="S4.E10.m1.2.2.2.4.2" xref="S4.E10.m1.2.2.2.4.2.cmml">η</mi><mn id="S4.E10.m1.2.2.2.4.3" xref="S4.E10.m1.2.2.2.4.3.cmml">2</mn></msup><mo id="S4.E10.m1.2.2.2.2a" xref="S4.E10.m1.2.2.2.2.cmml">⁢</mo><mi id="S4.E10.m1.2.2.2.5" xref="S4.E10.m1.2.2.2.5.cmml">L</mi><mo id="S4.E10.m1.2.2.2.2b" xref="S4.E10.m1.2.2.2.2.cmml">⁢</mo><mrow id="S4.E10.m1.2.2.2.1.1" xref="S4.E10.m1.2.2.2.1.1.1.cmml"><mo id="S4.E10.m1.2.2.2.1.1.2" xref="S4.E10.m1.2.2.2.1.1.1.cmml">(</mo><mrow id="S4.E10.m1.2.2.2.1.1.1" xref="S4.E10.m1.2.2.2.1.1.1.cmml"><mrow id="S4.E10.m1.2.2.2.1.1.1.1" xref="S4.E10.m1.2.2.2.1.1.1.1.cmml"><msub id="S4.E10.m1.2.2.2.1.1.1.1.3" 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xref="S4.E10.m1.2.2.2.1.1.1.1.3.3">𝑘</ci></apply><apply id="S4.E10.m1.2.2.2.1.1.1.1.1.1.1.cmml" xref="S4.E10.m1.2.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E10.m1.2.2.2.1.1.1.1.1.1.1.1.cmml" xref="S4.E10.m1.2.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S4.E10.m1.2.2.2.1.1.1.1.1.1.1.2.cmml" xref="S4.E10.m1.2.2.2.1.1.1.1.1.1.1.2">𝑤</ci><ci id="S4.E10.m1.2.2.2.1.1.1.1.1.1.1.3.cmml" xref="S4.E10.m1.2.2.2.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply><apply id="S4.E10.m1.2.2.2.1.1.1.3.cmml" xref="S4.E10.m1.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S4.E10.m1.2.2.2.1.1.1.3.1.cmml" xref="S4.E10.m1.2.2.2.1.1.1.3">subscript</csymbol><apply id="S4.E10.m1.2.2.2.1.1.1.3.2.cmml" xref="S4.E10.m1.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S4.E10.m1.2.2.2.1.1.1.3.2.1.cmml" xref="S4.E10.m1.2.2.2.1.1.1.3">superscript</csymbol><ci id="S4.E10.m1.2.2.2.1.1.1.3.2.2.cmml" xref="S4.E10.m1.2.2.2.1.1.1.3.2.2">𝐹</ci><ci id="S4.E10.m1.2.2.2.1.1.1.3.2.3.cmml" xref="S4.E10.m1.2.2.2.1.1.1.3.2.3">∗</ci></apply><ci id="S4.E10.m1.2.2.2.1.1.1.3.3.cmml" xref="S4.E10.m1.2.2.2.1.1.1.3.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E10.m1.2c">\eta^{2}_{k}||\nabla F_{k}\left(w_{k}\right)||^{2}\leq 2\eta^{2}L\left(F_{k}% \left(w_{k}\right)-F^{\ast}_{k}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.E10.m1.2d">italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | | ∇ italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ 2 italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_L ( italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(10)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS1.p5.3">Then, we do the bonding at <math alttext="-2\eta_{k}<w_{k-1}-w^{\ast},\bar{A}_{k}>" class="ltx_math_unparsed" display="inline" id="S4.SS2.SSS1.p5.3.m1.1"><semantics id="S4.SS2.SSS1.p5.3.m1.1a"><mrow id="S4.SS2.SSS1.p5.3.m1.1b"><mo id="S4.SS2.SSS1.p5.3.m1.1.1">−</mo><mn id="S4.SS2.SSS1.p5.3.m1.1.2">2</mn><msub id="S4.SS2.SSS1.p5.3.m1.1.3"><mi id="S4.SS2.SSS1.p5.3.m1.1.3.2">η</mi><mi id="S4.SS2.SSS1.p5.3.m1.1.3.3">k</mi></msub><mo id="S4.SS2.SSS1.p5.3.m1.1.4"><</mo><msub id="S4.SS2.SSS1.p5.3.m1.1.5"><mi id="S4.SS2.SSS1.p5.3.m1.1.5.2">w</mi><mrow id="S4.SS2.SSS1.p5.3.m1.1.5.3"><mi id="S4.SS2.SSS1.p5.3.m1.1.5.3.2">k</mi><mo id="S4.SS2.SSS1.p5.3.m1.1.5.3.1">−</mo><mn id="S4.SS2.SSS1.p5.3.m1.1.5.3.3">1</mn></mrow></msub><mo id="S4.SS2.SSS1.p5.3.m1.1.6">−</mo><msup id="S4.SS2.SSS1.p5.3.m1.1.7"><mi id="S4.SS2.SSS1.p5.3.m1.1.7.2">w</mi><mo id="S4.SS2.SSS1.p5.3.m1.1.7.3">∗</mo></msup><mo id="S4.SS2.SSS1.p5.3.m1.1.8">,</mo><msub id="S4.SS2.SSS1.p5.3.m1.1.9"><mover accent="true" id="S4.SS2.SSS1.p5.3.m1.1.9.2"><mi id="S4.SS2.SSS1.p5.3.m1.1.9.2.2">A</mi><mo id="S4.SS2.SSS1.p5.3.m1.1.9.2.1">¯</mo></mover><mi id="S4.SS2.SSS1.p5.3.m1.1.9.3">k</mi></msub><mo id="S4.SS2.SSS1.p5.3.m1.1.10">></mo></mrow><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p5.3.m1.1c">-2\eta_{k}<w_{k-1}-w^{\ast},\bar{A}_{k}></annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p5.3.m1.1d">- 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT < italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ></annotation></semantics></math>, by the definition of the strongly convex:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S8.EGx1"> <tbody id="S4.Ex8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle-2\eta_{k}\langle w_{k-1}-w^{\ast},\nabla F_{k}\left(w_{k-1}% \right)\rangle" class="ltx_Math" display="inline" id="S4.Ex8.m1.2"><semantics id="S4.Ex8.m1.2a"><mrow id="S4.Ex8.m1.2.2" xref="S4.Ex8.m1.2.2.cmml"><mo id="S4.Ex8.m1.2.2a" xref="S4.Ex8.m1.2.2.cmml">−</mo><mrow id="S4.Ex8.m1.2.2.2" xref="S4.Ex8.m1.2.2.2.cmml"><mn id="S4.Ex8.m1.2.2.2.4" xref="S4.Ex8.m1.2.2.2.4.cmml">2</mn><mo id="S4.Ex8.m1.2.2.2.3" xref="S4.Ex8.m1.2.2.2.3.cmml">⁢</mo><msub id="S4.Ex8.m1.2.2.2.5" xref="S4.Ex8.m1.2.2.2.5.cmml"><mi id="S4.Ex8.m1.2.2.2.5.2" xref="S4.Ex8.m1.2.2.2.5.2.cmml">η</mi><mi id="S4.Ex8.m1.2.2.2.5.3" xref="S4.Ex8.m1.2.2.2.5.3.cmml">k</mi></msub><mo id="S4.Ex8.m1.2.2.2.3a" 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id="S4.Ex8.m1.2c">\displaystyle-2\eta_{k}\langle w_{k-1}-w^{\ast},\nabla F_{k}\left(w_{k-1}% \right)\rangle</annotation><annotation encoding="application/x-llamapun" id="S4.Ex8.m1.2d">- 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⟨ italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , ∇ italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) ⟩</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq-2\eta_{k}\left(F_{k}\left(w_{k-1}\right)-F_{k}\left(w^{\ast}% \right)\right." class="ltx_math_unparsed" display="inline" id="S4.Ex8.m2.1"><semantics id="S4.Ex8.m2.1a"><mrow id="S4.Ex8.m2.1b"><mo id="S4.Ex8.m2.1.1" rspace="0em">≤</mo><mo id="S4.Ex8.m2.1.2" lspace="0em">−</mo><mn id="S4.Ex8.m2.1.3">2</mn><msub id="S4.Ex8.m2.1.4"><mi id="S4.Ex8.m2.1.4.2">η</mi><mi id="S4.Ex8.m2.1.4.3">k</mi></msub><mrow id="S4.Ex8.m2.1.5"><mo id="S4.Ex8.m2.1.5.1">(</mo><msub id="S4.Ex8.m2.1.5.2"><mi id="S4.Ex8.m2.1.5.2.2">F</mi><mi id="S4.Ex8.m2.1.5.2.3">k</mi></msub><mrow id="S4.Ex8.m2.1.5.3"><mo id="S4.Ex8.m2.1.5.3.1">(</mo><msub id="S4.Ex8.m2.1.5.3.2"><mi id="S4.Ex8.m2.1.5.3.2.2">w</mi><mrow id="S4.Ex8.m2.1.5.3.2.3"><mi id="S4.Ex8.m2.1.5.3.2.3.2">k</mi><mo id="S4.Ex8.m2.1.5.3.2.3.1">−</mo><mn id="S4.Ex8.m2.1.5.3.2.3.3">1</mn></mrow></msub><mo id="S4.Ex8.m2.1.5.3.3">)</mo></mrow><mo id="S4.Ex8.m2.1.5.4">−</mo><msub id="S4.Ex8.m2.1.5.5"><mi id="S4.Ex8.m2.1.5.5.2">F</mi><mi id="S4.Ex8.m2.1.5.5.3">k</mi></msub><mrow id="S4.Ex8.m2.1.5.6"><mo id="S4.Ex8.m2.1.5.6.1">(</mo><msup id="S4.Ex8.m2.1.5.6.2"><mi id="S4.Ex8.m2.1.5.6.2.2">w</mi><mo id="S4.Ex8.m2.1.5.6.2.3">∗</mo></msup><mo id="S4.Ex8.m2.1.5.6.3">)</mo></mrow></mrow></mrow><annotation encoding="application/x-tex" id="S4.Ex8.m2.1c">\displaystyle\leq-2\eta_{k}\left(F_{k}\left(w_{k-1}\right)-F_{k}\left(w^{\ast}% \right)\right.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex8.m2.1d">≤ - 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) - italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.E11"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\quad\left.-\frac{\mu}{2}\parallel w_{k-1}-w^{\ast}\parallel^{2}_% {2}\right)" class="ltx_math_unparsed" display="inline" id="S4.E11.m1.1"><semantics id="S4.E11.m1.1a"><mrow id="S4.E11.m1.1b"><mo id="S4.E11.m1.1.1">−</mo><mstyle displaystyle="true" id="S4.E11.m1.1.2"><mfrac id="S4.E11.m1.1.2a"><mi id="S4.E11.m1.1.2.2">μ</mi><mn id="S4.E11.m1.1.2.3">2</mn></mfrac></mstyle><mo id="S4.E11.m1.1.3" lspace="0em" rspace="0.167em">∥</mo><msub id="S4.E11.m1.1.4"><mi id="S4.E11.m1.1.4.2">w</mi><mrow id="S4.E11.m1.1.4.3"><mi id="S4.E11.m1.1.4.3.2">k</mi><mo id="S4.E11.m1.1.4.3.1">−</mo><mn id="S4.E11.m1.1.4.3.3">1</mn></mrow></msub><mo id="S4.E11.m1.1.5">−</mo><msup id="S4.E11.m1.1.6"><mi id="S4.E11.m1.1.6.2">w</mi><mo id="S4.E11.m1.1.6.3">∗</mo></msup><msubsup id="S4.E11.m1.1.7"><mo id="S4.E11.m1.1.7.2.2" lspace="0em" rspace="0.167em">∥</mo><mn id="S4.E11.m1.1.7.3">2</mn><mn id="S4.E11.m1.1.7.2.3">2</mn></msubsup><mo id="S4.E11.m1.1.8">)</mo></mrow><annotation encoding="application/x-tex" id="S4.E11.m1.1c">\displaystyle\quad\left.-\frac{\mu}{2}\parallel w_{k-1}-w^{\ast}\parallel^{2}_% {2}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.E11.m1.1d">- divide start_ARG italic_μ end_ARG start_ARG 2 end_ARG ∥ italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(11)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS1.p6"> <p class="ltx_p" id="S4.SS2.SSS1.p6.1">Then, we substitute the above two equations into Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4.E9" title="In IV-B1 Continuous Linear ‣ IV-B Convergence Analysis ‣ IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">9</span></a>:</p> <table class="ltx_equation ltx_eqn_table" id="S4.E12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{split}||w_{k-1}-\eta_{k}\bar{A}_{k}-w^{\ast}||^{2}\leq||w_{k-1}-w^{\ast% }||^{2}-2\eta_{k}(F_{k}\left(w_{k-1}\right)\\ -F_{k}\left(w^{\ast}\right)-\frac{\mu}{2}\parallel w_{k-1}-w^{\ast}\parallel^{% 2}_{2})+2\eta^{2}_{k}L\left(F_{k}\left(w_{k}\right)-F^{\ast}_{k}\right)\end{split}" class="ltx_Math" display="block" id="S4.E12.m1.78"><semantics id="S4.E12.m1.78a"><mtable displaystyle="true" id="S4.E12.m1.74.74" rowspacing="0pt" xref="S4.E12.m1.78.78.4.cmml"><mtr id="S4.E12.m1.74.74a" xref="S4.E12.m1.78.78.4.cmml"><mtd class="ltx_align_right" columnalign="right" id="S4.E12.m1.74.74b" xref="S4.E12.m1.78.78.4.cmml"><mrow id="S4.E12.m1.37.37.37.37.37" xref="S4.E12.m1.78.78.4.cmml"><mo fence="false" id="S4.E12.m1.1.1.1.1.1.1" rspace="0.167em" stretchy="false" xref="S4.E12.m1.78.78.4.cmml">|</mo><mo fence="false" id="S4.E12.m1.2.2.2.2.2.2" rspace="0.167em" 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xref="S4.E12.m1.73.73.73.36.36.36.1">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E12.m1.78c">\begin{split}||w_{k-1}-\eta_{k}\bar{A}_{k}-w^{\ast}||^{2}\leq||w_{k-1}-w^{\ast% }||^{2}-2\eta_{k}(F_{k}\left(w_{k-1}\right)\\ -F_{k}\left(w^{\ast}\right)-\frac{\mu}{2}\parallel w_{k-1}-w^{\ast}\parallel^{% 2}_{2})+2\eta^{2}_{k}L\left(F_{k}\left(w_{k}\right)-F^{\ast}_{k}\right)\end{split}</annotation><annotation encoding="application/x-llamapun" id="S4.E12.m1.78d">start_ROW start_CELL | | italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ | | italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL - italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) - divide start_ARG italic_μ end_ARG start_ARG 2 end_ARG ∥ italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + 2 italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_L ( italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(12)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS1.p7"> <table class="ltx_equation ltx_eqn_table" id="S4.E13"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{split}||w_{k-1}-\eta_{k}\bar{A}_{k}-w^{\ast}||^{2}\leq||w_{k-1}-w^{\ast% }||^{2}-2\eta_{k}[F_{k}\left(w_{k-1}\right)\\ -F_{k}\left(w^{\ast}\right)-\frac{\mu}{2}\parallel w_{k-1}-w^{\ast}\parallel^{% 2}]+(2\eta_{k})^{2}L\left(F_{k}\left(w_{k}\right)-F^{\ast}_{k}\right)\\ \leq||w_{k-1}-w^{\ast}||^{2}-2\eta_{k}F_{k}\left(w_{k-1}\right)+2\eta_{k}F_{k}% \left(w^{\ast}\right)\\ +\mu\eta_{k}\parallel w_{k-1}-w^{\ast}\parallel^{2}+(2\eta_{k})^{2}L\left(F_{k% }\left(w_{k}\right)-F^{\ast}_{k}\right)\\ =\left(1+\mu\eta_{k}\right)||w_{k-1}-w^{\ast}||^{2}-2\eta_{k}\left(F_{k}\left(% w_{k-1}\right)-F^{\ast}_{k}\right)\\ +(2\eta_{k})^{2}L\left(F_{k}\left(w_{k}\right)-F^{\ast}_{k}\right)\end{split}" class="ltx_Math" display="block" id="S4.E13.m1.219"><semantics id="S4.E13.m1.219a"><mtable displaystyle="true" id="S4.E13.m1.219.219.27" rowspacing="0pt" xref="S4.E13.m1.208.208.16.cmml"><mtr id="S4.E13.m1.219.219.27a" xref="S4.E13.m1.208.208.16.cmml"><mtd class="ltx_align_right" columnalign="right" id="S4.E13.m1.219.219.27b" xref="S4.E13.m1.208.208.16.cmml"><mrow id="S4.E13.m1.37.37.37.37.37" xref="S4.E13.m1.208.208.16.cmml"><mo fence="false" id="S4.E13.m1.1.1.1.1.1.1" rspace="0.167em" stretchy="false" xref="S4.E13.m1.208.208.16a.cmml">|</mo><mo fence="false" id="S4.E13.m1.2.2.2.2.2.2" rspace="0.167em" stretchy="false" xref="S4.E13.m1.208.208.16a.cmml">|</mo><msub 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xref="S4.E13.m1.208.208.16a.cmml">(</mo><msub id="S4.E13.m1.75.75.75.38.38.46" xref="S4.E13.m1.208.208.16.cmml"><mi id="S4.E13.m1.65.65.65.28.28.28" xref="S4.E13.m1.65.65.65.28.28.28.cmml">F</mi><mi id="S4.E13.m1.66.66.66.29.29.29.1" xref="S4.E13.m1.66.66.66.29.29.29.1.cmml">k</mi></msub><mrow id="S4.E13.m1.75.75.75.38.38.47" xref="S4.E13.m1.208.208.16.cmml"><mo id="S4.E13.m1.67.67.67.30.30.30" xref="S4.E13.m1.208.208.16a.cmml">(</mo><msub id="S4.E13.m1.75.75.75.38.38.47.1" xref="S4.E13.m1.208.208.16.cmml"><mi id="S4.E13.m1.68.68.68.31.31.31" xref="S4.E13.m1.68.68.68.31.31.31.cmml">w</mi><mi id="S4.E13.m1.69.69.69.32.32.32.1" xref="S4.E13.m1.69.69.69.32.32.32.1.cmml">k</mi></msub><mo id="S4.E13.m1.70.70.70.33.33.33" xref="S4.E13.m1.208.208.16a.cmml">)</mo></mrow><mo id="S4.E13.m1.71.71.71.34.34.34" xref="S4.E13.m1.71.71.71.34.34.34.cmml">−</mo><msubsup id="S4.E13.m1.75.75.75.38.38.48" xref="S4.E13.m1.208.208.16.cmml"><mi id="S4.E13.m1.72.72.72.35.35.35" xref="S4.E13.m1.72.72.72.35.35.35.cmml">F</mi><mi id="S4.E13.m1.74.74.74.37.37.37.1" xref="S4.E13.m1.74.74.74.37.37.37.1.cmml">k</mi><mo id="S4.E13.m1.73.73.73.36.36.36.1" xref="S4.E13.m1.73.73.73.36.36.36.1.cmml">∗</mo></msubsup><mo id="S4.E13.m1.75.75.75.38.38.38" xref="S4.E13.m1.208.208.16a.cmml">)</mo></mrow></mtd></mtr><mtr id="S4.E13.m1.219.219.27e" xref="S4.E13.m1.208.208.16.cmml"><mtd class="ltx_align_right" columnalign="right" id="S4.E13.m1.219.219.27f" xref="S4.E13.m1.208.208.16.cmml"><mrow id="S4.E13.m1.211.211.19.195.34.34" xref="S4.E13.m1.208.208.16.cmml"><mi id="S4.E13.m1.211.211.19.195.34.34.35" xref="S4.E13.m1.208.208.16a.cmml"></mi><mo id="S4.E13.m1.76.76.76.1.1.1" xref="S4.E13.m1.76.76.76.1.1.1.cmml">≤</mo><mrow id="S4.E13.m1.211.211.19.195.34.34.34" xref="S4.E13.m1.208.208.16.cmml"><mrow id="S4.E13.m1.210.210.18.194.33.33.33.2" xref="S4.E13.m1.208.208.16.cmml"><msup id="S4.E13.m1.209.209.17.193.32.32.32.1.1" xref="S4.E13.m1.208.208.16.cmml"><mrow id="S4.E13.m1.209.209.17.193.32.32.32.1.1.1.1" xref="S4.E13.m1.208.208.16.cmml"><mo id="S4.E13.m1.77.77.77.2.2.2b" stretchy="false" xref="S4.E13.m1.208.208.16a.cmml">‖</mo><mrow id="S4.E13.m1.209.209.17.193.32.32.32.1.1.1.1.1" xref="S4.E13.m1.208.208.16.cmml"><msub id="S4.E13.m1.209.209.17.193.32.32.32.1.1.1.1.1.1" xref="S4.E13.m1.208.208.16.cmml"><mi id="S4.E13.m1.79.79.79.4.4.4" xref="S4.E13.m1.79.79.79.4.4.4.cmml">w</mi><mrow id="S4.E13.m1.80.80.80.5.5.5.1" xref="S4.E13.m1.80.80.80.5.5.5.1.cmml"><mi id="S4.E13.m1.80.80.80.5.5.5.1.2" xref="S4.E13.m1.80.80.80.5.5.5.1.2.cmml">k</mi><mo id="S4.E13.m1.80.80.80.5.5.5.1.1" xref="S4.E13.m1.80.80.80.5.5.5.1.1.cmml">−</mo><mn id="S4.E13.m1.80.80.80.5.5.5.1.3" xref="S4.E13.m1.80.80.80.5.5.5.1.3.cmml">1</mn></mrow></msub><mo id="S4.E13.m1.81.81.81.6.6.6" xref="S4.E13.m1.81.81.81.6.6.6.cmml">−</mo><msup id="S4.E13.m1.209.209.17.193.32.32.32.1.1.1.1.1.2" xref="S4.E13.m1.208.208.16.cmml"><mi id="S4.E13.m1.82.82.82.7.7.7" xref="S4.E13.m1.82.82.82.7.7.7.cmml">w</mi><mo id="S4.E13.m1.83.83.83.8.8.8.1" xref="S4.E13.m1.83.83.83.8.8.8.1.cmml">∗</mo></msup></mrow><mo id="S4.E13.m1.84.84.84.9.9.9b" stretchy="false" xref="S4.E13.m1.208.208.16a.cmml">‖</mo></mrow><mn id="S4.E13.m1.86.86.86.11.11.11.1" xref="S4.E13.m1.86.86.86.11.11.11.1.cmml">2</mn></msup><mo id="S4.E13.m1.87.87.87.12.12.12" xref="S4.E13.m1.87.87.87.12.12.12.cmml">−</mo><mrow id="S4.E13.m1.210.210.18.194.33.33.33.2.2" xref="S4.E13.m1.208.208.16.cmml"><mn id="S4.E13.m1.88.88.88.13.13.13" xref="S4.E13.m1.88.88.88.13.13.13.cmml">2</mn><mo id="S4.E13.m1.210.210.18.194.33.33.33.2.2.2" xref="S4.E13.m1.208.208.16a.cmml">⁢</mo><msub id="S4.E13.m1.210.210.18.194.33.33.33.2.2.3" xref="S4.E13.m1.208.208.16.cmml"><mi id="S4.E13.m1.89.89.89.14.14.14" xref="S4.E13.m1.89.89.89.14.14.14.cmml">η</mi><mi id="S4.E13.m1.90.90.90.15.15.15.1" xref="S4.E13.m1.90.90.90.15.15.15.1.cmml">k</mi></msub><mo id="S4.E13.m1.210.210.18.194.33.33.33.2.2.2a" xref="S4.E13.m1.208.208.16a.cmml">⁢</mo><msub id="S4.E13.m1.210.210.18.194.33.33.33.2.2.4" xref="S4.E13.m1.208.208.16.cmml"><mi id="S4.E13.m1.91.91.91.16.16.16" xref="S4.E13.m1.91.91.91.16.16.16.cmml">F</mi><mi id="S4.E13.m1.92.92.92.17.17.17.1" xref="S4.E13.m1.92.92.92.17.17.17.1.cmml">k</mi></msub><mo id="S4.E13.m1.210.210.18.194.33.33.33.2.2.2b" xref="S4.E13.m1.208.208.16a.cmml">⁢</mo><mrow id="S4.E13.m1.210.210.18.194.33.33.33.2.2.1.1" xref="S4.E13.m1.208.208.16.cmml"><mo id="S4.E13.m1.93.93.93.18.18.18" xref="S4.E13.m1.208.208.16a.cmml">(</mo><msub id="S4.E13.m1.210.210.18.194.33.33.33.2.2.1.1.1" xref="S4.E13.m1.208.208.16.cmml"><mi id="S4.E13.m1.94.94.94.19.19.19" xref="S4.E13.m1.94.94.94.19.19.19.cmml">w</mi><mrow id="S4.E13.m1.95.95.95.20.20.20.1" xref="S4.E13.m1.95.95.95.20.20.20.1.cmml"><mi id="S4.E13.m1.95.95.95.20.20.20.1.2" xref="S4.E13.m1.95.95.95.20.20.20.1.2.cmml">k</mi><mo id="S4.E13.m1.95.95.95.20.20.20.1.1" xref="S4.E13.m1.95.95.95.20.20.20.1.1.cmml">−</mo><mn id="S4.E13.m1.95.95.95.20.20.20.1.3" xref="S4.E13.m1.95.95.95.20.20.20.1.3.cmml">1</mn></mrow></msub><mo id="S4.E13.m1.96.96.96.21.21.21" xref="S4.E13.m1.208.208.16a.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E13.m1.97.97.97.22.22.22" xref="S4.E13.m1.97.97.97.22.22.22.cmml">+</mo><mrow id="S4.E13.m1.211.211.19.195.34.34.34.3" xref="S4.E13.m1.208.208.16.cmml"><mn id="S4.E13.m1.98.98.98.23.23.23" xref="S4.E13.m1.98.98.98.23.23.23.cmml">2</mn><mo id="S4.E13.m1.211.211.19.195.34.34.34.3.2" xref="S4.E13.m1.208.208.16a.cmml">⁢</mo><msub id="S4.E13.m1.211.211.19.195.34.34.34.3.3" xref="S4.E13.m1.208.208.16.cmml"><mi id="S4.E13.m1.99.99.99.24.24.24" xref="S4.E13.m1.99.99.99.24.24.24.cmml">η</mi><mi id="S4.E13.m1.100.100.100.25.25.25.1" xref="S4.E13.m1.100.100.100.25.25.25.1.cmml">k</mi></msub><mo id="S4.E13.m1.211.211.19.195.34.34.34.3.2a" xref="S4.E13.m1.208.208.16a.cmml">⁢</mo><msub id="S4.E13.m1.211.211.19.195.34.34.34.3.4" xref="S4.E13.m1.208.208.16.cmml"><mi id="S4.E13.m1.101.101.101.26.26.26" xref="S4.E13.m1.101.101.101.26.26.26.cmml">F</mi><mi id="S4.E13.m1.102.102.102.27.27.27.1" xref="S4.E13.m1.102.102.102.27.27.27.1.cmml">k</mi></msub><mo id="S4.E13.m1.211.211.19.195.34.34.34.3.2b" xref="S4.E13.m1.208.208.16a.cmml">⁢</mo><mrow id="S4.E13.m1.211.211.19.195.34.34.34.3.1.1" xref="S4.E13.m1.208.208.16.cmml"><mo id="S4.E13.m1.103.103.103.28.28.28" xref="S4.E13.m1.208.208.16a.cmml">(</mo><msup id="S4.E13.m1.211.211.19.195.34.34.34.3.1.1.1" xref="S4.E13.m1.208.208.16.cmml"><mi id="S4.E13.m1.104.104.104.29.29.29" xref="S4.E13.m1.104.104.104.29.29.29.cmml">w</mi><mo id="S4.E13.m1.105.105.105.30.30.30.1" xref="S4.E13.m1.105.105.105.30.30.30.1.cmml">∗</mo></msup><mo id="S4.E13.m1.106.106.106.31.31.31" xref="S4.E13.m1.208.208.16a.cmml">)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr id="S4.E13.m1.219.219.27g" xref="S4.E13.m1.208.208.16.cmml"><mtd class="ltx_align_right" columnalign="right" id="S4.E13.m1.219.219.27h" xref="S4.E13.m1.208.208.16.cmml"><mrow id="S4.E13.m1.214.214.22.198.35.35" xref="S4.E13.m1.208.208.16.cmml"><mrow id="S4.E13.m1.212.212.20.196.33.33.33" xref="S4.E13.m1.208.208.16.cmml"><mo id="S4.E13.m1.212.212.20.196.33.33.33a" xref="S4.E13.m1.208.208.16a.cmml">+</mo><mrow id="S4.E13.m1.212.212.20.196.33.33.33.1" xref="S4.E13.m1.208.208.16.cmml"><mi id="S4.E13.m1.108.108.108.2.2.2" xref="S4.E13.m1.108.108.108.2.2.2.cmml">μ</mi><mo id="S4.E13.m1.212.212.20.196.33.33.33.1.2" xref="S4.E13.m1.208.208.16a.cmml">⁢</mo><msub id="S4.E13.m1.212.212.20.196.33.33.33.1.3" xref="S4.E13.m1.208.208.16.cmml"><mi id="S4.E13.m1.109.109.109.3.3.3" xref="S4.E13.m1.109.109.109.3.3.3.cmml">η</mi><mi id="S4.E13.m1.110.110.110.4.4.4.1" xref="S4.E13.m1.110.110.110.4.4.4.1.cmml">k</mi></msub><mo id="S4.E13.m1.212.212.20.196.33.33.33.1.2a" xref="S4.E13.m1.208.208.16a.cmml">⁢</mo><msup id="S4.E13.m1.212.212.20.196.33.33.33.1.1" xref="S4.E13.m1.208.208.16.cmml"><mrow id="S4.E13.m1.212.212.20.196.33.33.33.1.1.1.1" xref="S4.E13.m1.208.208.16.cmml"><mo id="S4.E13.m1.111.111.111.5.5.5" stretchy="false" xref="S4.E13.m1.208.208.16a.cmml">‖</mo><mrow id="S4.E13.m1.212.212.20.196.33.33.33.1.1.1.1.1" xref="S4.E13.m1.208.208.16.cmml"><msub id="S4.E13.m1.212.212.20.196.33.33.33.1.1.1.1.1.1" xref="S4.E13.m1.208.208.16.cmml"><mi id="S4.E13.m1.112.112.112.6.6.6" xref="S4.E13.m1.112.112.112.6.6.6.cmml">w</mi><mrow id="S4.E13.m1.113.113.113.7.7.7.1" xref="S4.E13.m1.113.113.113.7.7.7.1.cmml"><mi id="S4.E13.m1.113.113.113.7.7.7.1.2" xref="S4.E13.m1.113.113.113.7.7.7.1.2.cmml">k</mi><mo id="S4.E13.m1.113.113.113.7.7.7.1.1" xref="S4.E13.m1.113.113.113.7.7.7.1.1.cmml">−</mo><mn id="S4.E13.m1.113.113.113.7.7.7.1.3" xref="S4.E13.m1.113.113.113.7.7.7.1.3.cmml">1</mn></mrow></msub><mo id="S4.E13.m1.114.114.114.8.8.8" xref="S4.E13.m1.114.114.114.8.8.8.cmml">−</mo><msup id="S4.E13.m1.212.212.20.196.33.33.33.1.1.1.1.1.2" xref="S4.E13.m1.208.208.16.cmml"><mi 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id="S4.E13.m1.208.208.16.16.5.2.1.1.3.1.cmml" xref="S4.E13.m1.219.219.27">subscript</csymbol><apply id="S4.E13.m1.208.208.16.16.5.2.1.1.3.2.cmml" xref="S4.E13.m1.219.219.27"><csymbol cd="ambiguous" id="S4.E13.m1.208.208.16.16.5.2.1.1.3.2.1.cmml" xref="S4.E13.m1.219.219.27">superscript</csymbol><ci id="S4.E13.m1.189.189.189.17.17.17.cmml" xref="S4.E13.m1.189.189.189.17.17.17">𝐹</ci><ci id="S4.E13.m1.190.190.190.18.18.18.1.cmml" xref="S4.E13.m1.190.190.190.18.18.18.1">∗</ci></apply><ci id="S4.E13.m1.191.191.191.19.19.19.1.cmml" xref="S4.E13.m1.191.191.191.19.19.19.1">𝑘</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E13.m1.219c">\begin{split}||w_{k-1}-\eta_{k}\bar{A}_{k}-w^{\ast}||^{2}\leq||w_{k-1}-w^{\ast% }||^{2}-2\eta_{k}[F_{k}\left(w_{k-1}\right)\\ -F_{k}\left(w^{\ast}\right)-\frac{\mu}{2}\parallel w_{k-1}-w^{\ast}\parallel^{% 2}]+(2\eta_{k})^{2}L\left(F_{k}\left(w_{k}\right)-F^{\ast}_{k}\right)\\ \leq||w_{k-1}-w^{\ast}||^{2}-2\eta_{k}F_{k}\left(w_{k-1}\right)+2\eta_{k}F_{k}% \left(w^{\ast}\right)\\ +\mu\eta_{k}\parallel w_{k-1}-w^{\ast}\parallel^{2}+(2\eta_{k})^{2}L\left(F_{k% }\left(w_{k}\right)-F^{\ast}_{k}\right)\\ =\left(1+\mu\eta_{k}\right)||w_{k-1}-w^{\ast}||^{2}-2\eta_{k}\left(F_{k}\left(% w_{k-1}\right)-F^{\ast}_{k}\right)\\ +(2\eta_{k})^{2}L\left(F_{k}\left(w_{k}\right)-F^{\ast}_{k}\right)\end{split}</annotation><annotation encoding="application/x-llamapun" id="S4.E13.m1.219d">start_ROW start_CELL | | italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ | | italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT [ italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL - italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) - divide start_ARG italic_μ end_ARG start_ARG 2 end_ARG ∥ italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] + ( 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_L ( italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL ≤ | | italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) + 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) end_CELL end_ROW start_ROW start_CELL + italic_μ italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_L ( italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL = ( 1 + italic_μ italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) | | italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL + ( 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_L ( italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(13)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS1.p7.1">Then, we let the <math alttext="J=-2\eta_{k}\left(F_{k}\left(w_{k-1}\right)-F^{\ast}_{k}\right)+(2\eta_{k})^{2% }L\left(F_{k}\left(w_{k}\right)-F^{\ast}_{k}\right))" class="ltx_math_unparsed" display="inline" id="S4.SS2.SSS1.p7.1.m1.1"><semantics id="S4.SS2.SSS1.p7.1.m1.1a"><mrow id="S4.SS2.SSS1.p7.1.m1.1b"><mi id="S4.SS2.SSS1.p7.1.m1.1.1">J</mi><mo id="S4.SS2.SSS1.p7.1.m1.1.2" rspace="0em">=</mo><mo id="S4.SS2.SSS1.p7.1.m1.1.3" lspace="0em">−</mo><mn id="S4.SS2.SSS1.p7.1.m1.1.4">2</mn><msub id="S4.SS2.SSS1.p7.1.m1.1.5"><mi id="S4.SS2.SSS1.p7.1.m1.1.5.2">η</mi><mi id="S4.SS2.SSS1.p7.1.m1.1.5.3">k</mi></msub><mrow id="S4.SS2.SSS1.p7.1.m1.1.6"><mo id="S4.SS2.SSS1.p7.1.m1.1.6.1">(</mo><msub id="S4.SS2.SSS1.p7.1.m1.1.6.2"><mi id="S4.SS2.SSS1.p7.1.m1.1.6.2.2">F</mi><mi id="S4.SS2.SSS1.p7.1.m1.1.6.2.3">k</mi></msub><mrow id="S4.SS2.SSS1.p7.1.m1.1.6.3"><mo id="S4.SS2.SSS1.p7.1.m1.1.6.3.1">(</mo><msub id="S4.SS2.SSS1.p7.1.m1.1.6.3.2"><mi id="S4.SS2.SSS1.p7.1.m1.1.6.3.2.2">w</mi><mrow id="S4.SS2.SSS1.p7.1.m1.1.6.3.2.3"><mi id="S4.SS2.SSS1.p7.1.m1.1.6.3.2.3.2">k</mi><mo id="S4.SS2.SSS1.p7.1.m1.1.6.3.2.3.1">−</mo><mn id="S4.SS2.SSS1.p7.1.m1.1.6.3.2.3.3">1</mn></mrow></msub><mo id="S4.SS2.SSS1.p7.1.m1.1.6.3.3">)</mo></mrow><mo id="S4.SS2.SSS1.p7.1.m1.1.6.4">−</mo><msubsup id="S4.SS2.SSS1.p7.1.m1.1.6.5"><mi id="S4.SS2.SSS1.p7.1.m1.1.6.5.2.2">F</mi><mi id="S4.SS2.SSS1.p7.1.m1.1.6.5.3">k</mi><mo id="S4.SS2.SSS1.p7.1.m1.1.6.5.2.3">∗</mo></msubsup><mo id="S4.SS2.SSS1.p7.1.m1.1.6.6">)</mo></mrow><mo id="S4.SS2.SSS1.p7.1.m1.1.7">+</mo><msup id="S4.SS2.SSS1.p7.1.m1.1.8"><mrow id="S4.SS2.SSS1.p7.1.m1.1.8.2"><mo id="S4.SS2.SSS1.p7.1.m1.1.8.2.1" stretchy="false">(</mo><mn id="S4.SS2.SSS1.p7.1.m1.1.8.2.2">2</mn><msub id="S4.SS2.SSS1.p7.1.m1.1.8.2.3"><mi id="S4.SS2.SSS1.p7.1.m1.1.8.2.3.2">η</mi><mi id="S4.SS2.SSS1.p7.1.m1.1.8.2.3.3">k</mi></msub><mo id="S4.SS2.SSS1.p7.1.m1.1.8.2.4" stretchy="false">)</mo></mrow><mn id="S4.SS2.SSS1.p7.1.m1.1.8.3">2</mn></msup><mi id="S4.SS2.SSS1.p7.1.m1.1.9">L</mi><mrow id="S4.SS2.SSS1.p7.1.m1.1.10"><mo id="S4.SS2.SSS1.p7.1.m1.1.10.1">(</mo><msub id="S4.SS2.SSS1.p7.1.m1.1.10.2"><mi id="S4.SS2.SSS1.p7.1.m1.1.10.2.2">F</mi><mi id="S4.SS2.SSS1.p7.1.m1.1.10.2.3">k</mi></msub><mrow id="S4.SS2.SSS1.p7.1.m1.1.10.3"><mo id="S4.SS2.SSS1.p7.1.m1.1.10.3.1">(</mo><msub id="S4.SS2.SSS1.p7.1.m1.1.10.3.2"><mi id="S4.SS2.SSS1.p7.1.m1.1.10.3.2.2">w</mi><mi id="S4.SS2.SSS1.p7.1.m1.1.10.3.2.3">k</mi></msub><mo id="S4.SS2.SSS1.p7.1.m1.1.10.3.3">)</mo></mrow><mo id="S4.SS2.SSS1.p7.1.m1.1.10.4">−</mo><msubsup id="S4.SS2.SSS1.p7.1.m1.1.10.5"><mi id="S4.SS2.SSS1.p7.1.m1.1.10.5.2.2">F</mi><mi id="S4.SS2.SSS1.p7.1.m1.1.10.5.3">k</mi><mo id="S4.SS2.SSS1.p7.1.m1.1.10.5.2.3">∗</mo></msubsup><mo id="S4.SS2.SSS1.p7.1.m1.1.10.6">)</mo></mrow><mo id="S4.SS2.SSS1.p7.1.m1.1.11" stretchy="false">)</mo></mrow><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p7.1.m1.1c">J=-2\eta_{k}\left(F_{k}\left(w_{k-1}\right)-F^{\ast}_{k}\right)+(2\eta_{k})^{2% }L\left(F_{k}\left(w_{k}\right)-F^{\ast}_{k}\right))</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p7.1.m1.1d">italic_J = - 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) + ( 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_L ( italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.SSS1.p8"> <p class="ltx_p" id="S4.SS2.SSS1.p8.9">Since <math alttext="F^{\ast}_{k}-F_{k-1}\leq F^{\ast}-F_{k-1}\leq F^{\ast}-F^{\ast}_{k}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p8.1.m1.1"><semantics id="S4.SS2.SSS1.p8.1.m1.1a"><mrow id="S4.SS2.SSS1.p8.1.m1.1.1" xref="S4.SS2.SSS1.p8.1.m1.1.1.cmml"><mrow id="S4.SS2.SSS1.p8.1.m1.1.1.2" xref="S4.SS2.SSS1.p8.1.m1.1.1.2.cmml"><msubsup id="S4.SS2.SSS1.p8.1.m1.1.1.2.2" xref="S4.SS2.SSS1.p8.1.m1.1.1.2.2.cmml"><mi id="S4.SS2.SSS1.p8.1.m1.1.1.2.2.2.2" xref="S4.SS2.SSS1.p8.1.m1.1.1.2.2.2.2.cmml">F</mi><mi id="S4.SS2.SSS1.p8.1.m1.1.1.2.2.3" xref="S4.SS2.SSS1.p8.1.m1.1.1.2.2.3.cmml">k</mi><mo id="S4.SS2.SSS1.p8.1.m1.1.1.2.2.2.3" xref="S4.SS2.SSS1.p8.1.m1.1.1.2.2.2.3.cmml">∗</mo></msubsup><mo id="S4.SS2.SSS1.p8.1.m1.1.1.2.1" xref="S4.SS2.SSS1.p8.1.m1.1.1.2.1.cmml">−</mo><msub id="S4.SS2.SSS1.p8.1.m1.1.1.2.3" xref="S4.SS2.SSS1.p8.1.m1.1.1.2.3.cmml"><mi id="S4.SS2.SSS1.p8.1.m1.1.1.2.3.2" xref="S4.SS2.SSS1.p8.1.m1.1.1.2.3.2.cmml">F</mi><mrow id="S4.SS2.SSS1.p8.1.m1.1.1.2.3.3" xref="S4.SS2.SSS1.p8.1.m1.1.1.2.3.3.cmml"><mi id="S4.SS2.SSS1.p8.1.m1.1.1.2.3.3.2" xref="S4.SS2.SSS1.p8.1.m1.1.1.2.3.3.2.cmml">k</mi><mo id="S4.SS2.SSS1.p8.1.m1.1.1.2.3.3.1" xref="S4.SS2.SSS1.p8.1.m1.1.1.2.3.3.1.cmml">−</mo><mn id="S4.SS2.SSS1.p8.1.m1.1.1.2.3.3.3" xref="S4.SS2.SSS1.p8.1.m1.1.1.2.3.3.3.cmml">1</mn></mrow></msub></mrow><mo id="S4.SS2.SSS1.p8.1.m1.1.1.3" xref="S4.SS2.SSS1.p8.1.m1.1.1.3.cmml">≤</mo><mrow id="S4.SS2.SSS1.p8.1.m1.1.1.4" xref="S4.SS2.SSS1.p8.1.m1.1.1.4.cmml"><msup id="S4.SS2.SSS1.p8.1.m1.1.1.4.2" xref="S4.SS2.SSS1.p8.1.m1.1.1.4.2.cmml"><mi id="S4.SS2.SSS1.p8.1.m1.1.1.4.2.2" xref="S4.SS2.SSS1.p8.1.m1.1.1.4.2.2.cmml">F</mi><mo id="S4.SS2.SSS1.p8.1.m1.1.1.4.2.3" 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href="https://arxiv.org/html/2503.11828v1#S4.SS2.SSS1.p8.1.m1.1.1.4.cmml" id="S4.SS2.SSS1.p8.1.m1.1.1d.cmml" xref="S4.SS2.SSS1.p8.1.m1.1.1"></share><apply id="S4.SS2.SSS1.p8.1.m1.1.1.6.cmml" xref="S4.SS2.SSS1.p8.1.m1.1.1.6"><minus id="S4.SS2.SSS1.p8.1.m1.1.1.6.1.cmml" xref="S4.SS2.SSS1.p8.1.m1.1.1.6.1"></minus><apply id="S4.SS2.SSS1.p8.1.m1.1.1.6.2.cmml" xref="S4.SS2.SSS1.p8.1.m1.1.1.6.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p8.1.m1.1.1.6.2.1.cmml" xref="S4.SS2.SSS1.p8.1.m1.1.1.6.2">superscript</csymbol><ci id="S4.SS2.SSS1.p8.1.m1.1.1.6.2.2.cmml" xref="S4.SS2.SSS1.p8.1.m1.1.1.6.2.2">𝐹</ci><ci id="S4.SS2.SSS1.p8.1.m1.1.1.6.2.3.cmml" xref="S4.SS2.SSS1.p8.1.m1.1.1.6.2.3">∗</ci></apply><apply id="S4.SS2.SSS1.p8.1.m1.1.1.6.3.cmml" xref="S4.SS2.SSS1.p8.1.m1.1.1.6.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p8.1.m1.1.1.6.3.1.cmml" xref="S4.SS2.SSS1.p8.1.m1.1.1.6.3">subscript</csymbol><apply id="S4.SS2.SSS1.p8.1.m1.1.1.6.3.2.cmml" xref="S4.SS2.SSS1.p8.1.m1.1.1.6.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p8.1.m1.1.1.6.3.2.1.cmml" xref="S4.SS2.SSS1.p8.1.m1.1.1.6.3">superscript</csymbol><ci id="S4.SS2.SSS1.p8.1.m1.1.1.6.3.2.2.cmml" xref="S4.SS2.SSS1.p8.1.m1.1.1.6.3.2.2">𝐹</ci><ci id="S4.SS2.SSS1.p8.1.m1.1.1.6.3.2.3.cmml" xref="S4.SS2.SSS1.p8.1.m1.1.1.6.3.2.3">∗</ci></apply><ci id="S4.SS2.SSS1.p8.1.m1.1.1.6.3.3.cmml" xref="S4.SS2.SSS1.p8.1.m1.1.1.6.3.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p8.1.m1.1c">F^{\ast}_{k}-F_{k-1}\leq F^{\ast}-F_{k-1}\leq F^{\ast}-F^{\ast}_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p8.1.m1.1d">italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_F start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ≤ italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT - italic_F start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ≤ italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, therefore, <math alttext="J\leq 2\eta_{t}\left(F^{\ast}-F^{\ast}_{k}\right)+2\eta^{2}_{t}L\left(F_{k}-F^% {\ast}_{k}\right)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p8.2.m2.2"><semantics id="S4.SS2.SSS1.p8.2.m2.2a"><mrow id="S4.SS2.SSS1.p8.2.m2.2.2" xref="S4.SS2.SSS1.p8.2.m2.2.2.cmml"><mi id="S4.SS2.SSS1.p8.2.m2.2.2.4" xref="S4.SS2.SSS1.p8.2.m2.2.2.4.cmml">J</mi><mo id="S4.SS2.SSS1.p8.2.m2.2.2.3" xref="S4.SS2.SSS1.p8.2.m2.2.2.3.cmml">≤</mo><mrow id="S4.SS2.SSS1.p8.2.m2.2.2.2" xref="S4.SS2.SSS1.p8.2.m2.2.2.2.cmml"><mrow id="S4.SS2.SSS1.p8.2.m2.1.1.1.1" xref="S4.SS2.SSS1.p8.2.m2.1.1.1.1.cmml"><mn id="S4.SS2.SSS1.p8.2.m2.1.1.1.1.3" xref="S4.SS2.SSS1.p8.2.m2.1.1.1.1.3.cmml">2</mn><mo id="S4.SS2.SSS1.p8.2.m2.1.1.1.1.2" xref="S4.SS2.SSS1.p8.2.m2.1.1.1.1.2.cmml">⁢</mo><msub id="S4.SS2.SSS1.p8.2.m2.1.1.1.1.4" xref="S4.SS2.SSS1.p8.2.m2.1.1.1.1.4.cmml"><mi 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xref="S4.SS2.SSS1.p8.2.m2.2.2.2.2.1.1.1.2.2">𝐹</ci><ci id="S4.SS2.SSS1.p8.2.m2.2.2.2.2.1.1.1.2.3.cmml" xref="S4.SS2.SSS1.p8.2.m2.2.2.2.2.1.1.1.2.3">𝑘</ci></apply><apply id="S4.SS2.SSS1.p8.2.m2.2.2.2.2.1.1.1.3.cmml" xref="S4.SS2.SSS1.p8.2.m2.2.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p8.2.m2.2.2.2.2.1.1.1.3.1.cmml" xref="S4.SS2.SSS1.p8.2.m2.2.2.2.2.1.1.1.3">subscript</csymbol><apply id="S4.SS2.SSS1.p8.2.m2.2.2.2.2.1.1.1.3.2.cmml" xref="S4.SS2.SSS1.p8.2.m2.2.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p8.2.m2.2.2.2.2.1.1.1.3.2.1.cmml" xref="S4.SS2.SSS1.p8.2.m2.2.2.2.2.1.1.1.3">superscript</csymbol><ci id="S4.SS2.SSS1.p8.2.m2.2.2.2.2.1.1.1.3.2.2.cmml" xref="S4.SS2.SSS1.p8.2.m2.2.2.2.2.1.1.1.3.2.2">𝐹</ci><ci id="S4.SS2.SSS1.p8.2.m2.2.2.2.2.1.1.1.3.2.3.cmml" xref="S4.SS2.SSS1.p8.2.m2.2.2.2.2.1.1.1.3.2.3">∗</ci></apply><ci id="S4.SS2.SSS1.p8.2.m2.2.2.2.2.1.1.1.3.3.cmml" xref="S4.SS2.SSS1.p8.2.m2.2.2.2.2.1.1.1.3.3">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p8.2.m2.2c">J\leq 2\eta_{t}\left(F^{\ast}-F^{\ast}_{k}\right)+2\eta^{2}_{t}L\left(F_{k}-F^% {\ast}_{k}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p8.2.m2.2d">italic_J ≤ 2 italic_η start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) + 2 italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_L ( italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math>. To bound <math alttext="(F_{k}-F^{\ast}_{k})" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p8.3.m3.1"><semantics id="S4.SS2.SSS1.p8.3.m3.1a"><mrow id="S4.SS2.SSS1.p8.3.m3.1.1.1" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.cmml"><mo id="S4.SS2.SSS1.p8.3.m3.1.1.1.2" stretchy="false" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS1.p8.3.m3.1.1.1.1" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.cmml"><msub id="S4.SS2.SSS1.p8.3.m3.1.1.1.1.2" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.2.cmml"><mi id="S4.SS2.SSS1.p8.3.m3.1.1.1.1.2.2" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.2.2.cmml">F</mi><mi id="S4.SS2.SSS1.p8.3.m3.1.1.1.1.2.3" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.2.3.cmml">k</mi></msub><mo id="S4.SS2.SSS1.p8.3.m3.1.1.1.1.1" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.1.cmml">−</mo><msubsup id="S4.SS2.SSS1.p8.3.m3.1.1.1.1.3" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.3.cmml"><mi id="S4.SS2.SSS1.p8.3.m3.1.1.1.1.3.2.2" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.3.2.2.cmml">F</mi><mi id="S4.SS2.SSS1.p8.3.m3.1.1.1.1.3.3" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.3.3.cmml">k</mi><mo id="S4.SS2.SSS1.p8.3.m3.1.1.1.1.3.2.3" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.3.2.3.cmml">∗</mo></msubsup></mrow><mo id="S4.SS2.SSS1.p8.3.m3.1.1.1.3" stretchy="false" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p8.3.m3.1b"><apply id="S4.SS2.SSS1.p8.3.m3.1.1.1.1.cmml" xref="S4.SS2.SSS1.p8.3.m3.1.1.1"><minus id="S4.SS2.SSS1.p8.3.m3.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.1"></minus><apply id="S4.SS2.SSS1.p8.3.m3.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p8.3.m3.1.1.1.1.2.1.cmml" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS1.p8.3.m3.1.1.1.1.2.2.cmml" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.2.2">𝐹</ci><ci id="S4.SS2.SSS1.p8.3.m3.1.1.1.1.2.3.cmml" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.2.3">𝑘</ci></apply><apply id="S4.SS2.SSS1.p8.3.m3.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p8.3.m3.1.1.1.1.3.1.cmml" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.3">subscript</csymbol><apply id="S4.SS2.SSS1.p8.3.m3.1.1.1.1.3.2.cmml" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p8.3.m3.1.1.1.1.3.2.1.cmml" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.3">superscript</csymbol><ci id="S4.SS2.SSS1.p8.3.m3.1.1.1.1.3.2.2.cmml" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.3.2.2">𝐹</ci><ci id="S4.SS2.SSS1.p8.3.m3.1.1.1.1.3.2.3.cmml" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.3.2.3">∗</ci></apply><ci id="S4.SS2.SSS1.p8.3.m3.1.1.1.1.3.3.cmml" xref="S4.SS2.SSS1.p8.3.m3.1.1.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p8.3.m3.1c">(F_{k}-F^{\ast}_{k})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p8.3.m3.1d">( italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math>, since <math alttext="F_{k}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p8.4.m4.1"><semantics id="S4.SS2.SSS1.p8.4.m4.1a"><msub id="S4.SS2.SSS1.p8.4.m4.1.1" xref="S4.SS2.SSS1.p8.4.m4.1.1.cmml"><mi id="S4.SS2.SSS1.p8.4.m4.1.1.2" xref="S4.SS2.SSS1.p8.4.m4.1.1.2.cmml">F</mi><mi id="S4.SS2.SSS1.p8.4.m4.1.1.3" xref="S4.SS2.SSS1.p8.4.m4.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p8.4.m4.1b"><apply id="S4.SS2.SSS1.p8.4.m4.1.1.cmml" xref="S4.SS2.SSS1.p8.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p8.4.m4.1.1.1.cmml" xref="S4.SS2.SSS1.p8.4.m4.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p8.4.m4.1.1.2.cmml" xref="S4.SS2.SSS1.p8.4.m4.1.1.2">𝐹</ci><ci id="S4.SS2.SSS1.p8.4.m4.1.1.3.cmml" xref="S4.SS2.SSS1.p8.4.m4.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p8.4.m4.1c">F_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p8.4.m4.1d">italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is <math alttext="L" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p8.5.m5.1"><semantics id="S4.SS2.SSS1.p8.5.m5.1a"><mi id="S4.SS2.SSS1.p8.5.m5.1.1" xref="S4.SS2.SSS1.p8.5.m5.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p8.5.m5.1b"><ci id="S4.SS2.SSS1.p8.5.m5.1.1.cmml" xref="S4.SS2.SSS1.p8.5.m5.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p8.5.m5.1c">L</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p8.5.m5.1d">italic_L</annotation></semantics></math>-smooth, we can get <math alttext="\left(F_{k}-F^{\ast}_{k}\right)\leq\frac{L}{2}\parallel w_{k}-W^{\ast}% \parallel^{2}+\left(w_{k}-W^{\ast}\right)^{T}\nabla F_{k}\left(W^{\ast}\right)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p8.6.m6.4"><semantics id="S4.SS2.SSS1.p8.6.m6.4a"><mrow id="S4.SS2.SSS1.p8.6.m6.4.4" xref="S4.SS2.SSS1.p8.6.m6.4.4.cmml"><mrow id="S4.SS2.SSS1.p8.6.m6.1.1.1.1" xref="S4.SS2.SSS1.p8.6.m6.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS1.p8.6.m6.1.1.1.1.2" xref="S4.SS2.SSS1.p8.6.m6.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS1.p8.6.m6.1.1.1.1.1" xref="S4.SS2.SSS1.p8.6.m6.1.1.1.1.1.cmml"><msub id="S4.SS2.SSS1.p8.6.m6.1.1.1.1.1.2" xref="S4.SS2.SSS1.p8.6.m6.1.1.1.1.1.2.cmml"><mi id="S4.SS2.SSS1.p8.6.m6.1.1.1.1.1.2.2" xref="S4.SS2.SSS1.p8.6.m6.1.1.1.1.1.2.2.cmml">F</mi><mi id="S4.SS2.SSS1.p8.6.m6.1.1.1.1.1.2.3" xref="S4.SS2.SSS1.p8.6.m6.1.1.1.1.1.2.3.cmml">k</mi></msub><mo id="S4.SS2.SSS1.p8.6.m6.1.1.1.1.1.1" xref="S4.SS2.SSS1.p8.6.m6.1.1.1.1.1.1.cmml">−</mo><msubsup id="S4.SS2.SSS1.p8.6.m6.1.1.1.1.1.3" xref="S4.SS2.SSS1.p8.6.m6.1.1.1.1.1.3.cmml"><mi id="S4.SS2.SSS1.p8.6.m6.1.1.1.1.1.3.2.2" xref="S4.SS2.SSS1.p8.6.m6.1.1.1.1.1.3.2.2.cmml">F</mi><mi id="S4.SS2.SSS1.p8.6.m6.1.1.1.1.1.3.3" xref="S4.SS2.SSS1.p8.6.m6.1.1.1.1.1.3.3.cmml">k</mi><mo id="S4.SS2.SSS1.p8.6.m6.1.1.1.1.1.3.2.3" xref="S4.SS2.SSS1.p8.6.m6.1.1.1.1.1.3.2.3.cmml">∗</mo></msubsup></mrow><mo id="S4.SS2.SSS1.p8.6.m6.1.1.1.1.3" xref="S4.SS2.SSS1.p8.6.m6.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS2.SSS1.p8.6.m6.4.4.5" xref="S4.SS2.SSS1.p8.6.m6.4.4.5.cmml">≤</mo><mrow id="S4.SS2.SSS1.p8.6.m6.4.4.4" xref="S4.SS2.SSS1.p8.6.m6.4.4.4.cmml"><mrow id="S4.SS2.SSS1.p8.6.m6.2.2.2.1" xref="S4.SS2.SSS1.p8.6.m6.2.2.2.1.cmml"><mfrac id="S4.SS2.SSS1.p8.6.m6.2.2.2.1.3" xref="S4.SS2.SSS1.p8.6.m6.2.2.2.1.3.cmml"><mi id="S4.SS2.SSS1.p8.6.m6.2.2.2.1.3.2" xref="S4.SS2.SSS1.p8.6.m6.2.2.2.1.3.2.cmml">L</mi><mn id="S4.SS2.SSS1.p8.6.m6.2.2.2.1.3.3" xref="S4.SS2.SSS1.p8.6.m6.2.2.2.1.3.3.cmml">2</mn></mfrac><mo id="S4.SS2.SSS1.p8.6.m6.2.2.2.1.2" xref="S4.SS2.SSS1.p8.6.m6.2.2.2.1.2.cmml">⁢</mo><msup id="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1" xref="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1.cmml"><mrow id="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1.1.1" xref="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1.1.2.cmml"><mo id="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1.1.2.1.cmml">‖</mo><mrow id="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1.1.1.1" xref="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1.1.1.1.cmml"><msub id="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1.1.1.1.2" xref="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1.1.1.1.2.cmml"><mi id="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1.1.1.1.2.2" xref="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1.1.1.1.2.2.cmml">w</mi><mi id="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1.1.1.1.2.3" xref="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1.1.1.1.2.3.cmml">k</mi></msub><mo id="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1.1.1.1.1" xref="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1.1.1.1.1.cmml">−</mo><msup id="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1.1.1.1.3" xref="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1.1.1.1.3.cmml"><mi id="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1.1.1.1.3.2" xref="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1.1.1.1.3.2.cmml">W</mi><mo id="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1.1.1.1.3.3" xref="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1.1.1.1.3.3.cmml">∗</mo></msup></mrow><mo id="S4.SS2.SSS1.p8.6.m6.2.2.2.1.1.1.1.3" stretchy="false" 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xref="S4.SS2.SSS1.p8.6.m6.3.3.3.2.1.1.1.1.2.3.cmml">k</mi></msub><mo id="S4.SS2.SSS1.p8.6.m6.3.3.3.2.1.1.1.1.1" xref="S4.SS2.SSS1.p8.6.m6.3.3.3.2.1.1.1.1.1.cmml">−</mo><msup id="S4.SS2.SSS1.p8.6.m6.3.3.3.2.1.1.1.1.3" xref="S4.SS2.SSS1.p8.6.m6.3.3.3.2.1.1.1.1.3.cmml"><mi id="S4.SS2.SSS1.p8.6.m6.3.3.3.2.1.1.1.1.3.2" xref="S4.SS2.SSS1.p8.6.m6.3.3.3.2.1.1.1.1.3.2.cmml">W</mi><mo id="S4.SS2.SSS1.p8.6.m6.3.3.3.2.1.1.1.1.3.3" xref="S4.SS2.SSS1.p8.6.m6.3.3.3.2.1.1.1.1.3.3.cmml">∗</mo></msup></mrow><mo id="S4.SS2.SSS1.p8.6.m6.3.3.3.2.1.1.1.3" xref="S4.SS2.SSS1.p8.6.m6.3.3.3.2.1.1.1.1.cmml">)</mo></mrow><mi id="S4.SS2.SSS1.p8.6.m6.3.3.3.2.1.3" xref="S4.SS2.SSS1.p8.6.m6.3.3.3.2.1.3.cmml">T</mi></msup><mo id="S4.SS2.SSS1.p8.6.m6.4.4.4.3.3" lspace="0.167em" xref="S4.SS2.SSS1.p8.6.m6.4.4.4.3.3.cmml">⁢</mo><mrow id="S4.SS2.SSS1.p8.6.m6.4.4.4.3.4" xref="S4.SS2.SSS1.p8.6.m6.4.4.4.3.4.cmml"><mo id="S4.SS2.SSS1.p8.6.m6.4.4.4.3.4.1" rspace="0.167em" xref="S4.SS2.SSS1.p8.6.m6.4.4.4.3.4.1.cmml">∇</mo><msub 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id="S4.SS2.SSS1.p8.6.m6.4.4.4.3.4.2.1.cmml" xref="S4.SS2.SSS1.p8.6.m6.4.4.4.3.4.2">subscript</csymbol><ci id="S4.SS2.SSS1.p8.6.m6.4.4.4.3.4.2.2.cmml" xref="S4.SS2.SSS1.p8.6.m6.4.4.4.3.4.2.2">𝐹</ci><ci id="S4.SS2.SSS1.p8.6.m6.4.4.4.3.4.2.3.cmml" xref="S4.SS2.SSS1.p8.6.m6.4.4.4.3.4.2.3">𝑘</ci></apply></apply><apply id="S4.SS2.SSS1.p8.6.m6.4.4.4.3.2.1.1.cmml" xref="S4.SS2.SSS1.p8.6.m6.4.4.4.3.2.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p8.6.m6.4.4.4.3.2.1.1.1.cmml" xref="S4.SS2.SSS1.p8.6.m6.4.4.4.3.2.1">superscript</csymbol><ci id="S4.SS2.SSS1.p8.6.m6.4.4.4.3.2.1.1.2.cmml" xref="S4.SS2.SSS1.p8.6.m6.4.4.4.3.2.1.1.2">𝑊</ci><ci id="S4.SS2.SSS1.p8.6.m6.4.4.4.3.2.1.1.3.cmml" xref="S4.SS2.SSS1.p8.6.m6.4.4.4.3.2.1.1.3">∗</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p8.6.m6.4c">\left(F_{k}-F^{\ast}_{k}\right)\leq\frac{L}{2}\parallel w_{k}-W^{\ast}% \parallel^{2}+\left(w_{k}-W^{\ast}\right)^{T}\nabla F_{k}\left(W^{\ast}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p8.6.m6.4d">( italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ≤ divide start_ARG italic_L end_ARG start_ARG 2 end_ARG ∥ italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_W start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_W start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ∇ italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_W start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math>. Since <math alttext="\nabla F_{k}\left(W^{\ast}\right)=0" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p8.7.m7.1"><semantics id="S4.SS2.SSS1.p8.7.m7.1a"><mrow id="S4.SS2.SSS1.p8.7.m7.1.1" xref="S4.SS2.SSS1.p8.7.m7.1.1.cmml"><mrow id="S4.SS2.SSS1.p8.7.m7.1.1.1" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.cmml"><mrow id="S4.SS2.SSS1.p8.7.m7.1.1.1.3" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.3.cmml"><mo id="S4.SS2.SSS1.p8.7.m7.1.1.1.3.1" rspace="0.167em" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.3.1.cmml">∇</mo><msub id="S4.SS2.SSS1.p8.7.m7.1.1.1.3.2" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.3.2.cmml"><mi id="S4.SS2.SSS1.p8.7.m7.1.1.1.3.2.2" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.3.2.2.cmml">F</mi><mi id="S4.SS2.SSS1.p8.7.m7.1.1.1.3.2.3" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.3.2.3.cmml">k</mi></msub></mrow><mo id="S4.SS2.SSS1.p8.7.m7.1.1.1.2" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS1.p8.7.m7.1.1.1.1.1" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS1.p8.7.m7.1.1.1.1.1.2" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.1.1.1.cmml">(</mo><msup id="S4.SS2.SSS1.p8.7.m7.1.1.1.1.1.1" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.p8.7.m7.1.1.1.1.1.1.2" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.1.1.1.2.cmml">W</mi><mo id="S4.SS2.SSS1.p8.7.m7.1.1.1.1.1.1.3" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S4.SS2.SSS1.p8.7.m7.1.1.1.1.1.3" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS1.p8.7.m7.1.1.2" xref="S4.SS2.SSS1.p8.7.m7.1.1.2.cmml">=</mo><mn id="S4.SS2.SSS1.p8.7.m7.1.1.3" xref="S4.SS2.SSS1.p8.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p8.7.m7.1b"><apply id="S4.SS2.SSS1.p8.7.m7.1.1.cmml" xref="S4.SS2.SSS1.p8.7.m7.1.1"><eq id="S4.SS2.SSS1.p8.7.m7.1.1.2.cmml" xref="S4.SS2.SSS1.p8.7.m7.1.1.2"></eq><apply id="S4.SS2.SSS1.p8.7.m7.1.1.1.cmml" xref="S4.SS2.SSS1.p8.7.m7.1.1.1"><times id="S4.SS2.SSS1.p8.7.m7.1.1.1.2.cmml" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.2"></times><apply id="S4.SS2.SSS1.p8.7.m7.1.1.1.3.cmml" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.3"><ci id="S4.SS2.SSS1.p8.7.m7.1.1.1.3.1.cmml" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.3.1">∇</ci><apply id="S4.SS2.SSS1.p8.7.m7.1.1.1.3.2.cmml" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p8.7.m7.1.1.1.3.2.1.cmml" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.3.2">subscript</csymbol><ci id="S4.SS2.SSS1.p8.7.m7.1.1.1.3.2.2.cmml" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.3.2.2">𝐹</ci><ci id="S4.SS2.SSS1.p8.7.m7.1.1.1.3.2.3.cmml" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.3.2.3">𝑘</ci></apply></apply><apply id="S4.SS2.SSS1.p8.7.m7.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p8.7.m7.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS1.p8.7.m7.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.1.1.1.2">𝑊</ci><ci id="S4.SS2.SSS1.p8.7.m7.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.p8.7.m7.1.1.1.1.1.1.3">∗</ci></apply></apply><cn id="S4.SS2.SSS1.p8.7.m7.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p8.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p8.7.m7.1c">\nabla F_{k}\left(W^{\ast}\right)=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p8.7.m7.1d">∇ italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_W start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) = 0</annotation></semantics></math>, we have: <math alttext="J\leq 2\eta_{k}Z+\eta^{2}_{k}L^{2}\parallel w_{k}-w^{\ast}\parallel^{2}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p8.8.m8.1"><semantics id="S4.SS2.SSS1.p8.8.m8.1a"><mrow id="S4.SS2.SSS1.p8.8.m8.1.1" xref="S4.SS2.SSS1.p8.8.m8.1.1.cmml"><mi id="S4.SS2.SSS1.p8.8.m8.1.1.3" xref="S4.SS2.SSS1.p8.8.m8.1.1.3.cmml">J</mi><mo id="S4.SS2.SSS1.p8.8.m8.1.1.2" xref="S4.SS2.SSS1.p8.8.m8.1.1.2.cmml">≤</mo><mrow id="S4.SS2.SSS1.p8.8.m8.1.1.1" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.cmml"><mrow id="S4.SS2.SSS1.p8.8.m8.1.1.1.3" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.3.cmml"><mn id="S4.SS2.SSS1.p8.8.m8.1.1.1.3.2" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.3.2.cmml">2</mn><mo id="S4.SS2.SSS1.p8.8.m8.1.1.1.3.1" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.3.1.cmml">⁢</mo><msub id="S4.SS2.SSS1.p8.8.m8.1.1.1.3.3" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.3.3.cmml"><mi id="S4.SS2.SSS1.p8.8.m8.1.1.1.3.3.2" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.3.3.2.cmml">η</mi><mi id="S4.SS2.SSS1.p8.8.m8.1.1.1.3.3.3" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.3.3.3.cmml">k</mi></msub><mo id="S4.SS2.SSS1.p8.8.m8.1.1.1.3.1a" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.3.1.cmml">⁢</mo><mi id="S4.SS2.SSS1.p8.8.m8.1.1.1.3.4" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.3.4.cmml">Z</mi></mrow><mo id="S4.SS2.SSS1.p8.8.m8.1.1.1.2" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.2.cmml">+</mo><mrow id="S4.SS2.SSS1.p8.8.m8.1.1.1.1" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.cmml"><msubsup id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.3" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.3.cmml"><mi id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.3.2.2" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.3.2.2.cmml">η</mi><mi id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.3.3" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.3.3.cmml">k</mi><mn id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.3.2.3" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.3.2.3.cmml">2</mn></msubsup><mo id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.2" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.2.cmml">⁢</mo><msup id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.4" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.4.cmml"><mi id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.4.2" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.4.2.cmml">L</mi><mn id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.4.3" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.4.3.cmml">2</mn></msup><mo id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.2a" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.2.cmml">⁢</mo><msup id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.cmml"><mrow id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.2.cmml"><mo id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.1" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.1.cmml"><msub id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.1.2" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.1.2.cmml"><mi id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.1.2.2" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.1.2.2.cmml">w</mi><mi id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.1.2.3" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.1.2.3.cmml">k</mi></msub><mo id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.1.1" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.1.1.cmml">−</mo><msup id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.1.3" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.1.3.cmml"><mi id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.1.3.2" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.1.3.2.cmml">w</mi><mo id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.1.3.3" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.1.3.3.cmml">∗</mo></msup></mrow><mo id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mn 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id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.1.3.1.cmml" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.1.3">superscript</csymbol><ci id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.1.3.2.cmml" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.1.3.2">𝑤</ci><ci id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.1.3.3.cmml" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.1.1.1.3.3">∗</ci></apply></apply></apply><cn id="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p8.8.m8.1.1.1.1.1.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p8.8.m8.1c">J\leq 2\eta_{k}Z+\eta^{2}_{k}L^{2}\parallel w_{k}-w^{\ast}\parallel^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p8.8.m8.1d">italic_J ≤ 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_Z + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, and Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4.E13" title="In IV-B1 Continuous Linear ‣ IV-B Convergence Analysis ‣ IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">13</span></a> becomes <math alttext="\left(1+\mu\eta_{k}\right)||w_{k-1}-w^{\ast}||^{2}+2\eta_{k}Z+\eta^{2}_{k}L^{2% }\parallel w_{k}-w^{\ast}\parallel^{2}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p8.9.m9.3"><semantics id="S4.SS2.SSS1.p8.9.m9.3a"><mrow id="S4.SS2.SSS1.p8.9.m9.3.3" xref="S4.SS2.SSS1.p8.9.m9.3.3.cmml"><mrow id="S4.SS2.SSS1.p8.9.m9.2.2.2" xref="S4.SS2.SSS1.p8.9.m9.2.2.2.cmml"><mrow id="S4.SS2.SSS1.p8.9.m9.1.1.1.1.1" xref="S4.SS2.SSS1.p8.9.m9.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS1.p8.9.m9.1.1.1.1.1.2" 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id="S4.SS2.SSS1.p8.9.m9.3.3.3.1.1.1.1.3.3.cmml" xref="S4.SS2.SSS1.p8.9.m9.3.3.3.1.1.1.1.3.3">∗</ci></apply></apply></apply><cn id="S4.SS2.SSS1.p8.9.m9.3.3.3.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p8.9.m9.3.3.3.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p8.9.m9.3c">\left(1+\mu\eta_{k}\right)||w_{k-1}-w^{\ast}||^{2}+2\eta_{k}Z+\eta^{2}_{k}L^{2% }\parallel w_{k}-w^{\ast}\parallel^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p8.9.m9.3d">( 1 + italic_μ italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) | | italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_Z + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. Therefore, Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4.E8" title="In IV-B1 Continuous Linear ‣ IV-B Convergence Analysis ‣ IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">8</span></a> becomes:</p> </div> <div class="ltx_para" id="S4.SS2.SSS1.p9"> <table class="ltx_equation ltx_eqn_table" id="S4.E14"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{split}\parallel w_{k}-w^{\ast}\parallel^{2}\leq\left(1+\mu\eta_{k}% \right)||w_{k-1}-w^{\ast}||^{2}\\ +2\eta_{k}Z+\eta^{2}_{k}L^{2}\parallel w_{k}-w^{\ast}\parallel^{2}+\eta^{2}_{t% }\parallel A_{t}-\bar{A}_{t}\parallel^{2}\end{split}" class="ltx_Math" display="block" id="S4.E14.m1.67"><semantics id="S4.E14.m1.67a"><mtable displaystyle="true" id="S4.E14.m1.67.67.10" rowspacing="0pt" 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encoding="application/x-tex" id="S4.E14.m1.67c">\begin{split}\parallel w_{k}-w^{\ast}\parallel^{2}\leq\left(1+\mu\eta_{k}% \right)||w_{k-1}-w^{\ast}||^{2}\\ +2\eta_{k}Z+\eta^{2}_{k}L^{2}\parallel w_{k}-w^{\ast}\parallel^{2}+\eta^{2}_{t% }\parallel A_{t}-\bar{A}_{t}\parallel^{2}\end{split}</annotation><annotation encoding="application/x-llamapun" id="S4.E14.m1.67d">start_ROW start_CELL ∥ italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ ( 1 + italic_μ italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) | | italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL + 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_Z + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∥ italic_A start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT - over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(14)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS1.p10"> <p class="ltx_p" id="S4.SS2.SSS1.p10.1">By taking expectations on both sides of Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4.E14" title="In IV-B1 Continuous Linear ‣ IV-B Convergence Analysis ‣ IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">14</span></a> we get:</p> <table class="ltx_equation ltx_eqn_table" id="S4.E15"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{split}E\parallel w_{k}-w^{\ast}\parallel^{2}\leq\left(1+\mu\eta_{k}% \right)E||w_{k-1}-w^{\ast}||^{2}\\ +2\eta_{k}Z+\eta^{2}_{k}L^{2}E\parallel w_{k}-w^{\ast}\parallel^{2}+\eta^{2}_{% k}E\parallel A_{k}-\bar{A}_{k}\parallel^{2}\end{split}" class="ltx_Math" display="block" id="S4.E15.m1.71"><semantics id="S4.E15.m1.71a"><mtable displaystyle="true" id="S4.E15.m1.71.71.10" rowspacing="0pt" xref="S4.E15.m1.66.66.5.cmml"><mtr id="S4.E15.m1.71.71.10a" xref="S4.E15.m1.66.66.5.cmml"><mtd class="ltx_align_right" columnalign="right" 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xref="S4.E15.m1.59.59.59.31.31.31.1">𝑘</ci></apply></apply></apply><cn id="S4.E15.m1.61.61.61.33.33.33.1.cmml" type="integer" xref="S4.E15.m1.61.61.61.33.33.33.1">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E15.m1.71c">\begin{split}E\parallel w_{k}-w^{\ast}\parallel^{2}\leq\left(1+\mu\eta_{k}% \right)E||w_{k-1}-w^{\ast}||^{2}\\ +2\eta_{k}Z+\eta^{2}_{k}L^{2}E\parallel w_{k}-w^{\ast}\parallel^{2}+\eta^{2}_{% k}E\parallel A_{k}-\bar{A}_{k}\parallel^{2}\end{split}</annotation><annotation encoding="application/x-llamapun" id="S4.E15.m1.71d">start_ROW start_CELL italic_E ∥ italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ ( 1 + italic_μ italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) italic_E | | italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL + 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_Z + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_E ∥ italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_E ∥ italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(15)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS1.p11"> <p class="ltx_p" id="S4.SS2.SSS1.p11.2">We next bond the <math alttext="\eta^{2}_{t}E\parallel A_{t}-\bar{A}_{t}\parallel^{2}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p11.1.m1.1"><semantics id="S4.SS2.SSS1.p11.1.m1.1a"><mrow id="S4.SS2.SSS1.p11.1.m1.1.1" xref="S4.SS2.SSS1.p11.1.m1.1.1.cmml"><msubsup id="S4.SS2.SSS1.p11.1.m1.1.1.3" xref="S4.SS2.SSS1.p11.1.m1.1.1.3.cmml"><mi id="S4.SS2.SSS1.p11.1.m1.1.1.3.2.2" xref="S4.SS2.SSS1.p11.1.m1.1.1.3.2.2.cmml">η</mi><mi id="S4.SS2.SSS1.p11.1.m1.1.1.3.3" xref="S4.SS2.SSS1.p11.1.m1.1.1.3.3.cmml">t</mi><mn id="S4.SS2.SSS1.p11.1.m1.1.1.3.2.3" xref="S4.SS2.SSS1.p11.1.m1.1.1.3.2.3.cmml">2</mn></msubsup><mo id="S4.SS2.SSS1.p11.1.m1.1.1.2" xref="S4.SS2.SSS1.p11.1.m1.1.1.2.cmml">⁢</mo><mi id="S4.SS2.SSS1.p11.1.m1.1.1.4" xref="S4.SS2.SSS1.p11.1.m1.1.1.4.cmml">E</mi><mo id="S4.SS2.SSS1.p11.1.m1.1.1.2a" xref="S4.SS2.SSS1.p11.1.m1.1.1.2.cmml">⁢</mo><msup id="S4.SS2.SSS1.p11.1.m1.1.1.1" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.cmml"><mrow id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.2.cmml"><mo id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.2.1.cmml">‖</mo><mrow id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.cmml"><msub id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.2" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.2.cmml"><mi id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.2.2" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.2.2.cmml">A</mi><mi id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.2.3" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.2.3.cmml">t</mi></msub><mo id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.1" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.1.cmml">−</mo><msub id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.3" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.3.cmml"><mover accent="true" id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.3.2" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.3.2.cmml"><mi id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.3.2.2" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.3.2.2.cmml">A</mi><mo id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.3.2.1" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.3.2.1.cmml">¯</mo></mover><mi id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.3.3" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.3.3.cmml">t</mi></msub></mrow><mo id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S4.SS2.SSS1.p11.1.m1.1.1.1.3" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p11.1.m1.1b"><apply id="S4.SS2.SSS1.p11.1.m1.1.1.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1"><times id="S4.SS2.SSS1.p11.1.m1.1.1.2.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.2"></times><apply id="S4.SS2.SSS1.p11.1.m1.1.1.3.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p11.1.m1.1.1.3.1.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.3">subscript</csymbol><apply id="S4.SS2.SSS1.p11.1.m1.1.1.3.2.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p11.1.m1.1.1.3.2.1.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.3">superscript</csymbol><ci id="S4.SS2.SSS1.p11.1.m1.1.1.3.2.2.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.3.2.2">𝜂</ci><cn id="S4.SS2.SSS1.p11.1.m1.1.1.3.2.3.cmml" type="integer" xref="S4.SS2.SSS1.p11.1.m1.1.1.3.2.3">2</cn></apply><ci id="S4.SS2.SSS1.p11.1.m1.1.1.3.3.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.3.3">𝑡</ci></apply><ci id="S4.SS2.SSS1.p11.1.m1.1.1.4.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.4">𝐸</ci><apply id="S4.SS2.SSS1.p11.1.m1.1.1.1.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p11.1.m1.1.1.1.2.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.1">superscript</csymbol><apply id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1"><csymbol cd="latexml" id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.2.1.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.2">norm</csymbol><apply id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1"><minus id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.1"></minus><apply id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.2.1.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.2.2.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.2.2">𝐴</ci><ci id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.2.3.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.2.3">𝑡</ci></apply><apply id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.3.1.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.3">subscript</csymbol><apply id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.3.2.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.3.2"><ci id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.3.2.1.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.3.2.1">¯</ci><ci id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.3.2.2.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.3.2.2">𝐴</ci></apply><ci id="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.3.3.cmml" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply><cn id="S4.SS2.SSS1.p11.1.m1.1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p11.1.m1.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p11.1.m1.1c">\eta^{2}_{t}E\parallel A_{t}-\bar{A}_{t}\parallel^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p11.1.m1.1d">italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_E ∥ italic_A start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT - over¯ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, by the assumption, <math alttext="\eta^{2}_{t}E\parallel A_{t}-\bar{A}_{t}\parallel^{2}\leq\eta^{2}_{t}\sigma^{2% }_{k}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p11.2.m2.1"><semantics id="S4.SS2.SSS1.p11.2.m2.1a"><mrow id="S4.SS2.SSS1.p11.2.m2.1.1" xref="S4.SS2.SSS1.p11.2.m2.1.1.cmml"><mrow id="S4.SS2.SSS1.p11.2.m2.1.1.1" xref="S4.SS2.SSS1.p11.2.m2.1.1.1.cmml"><msubsup id="S4.SS2.SSS1.p11.2.m2.1.1.1.3" xref="S4.SS2.SSS1.p11.2.m2.1.1.1.3.cmml"><mi id="S4.SS2.SSS1.p11.2.m2.1.1.1.3.2.2" xref="S4.SS2.SSS1.p11.2.m2.1.1.1.3.2.2.cmml">η</mi><mi id="S4.SS2.SSS1.p11.2.m2.1.1.1.3.3" xref="S4.SS2.SSS1.p11.2.m2.1.1.1.3.3.cmml">t</mi><mn id="S4.SS2.SSS1.p11.2.m2.1.1.1.3.2.3" xref="S4.SS2.SSS1.p11.2.m2.1.1.1.3.2.3.cmml">2</mn></msubsup><mo id="S4.SS2.SSS1.p11.2.m2.1.1.1.2" xref="S4.SS2.SSS1.p11.2.m2.1.1.1.2.cmml">⁢</mo><mi id="S4.SS2.SSS1.p11.2.m2.1.1.1.4" xref="S4.SS2.SSS1.p11.2.m2.1.1.1.4.cmml">E</mi><mo id="S4.SS2.SSS1.p11.2.m2.1.1.1.2a" xref="S4.SS2.SSS1.p11.2.m2.1.1.1.2.cmml">⁢</mo><msup id="S4.SS2.SSS1.p11.2.m2.1.1.1.1" xref="S4.SS2.SSS1.p11.2.m2.1.1.1.1.cmml"><mrow id="S4.SS2.SSS1.p11.2.m2.1.1.1.1.1.1" xref="S4.SS2.SSS1.p11.2.m2.1.1.1.1.1.2.cmml"><mo id="S4.SS2.SSS1.p11.2.m2.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS1.p11.2.m2.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="S4.SS2.SSS1.p11.2.m2.1.1.1.1.1.1.1" xref="S4.SS2.SSS1.p11.2.m2.1.1.1.1.1.1.1.cmml"><msub id="S4.SS2.SSS1.p11.2.m2.1.1.1.1.1.1.1.2" xref="S4.SS2.SSS1.p11.2.m2.1.1.1.1.1.1.1.2.cmml"><mi id="S4.SS2.SSS1.p11.2.m2.1.1.1.1.1.1.1.2.2" xref="S4.SS2.SSS1.p11.2.m2.1.1.1.1.1.1.1.2.2.cmml">A</mi><mi id="S4.SS2.SSS1.p11.2.m2.1.1.1.1.1.1.1.2.3" xref="S4.SS2.SSS1.p11.2.m2.1.1.1.1.1.1.1.2.3.cmml">t</mi></msub><mo id="S4.SS2.SSS1.p11.2.m2.1.1.1.1.1.1.1.1" xref="S4.SS2.SSS1.p11.2.m2.1.1.1.1.1.1.1.1.cmml">−</mo><msub id="S4.SS2.SSS1.p11.2.m2.1.1.1.1.1.1.1.3" xref="S4.SS2.SSS1.p11.2.m2.1.1.1.1.1.1.1.3.cmml"><mover accent="true" 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start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.SSS1.p12"> <p class="ltx_p" id="S4.SS2.SSS1.p12.1">Thus, we can see that in the final Function <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4.Ex9" title="IV-B1 Continuous Linear ‣ IV-B Convergence Analysis ‣ IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag"><span class="ltx_text">IV-B</span>1</span></a>, if the <math alttext="Z" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p12.1.m1.1"><semantics id="S4.SS2.SSS1.p12.1.m1.1a"><mi id="S4.SS2.SSS1.p12.1.m1.1.1" xref="S4.SS2.SSS1.p12.1.m1.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p12.1.m1.1b"><ci id="S4.SS2.SSS1.p12.1.m1.1.1.cmml" xref="S4.SS2.SSS1.p12.1.m1.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p12.1.m1.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p12.1.m1.1d">italic_Z</annotation></semantics></math> (degree of non-IID) approaches zero, the left part will be bounded by a constant. This means that when the data distribution on each client is the same, the DFL model will converge. However, as the level of non-IID increases, the value on the right side of the equation increases, causing the gap between the actual loss and the ideal loss to increase. This leads to the model becoming more divergent, as follows:</p> </div> <div class="ltx_para" id="S4.SS2.SSS1.p13"> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S8.EGx2"> <tbody id="S4.Ex9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle E\left[F\left(w_{k}\right)-F^{\ast}\right]" class="ltx_Math" display="inline" id="S4.Ex9.m1.1"><semantics id="S4.Ex9.m1.1a"><mrow id="S4.Ex9.m1.1.1" xref="S4.Ex9.m1.1.1.cmml"><mi id="S4.Ex9.m1.1.1.3" xref="S4.Ex9.m1.1.1.3.cmml">E</mi><mo id="S4.Ex9.m1.1.1.2" xref="S4.Ex9.m1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex9.m1.1.1.1.1" xref="S4.Ex9.m1.1.1.1.2.cmml"><mo id="S4.Ex9.m1.1.1.1.1.2" xref="S4.Ex9.m1.1.1.1.2.1.cmml">[</mo><mrow id="S4.Ex9.m1.1.1.1.1.1" xref="S4.Ex9.m1.1.1.1.1.1.cmml"><mrow id="S4.Ex9.m1.1.1.1.1.1.1" xref="S4.Ex9.m1.1.1.1.1.1.1.cmml"><mi id="S4.Ex9.m1.1.1.1.1.1.1.3" xref="S4.Ex9.m1.1.1.1.1.1.1.3.cmml">F</mi><mo id="S4.Ex9.m1.1.1.1.1.1.1.2" xref="S4.Ex9.m1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex9.m1.1.1.1.1.1.1.1.1" xref="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex9.m1.1.1.1.1.1.1.1.1.2" xref="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.Ex9.m1.1.1.1.1.1.1.1.1.1" xref="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.2" xref="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.2.cmml">w</mi><mi id="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.3" xref="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S4.Ex9.m1.1.1.1.1.1.1.1.1.3" xref="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex9.m1.1.1.1.1.1.2" xref="S4.Ex9.m1.1.1.1.1.1.2.cmml">−</mo><msup id="S4.Ex9.m1.1.1.1.1.1.3" xref="S4.Ex9.m1.1.1.1.1.1.3.cmml"><mi id="S4.Ex9.m1.1.1.1.1.1.3.2" xref="S4.Ex9.m1.1.1.1.1.1.3.2.cmml">F</mi><mo id="S4.Ex9.m1.1.1.1.1.1.3.3" xref="S4.Ex9.m1.1.1.1.1.1.3.3.cmml">∗</mo></msup></mrow><mo id="S4.Ex9.m1.1.1.1.1.3" xref="S4.Ex9.m1.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex9.m1.1b"><apply id="S4.Ex9.m1.1.1.cmml" xref="S4.Ex9.m1.1.1"><times id="S4.Ex9.m1.1.1.2.cmml" xref="S4.Ex9.m1.1.1.2"></times><ci id="S4.Ex9.m1.1.1.3.cmml" xref="S4.Ex9.m1.1.1.3">𝐸</ci><apply id="S4.Ex9.m1.1.1.1.2.cmml" xref="S4.Ex9.m1.1.1.1.1"><csymbol cd="latexml" id="S4.Ex9.m1.1.1.1.2.1.cmml" xref="S4.Ex9.m1.1.1.1.1.2">delimited-[]</csymbol><apply id="S4.Ex9.m1.1.1.1.1.1.cmml" xref="S4.Ex9.m1.1.1.1.1.1"><minus id="S4.Ex9.m1.1.1.1.1.1.2.cmml" xref="S4.Ex9.m1.1.1.1.1.1.2"></minus><apply id="S4.Ex9.m1.1.1.1.1.1.1.cmml" xref="S4.Ex9.m1.1.1.1.1.1.1"><times id="S4.Ex9.m1.1.1.1.1.1.1.2.cmml" xref="S4.Ex9.m1.1.1.1.1.1.1.2"></times><ci id="S4.Ex9.m1.1.1.1.1.1.1.3.cmml" xref="S4.Ex9.m1.1.1.1.1.1.1.3">𝐹</ci><apply id="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex9.m1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex9.m1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.2">𝑤</ci><ci id="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply><apply id="S4.Ex9.m1.1.1.1.1.1.3.cmml" xref="S4.Ex9.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.Ex9.m1.1.1.1.1.1.3.1.cmml" xref="S4.Ex9.m1.1.1.1.1.1.3">superscript</csymbol><ci id="S4.Ex9.m1.1.1.1.1.1.3.2.cmml" xref="S4.Ex9.m1.1.1.1.1.1.3.2">𝐹</ci><ci id="S4.Ex9.m1.1.1.1.1.1.3.3.cmml" xref="S4.Ex9.m1.1.1.1.1.1.3.3">∗</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex9.m1.1c">\displaystyle E\left[F\left(w_{k}\right)-F^{\ast}\right]</annotation><annotation encoding="application/x-llamapun" id="S4.Ex9.m1.1d">italic_E [ italic_F ( italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ]</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\frac{L}{2}\left[\left(1+\mu\eta_{k}+\eta^{2}_{k}L^{2}\right)% E\|w^{0}-w^{\ast}\|^{2}\right." class="ltx_math_unparsed" display="inline" id="S4.Ex9.m2.1"><semantics id="S4.Ex9.m2.1a"><mrow id="S4.Ex9.m2.1b"><mo id="S4.Ex9.m2.1.1">≤</mo><mstyle displaystyle="true" id="S4.Ex9.m2.1.2"><mfrac id="S4.Ex9.m2.1.2a"><mi id="S4.Ex9.m2.1.2.2">L</mi><mn id="S4.Ex9.m2.1.2.3">2</mn></mfrac></mstyle><mrow id="S4.Ex9.m2.1.3"><mo id="S4.Ex9.m2.1.3.1">[</mo><mrow id="S4.Ex9.m2.1.3.2"><mo id="S4.Ex9.m2.1.3.2.1">(</mo><mn id="S4.Ex9.m2.1.3.2.2">1</mn><mo id="S4.Ex9.m2.1.3.2.3">+</mo><mi id="S4.Ex9.m2.1.3.2.4">μ</mi><msub id="S4.Ex9.m2.1.3.2.5"><mi id="S4.Ex9.m2.1.3.2.5.2">η</mi><mi id="S4.Ex9.m2.1.3.2.5.3">k</mi></msub><mo id="S4.Ex9.m2.1.3.2.6">+</mo><msubsup id="S4.Ex9.m2.1.3.2.7"><mi id="S4.Ex9.m2.1.3.2.7.2.2">η</mi><mi id="S4.Ex9.m2.1.3.2.7.3">k</mi><mn id="S4.Ex9.m2.1.3.2.7.2.3">2</mn></msubsup><msup id="S4.Ex9.m2.1.3.2.8"><mi id="S4.Ex9.m2.1.3.2.8.2">L</mi><mn id="S4.Ex9.m2.1.3.2.8.3">2</mn></msup><mo id="S4.Ex9.m2.1.3.2.9">)</mo></mrow><mi id="S4.Ex9.m2.1.3.3">E</mi><mo id="S4.Ex9.m2.1.3.4" lspace="0em" rspace="0.167em">∥</mo><msup id="S4.Ex9.m2.1.3.5"><mi id="S4.Ex9.m2.1.3.5.2">w</mi><mn id="S4.Ex9.m2.1.3.5.3">0</mn></msup><mo id="S4.Ex9.m2.1.3.6">−</mo><msup id="S4.Ex9.m2.1.3.7"><mi id="S4.Ex9.m2.1.3.7.2">w</mi><mo id="S4.Ex9.m2.1.3.7.3">∗</mo></msup><msup id="S4.Ex9.m2.1.3.8"><mo id="S4.Ex9.m2.1.3.8.2" lspace="0em">∥</mo><mn id="S4.Ex9.m2.1.3.8.3">2</mn></msup></mrow></mrow><annotation encoding="application/x-tex" id="S4.Ex9.m2.1c">\displaystyle\leq\frac{L}{2}\left[\left(1+\mu\eta_{k}+\eta^{2}_{k}L^{2}\right)% E\|w^{0}-w^{\ast}\|^{2}\right.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex9.m2.1d">≤ divide start_ARG italic_L end_ARG start_ARG 2 end_ARG [ ( 1 + italic_μ italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) italic_E ∥ italic_w start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.E16"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\quad\left.+2\eta_{k}Z+\eta^{2}_{k}\sigma^{2}_{k}\right]" class="ltx_math_unparsed" display="inline" id="S4.E16.m1.1"><semantics id="S4.E16.m1.1a"><mrow id="S4.E16.m1.1b"><mo id="S4.E16.m1.1.1">+</mo><mn id="S4.E16.m1.1.2">2</mn><msub id="S4.E16.m1.1.3"><mi id="S4.E16.m1.1.3.2">η</mi><mi id="S4.E16.m1.1.3.3">k</mi></msub><mi id="S4.E16.m1.1.4">Z</mi><mo id="S4.E16.m1.1.5">+</mo><msubsup id="S4.E16.m1.1.6"><mi id="S4.E16.m1.1.6.2.2">η</mi><mi id="S4.E16.m1.1.6.3">k</mi><mn id="S4.E16.m1.1.6.2.3">2</mn></msubsup><msubsup id="S4.E16.m1.1.7"><mi id="S4.E16.m1.1.7.2.2">σ</mi><mi id="S4.E16.m1.1.7.3">k</mi><mn id="S4.E16.m1.1.7.2.3">2</mn></msubsup><mo id="S4.E16.m1.1.8">]</mo></mrow><annotation encoding="application/x-tex" id="S4.E16.m1.1c">\displaystyle\quad\left.+2\eta_{k}Z+\eta^{2}_{k}\sigma^{2}_{k}\right]</annotation><annotation encoding="application/x-llamapun" id="S4.E16.m1.1d">+ 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_Z + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(16)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS1.p13.1">where <math alttext="w^{0}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p13.1.m1.1"><semantics id="S4.SS2.SSS1.p13.1.m1.1a"><msup id="S4.SS2.SSS1.p13.1.m1.1.1" xref="S4.SS2.SSS1.p13.1.m1.1.1.cmml"><mi id="S4.SS2.SSS1.p13.1.m1.1.1.2" xref="S4.SS2.SSS1.p13.1.m1.1.1.2.cmml">w</mi><mn id="S4.SS2.SSS1.p13.1.m1.1.1.3" xref="S4.SS2.SSS1.p13.1.m1.1.1.3.cmml">0</mn></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p13.1.m1.1b"><apply id="S4.SS2.SSS1.p13.1.m1.1.1.cmml" xref="S4.SS2.SSS1.p13.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p13.1.m1.1.1.1.cmml" xref="S4.SS2.SSS1.p13.1.m1.1.1">superscript</csymbol><ci id="S4.SS2.SSS1.p13.1.m1.1.1.2.cmml" xref="S4.SS2.SSS1.p13.1.m1.1.1.2">𝑤</ci><cn id="S4.SS2.SSS1.p13.1.m1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.p13.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p13.1.m1.1c">w^{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p13.1.m1.1d">italic_w start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT</annotation></semantics></math> represents the initial parameters.</p> </div> </section> <section class="ltx_subsubsection" id="S4.SS2.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection"><span class="ltx_text" id="S4.SS2.SSS2.4.1.1">IV-B</span>2 </span>Continuous Ring</h4> <div class="ltx_para" id="S4.SS2.SSS2.p1"> <p class="ltx_p" id="S4.SS2.SSS2.p1.1">The continuous ring DFL <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib5" title="">5</a>]</cite> comprises a network of devices interconnected in the form of a ring. We want to mention that the continuous ring is very similar to continuous linear because they have the same training strategy.</p> </div> <div class="ltx_para" id="S4.SS2.SSS2.p2"> <p class="ltx_p" id="S4.SS2.SSS2.p2.9">We define <math alttext="d" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p2.1.m1.1"><semantics id="S4.SS2.SSS2.p2.1.m1.1a"><mi id="S4.SS2.SSS2.p2.1.m1.1.1" xref="S4.SS2.SSS2.p2.1.m1.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p2.1.m1.1b"><ci id="S4.SS2.SSS2.p2.1.m1.1.1.cmml" xref="S4.SS2.SSS2.p2.1.m1.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p2.1.m1.1c">d</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p2.1.m1.1d">italic_d</annotation></semantics></math> as the cumulative number of times the model parameters are passed to all devices, which is an indicator of the flow of the model throughout the network, where <math alttext="d\in 1,...,N\ast t" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p2.2.m2.3"><semantics id="S4.SS2.SSS2.p2.2.m2.3a"><mrow id="S4.SS2.SSS2.p2.2.m2.3.3" xref="S4.SS2.SSS2.p2.2.m2.3.3.cmml"><mi id="S4.SS2.SSS2.p2.2.m2.3.3.3" xref="S4.SS2.SSS2.p2.2.m2.3.3.3.cmml">d</mi><mo id="S4.SS2.SSS2.p2.2.m2.3.3.2" xref="S4.SS2.SSS2.p2.2.m2.3.3.2.cmml">∈</mo><mrow id="S4.SS2.SSS2.p2.2.m2.3.3.1.1" xref="S4.SS2.SSS2.p2.2.m2.3.3.1.2.cmml"><mn id="S4.SS2.SSS2.p2.2.m2.1.1" xref="S4.SS2.SSS2.p2.2.m2.1.1.cmml">1</mn><mo id="S4.SS2.SSS2.p2.2.m2.3.3.1.1.2" xref="S4.SS2.SSS2.p2.2.m2.3.3.1.2.cmml">,</mo><mi id="S4.SS2.SSS2.p2.2.m2.2.2" mathvariant="normal" xref="S4.SS2.SSS2.p2.2.m2.2.2.cmml">…</mi><mo id="S4.SS2.SSS2.p2.2.m2.3.3.1.1.3" xref="S4.SS2.SSS2.p2.2.m2.3.3.1.2.cmml">,</mo><mrow id="S4.SS2.SSS2.p2.2.m2.3.3.1.1.1" xref="S4.SS2.SSS2.p2.2.m2.3.3.1.1.1.cmml"><mi id="S4.SS2.SSS2.p2.2.m2.3.3.1.1.1.2" xref="S4.SS2.SSS2.p2.2.m2.3.3.1.1.1.2.cmml">N</mi><mo id="S4.SS2.SSS2.p2.2.m2.3.3.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.SS2.SSS2.p2.2.m2.3.3.1.1.1.1.cmml">∗</mo><mi id="S4.SS2.SSS2.p2.2.m2.3.3.1.1.1.3" xref="S4.SS2.SSS2.p2.2.m2.3.3.1.1.1.3.cmml">t</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p2.2.m2.3b"><apply id="S4.SS2.SSS2.p2.2.m2.3.3.cmml" xref="S4.SS2.SSS2.p2.2.m2.3.3"><in id="S4.SS2.SSS2.p2.2.m2.3.3.2.cmml" xref="S4.SS2.SSS2.p2.2.m2.3.3.2"></in><ci id="S4.SS2.SSS2.p2.2.m2.3.3.3.cmml" xref="S4.SS2.SSS2.p2.2.m2.3.3.3">𝑑</ci><list id="S4.SS2.SSS2.p2.2.m2.3.3.1.2.cmml" xref="S4.SS2.SSS2.p2.2.m2.3.3.1.1"><cn id="S4.SS2.SSS2.p2.2.m2.1.1.cmml" type="integer" xref="S4.SS2.SSS2.p2.2.m2.1.1">1</cn><ci id="S4.SS2.SSS2.p2.2.m2.2.2.cmml" xref="S4.SS2.SSS2.p2.2.m2.2.2">…</ci><apply id="S4.SS2.SSS2.p2.2.m2.3.3.1.1.1.cmml" xref="S4.SS2.SSS2.p2.2.m2.3.3.1.1.1"><ci id="S4.SS2.SSS2.p2.2.m2.3.3.1.1.1.1.cmml" xref="S4.SS2.SSS2.p2.2.m2.3.3.1.1.1.1">∗</ci><ci id="S4.SS2.SSS2.p2.2.m2.3.3.1.1.1.2.cmml" xref="S4.SS2.SSS2.p2.2.m2.3.3.1.1.1.2">𝑁</ci><ci id="S4.SS2.SSS2.p2.2.m2.3.3.1.1.1.3.cmml" xref="S4.SS2.SSS2.p2.2.m2.3.3.1.1.1.3">𝑡</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p2.2.m2.3c">d\in 1,...,N\ast t</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p2.2.m2.3d">italic_d ∈ 1 , … , italic_N ∗ italic_t</annotation></semantics></math>. Assuming that device<math alttext="1" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p2.3.m3.1"><semantics id="S4.SS2.SSS2.p2.3.m3.1a"><mn id="S4.SS2.SSS2.p2.3.m3.1.1" xref="S4.SS2.SSS2.p2.3.m3.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p2.3.m3.1b"><cn id="S4.SS2.SSS2.p2.3.m3.1.1.cmml" type="integer" xref="S4.SS2.SSS2.p2.3.m3.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p2.3.m3.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p2.3.m3.1d">1</annotation></semantics></math> received initial parameters <math alttext="w^{0}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p2.4.m4.1"><semantics id="S4.SS2.SSS2.p2.4.m4.1a"><msup id="S4.SS2.SSS2.p2.4.m4.1.1" xref="S4.SS2.SSS2.p2.4.m4.1.1.cmml"><mi id="S4.SS2.SSS2.p2.4.m4.1.1.2" xref="S4.SS2.SSS2.p2.4.m4.1.1.2.cmml">w</mi><mn id="S4.SS2.SSS2.p2.4.m4.1.1.3" xref="S4.SS2.SSS2.p2.4.m4.1.1.3.cmml">0</mn></msup><annotation-xml 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id="S4.SS2.SSS2.p2.5.m5.1.1.cmml" xref="S4.SS2.SSS2.p2.5.m5.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p2.5.m5.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p2.5.m5.1d">italic_t</annotation></semantics></math> (<math alttext="t>0" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p2.6.m6.1"><semantics id="S4.SS2.SSS2.p2.6.m6.1a"><mrow id="S4.SS2.SSS2.p2.6.m6.1.1" xref="S4.SS2.SSS2.p2.6.m6.1.1.cmml"><mi id="S4.SS2.SSS2.p2.6.m6.1.1.2" xref="S4.SS2.SSS2.p2.6.m6.1.1.2.cmml">t</mi><mo id="S4.SS2.SSS2.p2.6.m6.1.1.1" xref="S4.SS2.SSS2.p2.6.m6.1.1.1.cmml">></mo><mn id="S4.SS2.SSS2.p2.6.m6.1.1.3" xref="S4.SS2.SSS2.p2.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p2.6.m6.1b"><apply id="S4.SS2.SSS2.p2.6.m6.1.1.cmml" xref="S4.SS2.SSS2.p2.6.m6.1.1"><gt id="S4.SS2.SSS2.p2.6.m6.1.1.1.cmml" xref="S4.SS2.SSS2.p2.6.m6.1.1.1"></gt><ci id="S4.SS2.SSS2.p2.6.m6.1.1.2.cmml" xref="S4.SS2.SSS2.p2.6.m6.1.1.2">𝑡</ci><cn id="S4.SS2.SSS2.p2.6.m6.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.p2.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p2.6.m6.1c">t>0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p2.6.m6.1d">italic_t > 0</annotation></semantics></math>), the <math alttext="k" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p2.7.m7.1"><semantics id="S4.SS2.SSS2.p2.7.m7.1a"><mi id="S4.SS2.SSS2.p2.7.m7.1.1" xref="S4.SS2.SSS2.p2.7.m7.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p2.7.m7.1b"><ci id="S4.SS2.SSS2.p2.7.m7.1.1.cmml" xref="S4.SS2.SSS2.p2.7.m7.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p2.7.m7.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p2.7.m7.1d">italic_k</annotation></semantics></math>-th (<math alttext="k=d-\left(t-1\right)N" class="ltx_Math" display="inline" 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italic_t - 1 ) italic_N</annotation></semantics></math>) device updates the initial model parameters using its data and then transmits the updated parameters to the (<math alttext="k+1" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p2.9.m9.1"><semantics id="S4.SS2.SSS2.p2.9.m9.1a"><mrow id="S4.SS2.SSS2.p2.9.m9.1.1" xref="S4.SS2.SSS2.p2.9.m9.1.1.cmml"><mi id="S4.SS2.SSS2.p2.9.m9.1.1.2" xref="S4.SS2.SSS2.p2.9.m9.1.1.2.cmml">k</mi><mo id="S4.SS2.SSS2.p2.9.m9.1.1.1" xref="S4.SS2.SSS2.p2.9.m9.1.1.1.cmml">+</mo><mn id="S4.SS2.SSS2.p2.9.m9.1.1.3" xref="S4.SS2.SSS2.p2.9.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p2.9.m9.1b"><apply id="S4.SS2.SSS2.p2.9.m9.1.1.cmml" xref="S4.SS2.SSS2.p2.9.m9.1.1"><plus id="S4.SS2.SSS2.p2.9.m9.1.1.1.cmml" xref="S4.SS2.SSS2.p2.9.m9.1.1.1"></plus><ci id="S4.SS2.SSS2.p2.9.m9.1.1.2.cmml" xref="S4.SS2.SSS2.p2.9.m9.1.1.2">𝑘</ci><cn id="S4.SS2.SSS2.p2.9.m9.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.p2.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p2.9.m9.1c">k+1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p2.9.m9.1d">italic_k + 1</annotation></semantics></math>)-th device.</p> </div> <div class="ltx_para" id="S4.SS2.SSS2.p3"> <p class="ltx_p" id="S4.SS2.SSS2.p3.1">The subsequent device considers the received model parameters from the previous device as the initial model and uses its data to update it. When the model is transmitted to the last device, that device sends the trained parameters back to the first device. Subsequently, the first device sets these parameters as the initial values and initiates training again. This cyclic process is mathematically represented by the following equation:</p> </div> <div class="ltx_para" id="S4.SS2.SSS2.p4"> <table class="ltx_equation ltx_eqn_table" id="S4.E17"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{split}\ w_{d}=w_{d-1}-\eta_{d}\nabla F_{d}\left(w_{d-1},\xi_{d}\right)% \\ k=d-\left(t-1\right)N\end{split}" class="ltx_Math" display="block" id="S4.E17.m1.34"><semantics id="S4.E17.m1.34a"><mtable displaystyle="true" id="S4.E17.m1.34.34.6" rowspacing="0pt" xref="S4.E17.m1.31.31.3.cmml"><mtr id="S4.E17.m1.34.34.6a" xref="S4.E17.m1.31.31.3.cmml"><mtd class="ltx_align_right" columnalign="right" id="S4.E17.m1.34.34.6b" xref="S4.E17.m1.31.31.3.cmml"><mrow id="S4.E17.m1.33.33.5.30.20.20" xref="S4.E17.m1.31.31.3.cmml"><msub id="S4.E17.m1.33.33.5.30.20.20.21" xref="S4.E17.m1.31.31.3.cmml"><mi id="S4.E17.m1.1.1.1.1.1.1" 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type="integer" xref="S4.E17.m1.26.26.26.8.8.8">1</cn></apply><ci id="S4.E17.m1.28.28.28.10.10.10.cmml" xref="S4.E17.m1.28.28.28.10.10.10">𝑁</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E17.m1.34c">\begin{split}\ w_{d}=w_{d-1}-\eta_{d}\nabla F_{d}\left(w_{d-1},\xi_{d}\right)% \\ k=d-\left(t-1\right)N\end{split}</annotation><annotation encoding="application/x-llamapun" id="S4.E17.m1.34d">start_ROW start_CELL italic_w start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = italic_w start_POSTSUBSCRIPT italic_d - 1 end_POSTSUBSCRIPT - italic_η start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ∇ italic_F start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT italic_d - 1 end_POSTSUBSCRIPT , italic_ξ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL italic_k = italic_d - ( italic_t - 1 ) italic_N end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(17)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS2.p5"> <p class="ltx_p" id="S4.SS2.SSS2.p5.1">Here, <math alttext="t" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p5.1.m1.1"><semantics id="S4.SS2.SSS2.p5.1.m1.1a"><mi id="S4.SS2.SSS2.p5.1.m1.1.1" xref="S4.SS2.SSS2.p5.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p5.1.m1.1b"><ci id="S4.SS2.SSS2.p5.1.m1.1.1.cmml" xref="S4.SS2.SSS2.p5.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p5.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p5.1.m1.1d">italic_t</annotation></semantics></math> represents the number of complete cycles from the first client to the last client and then back to the first client, constituting one full training cycle. Each complete cycle allows every device an opportunity to update and pass along the model parameters, considered as one complete training round. Therefore, the coverage analysis of ring continuous is similar to the linear continuous:</p> </div> <div class="ltx_para" id="S4.SS2.SSS2.p6"> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S8.EGx3"> <tbody id="S4.Ex10"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle E\left[F\left(w_{d}\right)-F^{\ast}\right]" class="ltx_Math" display="inline" id="S4.Ex10.m1.1"><semantics id="S4.Ex10.m1.1a"><mrow id="S4.Ex10.m1.1.1" xref="S4.Ex10.m1.1.1.cmml"><mi id="S4.Ex10.m1.1.1.3" xref="S4.Ex10.m1.1.1.3.cmml">E</mi><mo id="S4.Ex10.m1.1.1.2" xref="S4.Ex10.m1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex10.m1.1.1.1.1" xref="S4.Ex10.m1.1.1.1.2.cmml"><mo id="S4.Ex10.m1.1.1.1.1.2" xref="S4.Ex10.m1.1.1.1.2.1.cmml">[</mo><mrow id="S4.Ex10.m1.1.1.1.1.1" xref="S4.Ex10.m1.1.1.1.1.1.cmml"><mrow id="S4.Ex10.m1.1.1.1.1.1.1" xref="S4.Ex10.m1.1.1.1.1.1.1.cmml"><mi id="S4.Ex10.m1.1.1.1.1.1.1.3" xref="S4.Ex10.m1.1.1.1.1.1.1.3.cmml">F</mi><mo id="S4.Ex10.m1.1.1.1.1.1.1.2" xref="S4.Ex10.m1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex10.m1.1.1.1.1.1.1.1.1" xref="S4.Ex10.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex10.m1.1.1.1.1.1.1.1.1.2" xref="S4.Ex10.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.Ex10.m1.1.1.1.1.1.1.1.1.1" xref="S4.Ex10.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex10.m1.1.1.1.1.1.1.1.1.1.2" xref="S4.Ex10.m1.1.1.1.1.1.1.1.1.1.2.cmml">w</mi><mi id="S4.Ex10.m1.1.1.1.1.1.1.1.1.1.3" xref="S4.Ex10.m1.1.1.1.1.1.1.1.1.1.3.cmml">d</mi></msub><mo id="S4.Ex10.m1.1.1.1.1.1.1.1.1.3" xref="S4.Ex10.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex10.m1.1.1.1.1.1.2" xref="S4.Ex10.m1.1.1.1.1.1.2.cmml">−</mo><msup id="S4.Ex10.m1.1.1.1.1.1.3" xref="S4.Ex10.m1.1.1.1.1.1.3.cmml"><mi id="S4.Ex10.m1.1.1.1.1.1.3.2" xref="S4.Ex10.m1.1.1.1.1.1.3.2.cmml">F</mi><mo id="S4.Ex10.m1.1.1.1.1.1.3.3" xref="S4.Ex10.m1.1.1.1.1.1.3.3.cmml">∗</mo></msup></mrow><mo id="S4.Ex10.m1.1.1.1.1.3" xref="S4.Ex10.m1.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex10.m1.1b"><apply id="S4.Ex10.m1.1.1.cmml" xref="S4.Ex10.m1.1.1"><times id="S4.Ex10.m1.1.1.2.cmml" xref="S4.Ex10.m1.1.1.2"></times><ci id="S4.Ex10.m1.1.1.3.cmml" xref="S4.Ex10.m1.1.1.3">𝐸</ci><apply id="S4.Ex10.m1.1.1.1.2.cmml" xref="S4.Ex10.m1.1.1.1.1"><csymbol cd="latexml" id="S4.Ex10.m1.1.1.1.2.1.cmml" xref="S4.Ex10.m1.1.1.1.1.2">delimited-[]</csymbol><apply id="S4.Ex10.m1.1.1.1.1.1.cmml" xref="S4.Ex10.m1.1.1.1.1.1"><minus id="S4.Ex10.m1.1.1.1.1.1.2.cmml" xref="S4.Ex10.m1.1.1.1.1.1.2"></minus><apply id="S4.Ex10.m1.1.1.1.1.1.1.cmml" xref="S4.Ex10.m1.1.1.1.1.1.1"><times id="S4.Ex10.m1.1.1.1.1.1.1.2.cmml" xref="S4.Ex10.m1.1.1.1.1.1.1.2"></times><ci id="S4.Ex10.m1.1.1.1.1.1.1.3.cmml" xref="S4.Ex10.m1.1.1.1.1.1.1.3">𝐹</ci><apply id="S4.Ex10.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex10.m1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex10.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex10.m1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.Ex10.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.Ex10.m1.1.1.1.1.1.1.1.1.1.2">𝑤</ci><ci id="S4.Ex10.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.Ex10.m1.1.1.1.1.1.1.1.1.1.3">𝑑</ci></apply></apply><apply id="S4.Ex10.m1.1.1.1.1.1.3.cmml" xref="S4.Ex10.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.Ex10.m1.1.1.1.1.1.3.1.cmml" xref="S4.Ex10.m1.1.1.1.1.1.3">superscript</csymbol><ci id="S4.Ex10.m1.1.1.1.1.1.3.2.cmml" xref="S4.Ex10.m1.1.1.1.1.1.3.2">𝐹</ci><ci id="S4.Ex10.m1.1.1.1.1.1.3.3.cmml" xref="S4.Ex10.m1.1.1.1.1.1.3.3">∗</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex10.m1.1c">\displaystyle E\left[F\left(w_{d}\right)-F^{\ast}\right]</annotation><annotation encoding="application/x-llamapun" id="S4.Ex10.m1.1d">italic_E [ italic_F ( italic_w start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ]</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\frac{L}{2}\left[\left(1+\mu\eta_{d}+\eta^{2}_{d}L^{2}\right)% E||w^{0}-w^{\ast}||^{2}\right." class="ltx_math_unparsed" display="inline" id="S4.Ex10.m2.1"><semantics id="S4.Ex10.m2.1a"><mrow id="S4.Ex10.m2.1b"><mo id="S4.Ex10.m2.1.1">≤</mo><mstyle displaystyle="true" id="S4.Ex10.m2.1.2"><mfrac id="S4.Ex10.m2.1.2a"><mi id="S4.Ex10.m2.1.2.2">L</mi><mn id="S4.Ex10.m2.1.2.3">2</mn></mfrac></mstyle><mrow id="S4.Ex10.m2.1.3"><mo id="S4.Ex10.m2.1.3.1">[</mo><mrow id="S4.Ex10.m2.1.3.2"><mo id="S4.Ex10.m2.1.3.2.1">(</mo><mn id="S4.Ex10.m2.1.3.2.2">1</mn><mo id="S4.Ex10.m2.1.3.2.3">+</mo><mi id="S4.Ex10.m2.1.3.2.4">μ</mi><msub id="S4.Ex10.m2.1.3.2.5"><mi id="S4.Ex10.m2.1.3.2.5.2">η</mi><mi id="S4.Ex10.m2.1.3.2.5.3">d</mi></msub><mo id="S4.Ex10.m2.1.3.2.6">+</mo><msubsup id="S4.Ex10.m2.1.3.2.7"><mi id="S4.Ex10.m2.1.3.2.7.2.2">η</mi><mi id="S4.Ex10.m2.1.3.2.7.3">d</mi><mn id="S4.Ex10.m2.1.3.2.7.2.3">2</mn></msubsup><msup id="S4.Ex10.m2.1.3.2.8"><mi id="S4.Ex10.m2.1.3.2.8.2">L</mi><mn id="S4.Ex10.m2.1.3.2.8.3">2</mn></msup><mo id="S4.Ex10.m2.1.3.2.9">)</mo></mrow><mi id="S4.Ex10.m2.1.3.3">E</mi><mo fence="false" id="S4.Ex10.m2.1.3.4" rspace="0.167em" stretchy="false">|</mo><mo fence="false" id="S4.Ex10.m2.1.3.5" rspace="0.167em" stretchy="false">|</mo><msup id="S4.Ex10.m2.1.3.6"><mi id="S4.Ex10.m2.1.3.6.2">w</mi><mn id="S4.Ex10.m2.1.3.6.3">0</mn></msup><mo id="S4.Ex10.m2.1.3.7">−</mo><msup id="S4.Ex10.m2.1.3.8"><mi id="S4.Ex10.m2.1.3.8.2">w</mi><mo id="S4.Ex10.m2.1.3.8.3">∗</mo></msup><mo fence="false" id="S4.Ex10.m2.1.3.9" rspace="0.167em" stretchy="false">|</mo><msup id="S4.Ex10.m2.1.3.10"><mo fence="false" id="S4.Ex10.m2.1.3.10.2" stretchy="false">|</mo><mn id="S4.Ex10.m2.1.3.10.3">2</mn></msup></mrow></mrow><annotation encoding="application/x-tex" id="S4.Ex10.m2.1c">\displaystyle\leq\frac{L}{2}\left[\left(1+\mu\eta_{d}+\eta^{2}_{d}L^{2}\right)% E||w^{0}-w^{\ast}||^{2}\right.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex10.m2.1d">≤ divide start_ARG italic_L end_ARG start_ARG 2 end_ARG [ ( 1 + italic_μ italic_η start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) italic_E | | italic_w start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.E18"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\quad\left.+2\eta_{d}Z+\eta^{2}_{d}\sigma^{2}_{d}\right]" class="ltx_math_unparsed" display="inline" id="S4.E18.m1.1"><semantics id="S4.E18.m1.1a"><mrow id="S4.E18.m1.1b"><mo id="S4.E18.m1.1.1">+</mo><mn id="S4.E18.m1.1.2">2</mn><msub id="S4.E18.m1.1.3"><mi id="S4.E18.m1.1.3.2">η</mi><mi id="S4.E18.m1.1.3.3">d</mi></msub><mi id="S4.E18.m1.1.4">Z</mi><mo id="S4.E18.m1.1.5">+</mo><msubsup id="S4.E18.m1.1.6"><mi id="S4.E18.m1.1.6.2.2">η</mi><mi id="S4.E18.m1.1.6.3">d</mi><mn id="S4.E18.m1.1.6.2.3">2</mn></msubsup><msubsup id="S4.E18.m1.1.7"><mi id="S4.E18.m1.1.7.2.2">σ</mi><mi id="S4.E18.m1.1.7.3">d</mi><mn id="S4.E18.m1.1.7.2.3">2</mn></msubsup><mo id="S4.E18.m1.1.8">]</mo></mrow><annotation encoding="application/x-tex" id="S4.E18.m1.1c">\displaystyle\quad\left.+2\eta_{d}Z+\eta^{2}_{d}\sigma^{2}_{d}\right]</annotation><annotation encoding="application/x-llamapun" id="S4.E18.m1.1d">+ 2 italic_η start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT italic_Z + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(18)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS2.p7"> <p class="ltx_p" id="S4.SS2.SSS2.p7.1">Since C_ring and C_linear use the same aggregation strategy, their mathematical convergence analysis formulas are similar. Therefore, the derivation process is omitted. By observing the formula, we can see that as <math alttext="Z" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p7.1.m1.1"><semantics id="S4.SS2.SSS2.p7.1.m1.1a"><mi id="S4.SS2.SSS2.p7.1.m1.1.1" xref="S4.SS2.SSS2.p7.1.m1.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p7.1.m1.1b"><ci id="S4.SS2.SSS2.p7.1.m1.1.1.cmml" xref="S4.SS2.SSS2.p7.1.m1.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p7.1.m1.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p7.1.m1.1d">italic_Z</annotation></semantics></math> (the degree of non-IID) increases, the value on the left side becomes larger, and the gap between the optimal solution and the actual value on the right side also increases, causing the model to become more divergent.</p> </div> </section> <section class="ltx_subsubsection" id="S4.SS2.SSS3"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection"><span class="ltx_text" id="S4.SS2.SSS3.4.1.1">IV-B</span>3 </span>Aggregate Linear</h4> <div class="ltx_para" id="S4.SS2.SSS3.p1"> <p class="ltx_p" id="S4.SS2.SSS3.p1.1">Aggregate linear DFL refers to the use of a linear network topology, combined with an aggregation training strategy. Similar to the continuous linear, this deployment comprises a series of interconnected devices aligned linearly. Each device within this network possesses a unique dataset. After training on Device 1, it transmits its parameters to Device 2.</p> </div> <div class="ltx_para" id="S4.SS2.SSS3.p2"> <p class="ltx_p" id="S4.SS2.SSS3.p2.1">Following the same procedure as the continuous linear approach, Device 2 trains based on the parameters from Device 1. Device 2 then transmits both its own parameters and those from Device 1 to Device 3. Device 3 aggregates the parameters from both Device 1 and Device 2, and then trains based on the aggregated parameters. The aggregation follows a method similar to FedAvg.</p> </div> <div class="ltx_para" id="S4.SS2.SSS3.p3"> <p class="ltx_p" id="S4.SS2.SSS3.p3.6">Specifically, we calculate the cumulative sample count of Devices 1 and 2, termed as <math alttext="s_{sum}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p3.1.m1.1"><semantics id="S4.SS2.SSS3.p3.1.m1.1a"><msub id="S4.SS2.SSS3.p3.1.m1.1.1" xref="S4.SS2.SSS3.p3.1.m1.1.1.cmml"><mi id="S4.SS2.SSS3.p3.1.m1.1.1.2" xref="S4.SS2.SSS3.p3.1.m1.1.1.2.cmml">s</mi><mrow id="S4.SS2.SSS3.p3.1.m1.1.1.3" xref="S4.SS2.SSS3.p3.1.m1.1.1.3.cmml"><mi id="S4.SS2.SSS3.p3.1.m1.1.1.3.2" xref="S4.SS2.SSS3.p3.1.m1.1.1.3.2.cmml">s</mi><mo id="S4.SS2.SSS3.p3.1.m1.1.1.3.1" xref="S4.SS2.SSS3.p3.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S4.SS2.SSS3.p3.1.m1.1.1.3.3" xref="S4.SS2.SSS3.p3.1.m1.1.1.3.3.cmml">u</mi><mo id="S4.SS2.SSS3.p3.1.m1.1.1.3.1a" xref="S4.SS2.SSS3.p3.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S4.SS2.SSS3.p3.1.m1.1.1.3.4" xref="S4.SS2.SSS3.p3.1.m1.1.1.3.4.cmml">m</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p3.1.m1.1b"><apply id="S4.SS2.SSS3.p3.1.m1.1.1.cmml" xref="S4.SS2.SSS3.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.p3.1.m1.1.1.1.cmml" xref="S4.SS2.SSS3.p3.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.p3.1.m1.1.1.2.cmml" xref="S4.SS2.SSS3.p3.1.m1.1.1.2">𝑠</ci><apply id="S4.SS2.SSS3.p3.1.m1.1.1.3.cmml" xref="S4.SS2.SSS3.p3.1.m1.1.1.3"><times id="S4.SS2.SSS3.p3.1.m1.1.1.3.1.cmml" xref="S4.SS2.SSS3.p3.1.m1.1.1.3.1"></times><ci id="S4.SS2.SSS3.p3.1.m1.1.1.3.2.cmml" xref="S4.SS2.SSS3.p3.1.m1.1.1.3.2">𝑠</ci><ci id="S4.SS2.SSS3.p3.1.m1.1.1.3.3.cmml" xref="S4.SS2.SSS3.p3.1.m1.1.1.3.3">𝑢</ci><ci id="S4.SS2.SSS3.p3.1.m1.1.1.3.4.cmml" xref="S4.SS2.SSS3.p3.1.m1.1.1.3.4">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p3.1.m1.1c">s_{sum}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p3.1.m1.1d">italic_s start_POSTSUBSCRIPT italic_s italic_u italic_m end_POSTSUBSCRIPT</annotation></semantics></math>. Additionally, we denote the individual sample counts for Device1 and Device2 as <math alttext="s1" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p3.2.m2.1"><semantics id="S4.SS2.SSS3.p3.2.m2.1a"><mrow id="S4.SS2.SSS3.p3.2.m2.1.1" xref="S4.SS2.SSS3.p3.2.m2.1.1.cmml"><mi id="S4.SS2.SSS3.p3.2.m2.1.1.2" xref="S4.SS2.SSS3.p3.2.m2.1.1.2.cmml">s</mi><mo id="S4.SS2.SSS3.p3.2.m2.1.1.1" xref="S4.SS2.SSS3.p3.2.m2.1.1.1.cmml">⁢</mo><mn id="S4.SS2.SSS3.p3.2.m2.1.1.3" xref="S4.SS2.SSS3.p3.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p3.2.m2.1b"><apply id="S4.SS2.SSS3.p3.2.m2.1.1.cmml" xref="S4.SS2.SSS3.p3.2.m2.1.1"><times id="S4.SS2.SSS3.p3.2.m2.1.1.1.cmml" xref="S4.SS2.SSS3.p3.2.m2.1.1.1"></times><ci id="S4.SS2.SSS3.p3.2.m2.1.1.2.cmml" xref="S4.SS2.SSS3.p3.2.m2.1.1.2">𝑠</ci><cn id="S4.SS2.SSS3.p3.2.m2.1.1.3.cmml" type="integer" xref="S4.SS2.SSS3.p3.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p3.2.m2.1c">s1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p3.2.m2.1d">italic_s 1</annotation></semantics></math> and <math alttext="s2" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p3.3.m3.1"><semantics id="S4.SS2.SSS3.p3.3.m3.1a"><mrow id="S4.SS2.SSS3.p3.3.m3.1.1" xref="S4.SS2.SSS3.p3.3.m3.1.1.cmml"><mi id="S4.SS2.SSS3.p3.3.m3.1.1.2" xref="S4.SS2.SSS3.p3.3.m3.1.1.2.cmml">s</mi><mo id="S4.SS2.SSS3.p3.3.m3.1.1.1" xref="S4.SS2.SSS3.p3.3.m3.1.1.1.cmml">⁢</mo><mn id="S4.SS2.SSS3.p3.3.m3.1.1.3" xref="S4.SS2.SSS3.p3.3.m3.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p3.3.m3.1b"><apply id="S4.SS2.SSS3.p3.3.m3.1.1.cmml" xref="S4.SS2.SSS3.p3.3.m3.1.1"><times id="S4.SS2.SSS3.p3.3.m3.1.1.1.cmml" xref="S4.SS2.SSS3.p3.3.m3.1.1.1"></times><ci id="S4.SS2.SSS3.p3.3.m3.1.1.2.cmml" xref="S4.SS2.SSS3.p3.3.m3.1.1.2">𝑠</ci><cn id="S4.SS2.SSS3.p3.3.m3.1.1.3.cmml" type="integer" xref="S4.SS2.SSS3.p3.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p3.3.m3.1c">s2</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p3.3.m3.1d">italic_s 2</annotation></semantics></math>, respectively. The final parameter set transmitted from Device2 to Device3 is a weighted combination (<math alttext="w1" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p3.4.m4.1"><semantics id="S4.SS2.SSS3.p3.4.m4.1a"><mrow id="S4.SS2.SSS3.p3.4.m4.1.1" xref="S4.SS2.SSS3.p3.4.m4.1.1.cmml"><mi id="S4.SS2.SSS3.p3.4.m4.1.1.2" xref="S4.SS2.SSS3.p3.4.m4.1.1.2.cmml">w</mi><mo id="S4.SS2.SSS3.p3.4.m4.1.1.1" xref="S4.SS2.SSS3.p3.4.m4.1.1.1.cmml">⁢</mo><mn id="S4.SS2.SSS3.p3.4.m4.1.1.3" xref="S4.SS2.SSS3.p3.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p3.4.m4.1b"><apply id="S4.SS2.SSS3.p3.4.m4.1.1.cmml" xref="S4.SS2.SSS3.p3.4.m4.1.1"><times id="S4.SS2.SSS3.p3.4.m4.1.1.1.cmml" xref="S4.SS2.SSS3.p3.4.m4.1.1.1"></times><ci id="S4.SS2.SSS3.p3.4.m4.1.1.2.cmml" xref="S4.SS2.SSS3.p3.4.m4.1.1.2">𝑤</ci><cn id="S4.SS2.SSS3.p3.4.m4.1.1.3.cmml" type="integer" xref="S4.SS2.SSS3.p3.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p3.4.m4.1c">w1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p3.4.m4.1d">italic_w 1</annotation></semantics></math> is the weight from client1 to client2, <math alttext="w2" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p3.5.m5.1"><semantics id="S4.SS2.SSS3.p3.5.m5.1a"><mrow id="S4.SS2.SSS3.p3.5.m5.1.1" xref="S4.SS2.SSS3.p3.5.m5.1.1.cmml"><mi id="S4.SS2.SSS3.p3.5.m5.1.1.2" xref="S4.SS2.SSS3.p3.5.m5.1.1.2.cmml">w</mi><mo id="S4.SS2.SSS3.p3.5.m5.1.1.1" xref="S4.SS2.SSS3.p3.5.m5.1.1.1.cmml">⁢</mo><mn id="S4.SS2.SSS3.p3.5.m5.1.1.3" xref="S4.SS2.SSS3.p3.5.m5.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p3.5.m5.1b"><apply id="S4.SS2.SSS3.p3.5.m5.1.1.cmml" xref="S4.SS2.SSS3.p3.5.m5.1.1"><times id="S4.SS2.SSS3.p3.5.m5.1.1.1.cmml" xref="S4.SS2.SSS3.p3.5.m5.1.1.1"></times><ci id="S4.SS2.SSS3.p3.5.m5.1.1.2.cmml" xref="S4.SS2.SSS3.p3.5.m5.1.1.2">𝑤</ci><cn id="S4.SS2.SSS3.p3.5.m5.1.1.3.cmml" type="integer" xref="S4.SS2.SSS3.p3.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p3.5.m5.1c">w2</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p3.5.m5.1d">italic_w 2</annotation></semantics></math> is the weight after client2 trained its own dataset) defined as: <math alttext="\frac{s1}{s_{sum}}\times w_{1}+\frac{s2}{s_{sum}}\times w_{2}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p3.6.m6.1"><semantics id="S4.SS2.SSS3.p3.6.m6.1a"><mrow id="S4.SS2.SSS3.p3.6.m6.1.1" xref="S4.SS2.SSS3.p3.6.m6.1.1.cmml"><mrow id="S4.SS2.SSS3.p3.6.m6.1.1.2" xref="S4.SS2.SSS3.p3.6.m6.1.1.2.cmml"><mfrac id="S4.SS2.SSS3.p3.6.m6.1.1.2.2" xref="S4.SS2.SSS3.p3.6.m6.1.1.2.2.cmml"><mrow id="S4.SS2.SSS3.p3.6.m6.1.1.2.2.2" xref="S4.SS2.SSS3.p3.6.m6.1.1.2.2.2.cmml"><mi id="S4.SS2.SSS3.p3.6.m6.1.1.2.2.2.2" xref="S4.SS2.SSS3.p3.6.m6.1.1.2.2.2.2.cmml">s</mi><mo id="S4.SS2.SSS3.p3.6.m6.1.1.2.2.2.1" 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xref="S4.SS2.SSS3.p3.6.m6.1.1.2.1.cmml">×</mo><msub id="S4.SS2.SSS3.p3.6.m6.1.1.2.3" xref="S4.SS2.SSS3.p3.6.m6.1.1.2.3.cmml"><mi id="S4.SS2.SSS3.p3.6.m6.1.1.2.3.2" xref="S4.SS2.SSS3.p3.6.m6.1.1.2.3.2.cmml">w</mi><mn id="S4.SS2.SSS3.p3.6.m6.1.1.2.3.3" xref="S4.SS2.SSS3.p3.6.m6.1.1.2.3.3.cmml">1</mn></msub></mrow><mo id="S4.SS2.SSS3.p3.6.m6.1.1.1" xref="S4.SS2.SSS3.p3.6.m6.1.1.1.cmml">+</mo><mrow id="S4.SS2.SSS3.p3.6.m6.1.1.3" xref="S4.SS2.SSS3.p3.6.m6.1.1.3.cmml"><mfrac id="S4.SS2.SSS3.p3.6.m6.1.1.3.2" xref="S4.SS2.SSS3.p3.6.m6.1.1.3.2.cmml"><mrow id="S4.SS2.SSS3.p3.6.m6.1.1.3.2.2" xref="S4.SS2.SSS3.p3.6.m6.1.1.3.2.2.cmml"><mi id="S4.SS2.SSS3.p3.6.m6.1.1.3.2.2.2" xref="S4.SS2.SSS3.p3.6.m6.1.1.3.2.2.2.cmml">s</mi><mo id="S4.SS2.SSS3.p3.6.m6.1.1.3.2.2.1" xref="S4.SS2.SSS3.p3.6.m6.1.1.3.2.2.1.cmml">⁢</mo><mn id="S4.SS2.SSS3.p3.6.m6.1.1.3.2.2.3" xref="S4.SS2.SSS3.p3.6.m6.1.1.3.2.2.3.cmml">2</mn></mrow><msub id="S4.SS2.SSS3.p3.6.m6.1.1.3.2.3" xref="S4.SS2.SSS3.p3.6.m6.1.1.3.2.3.cmml"><mi 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id="S4.SS2.SSS3.p3.6.m6.1.1.3.3.2.cmml" xref="S4.SS2.SSS3.p3.6.m6.1.1.3.3.2">𝑤</ci><cn id="S4.SS2.SSS3.p3.6.m6.1.1.3.3.3.cmml" type="integer" xref="S4.SS2.SSS3.p3.6.m6.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p3.6.m6.1c">\frac{s1}{s_{sum}}\times w_{1}+\frac{s2}{s_{sum}}\times w_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p3.6.m6.1d">divide start_ARG italic_s 1 end_ARG start_ARG italic_s start_POSTSUBSCRIPT italic_s italic_u italic_m end_POSTSUBSCRIPT end_ARG × italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + divide start_ARG italic_s 2 end_ARG start_ARG italic_s start_POSTSUBSCRIPT italic_s italic_u italic_m end_POSTSUBSCRIPT end_ARG × italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.SSS3.p4"> <p class="ltx_p" id="S4.SS2.SSS3.p4.1">This process is mathematically represented by the following equation; 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ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(19)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS3.p5"> <p class="ltx_p" id="S4.SS2.SSS3.p5.1">When k is larger than 2, we get:</p> <table class="ltx_equation ltx_eqn_table" id="S4.E20"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{split}w_{k}=\left(\frac{\left(s_{k-1}\right)w_{k-1}+\left(s_{k-2}\right% )w_{k-2}}{\displaystyle\sum^{k-1}_{i=1}s_{i}}\right)-\eta_{k}\nabla F_{k}\\ \left(\left(\frac{\left(s_{k-1}\right)w_{k-1}+\left(s_{k-2}\right)w_{k-2}}{% \displaystyle\sum^{k-1}_{i=1}s_{i}}\right),\xi_{k}\right)\end{split}" class="ltx_Math" display="block" id="S4.E20.m1.24"><semantics id="S4.E20.m1.24a"><mtable displaystyle="true" id="S4.E20.m1.24.24.4" 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\displaystyle\sum^{k-1}_{i=1}s_{i}}\right),\xi_{k}\right)\end{split}</annotation><annotation encoding="application/x-llamapun" id="S4.E20.m1.24d">start_ROW start_CELL italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = ( divide start_ARG ( italic_s start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT + ( italic_s start_POSTSUBSCRIPT italic_k - 2 end_POSTSUBSCRIPT ) italic_w start_POSTSUBSCRIPT italic_k - 2 end_POSTSUBSCRIPT end_ARG start_ARG ∑ start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ) - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∇ italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL ( ( divide start_ARG ( italic_s start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT + ( italic_s start_POSTSUBSCRIPT italic_k - 2 end_POSTSUBSCRIPT ) italic_w start_POSTSUBSCRIPT italic_k - 2 end_POSTSUBSCRIPT end_ARG start_ARG ∑ start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ) , italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(20)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS3.p6"> <p class="ltx_p" id="S4.SS2.SSS3.p6.6">Next, we will analyze its convergence: For the case where <math alttext="w_{k}=M_{k}-\eta_{k}\nabla F_{k}\left(M_{k},\xi_{k}\right)" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p6.1.m1.2"><semantics id="S4.SS2.SSS3.p6.1.m1.2a"><mrow 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xref="S4.SS2.SSS3.p6.1.m1.2.2.2.2.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS3.p6.1.m1.2.2.2.2.2.2.2.2.cmml" xref="S4.SS2.SSS3.p6.1.m1.2.2.2.2.2.2.2.2">𝜉</ci><ci id="S4.SS2.SSS3.p6.1.m1.2.2.2.2.2.2.2.3.cmml" xref="S4.SS2.SSS3.p6.1.m1.2.2.2.2.2.2.2.3">𝑘</ci></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p6.1.m1.2c">w_{k}=M_{k}-\eta_{k}\nabla F_{k}\left(M_{k},\xi_{k}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p6.1.m1.2d">italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∇ italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> and <math 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id="S4.SS2.SSS3.p6.2.m2.1.1.3.2.1.cmml" xref="S4.SS2.SSS3.p6.2.m2.1.1.3.2">subscript</csymbol><ci id="S4.SS2.SSS3.p6.2.m2.1.1.3.2.2.cmml" xref="S4.SS2.SSS3.p6.2.m2.1.1.3.2.2">𝑠</ci><ci id="S4.SS2.SSS3.p6.2.m2.1.1.3.2.3.cmml" xref="S4.SS2.SSS3.p6.2.m2.1.1.3.2.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p6.2.m2.1c">M_{k}=\left(\frac{s_{k-1}w_{k-1}+s_{k-2}w_{k-2}}{\displaystyle\sum^{k-1}_{i=k}% s_{i}}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p6.2.m2.1d">italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = ( divide start_ARG italic_s start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT italic_k - 2 end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT italic_k - 2 end_POSTSUBSCRIPT end_ARG start_ARG ∑ start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = italic_k end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG )</annotation></semantics></math> we let <math alttext="H=\nabla F_{k}\left(M_{k},\xi_{k}\right)" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p6.3.m3.2"><semantics id="S4.SS2.SSS3.p6.3.m3.2a"><mrow id="S4.SS2.SSS3.p6.3.m3.2.2" xref="S4.SS2.SSS3.p6.3.m3.2.2.cmml"><mi id="S4.SS2.SSS3.p6.3.m3.2.2.4" xref="S4.SS2.SSS3.p6.3.m3.2.2.4.cmml">H</mi><mo id="S4.SS2.SSS3.p6.3.m3.2.2.3" xref="S4.SS2.SSS3.p6.3.m3.2.2.3.cmml">=</mo><mrow id="S4.SS2.SSS3.p6.3.m3.2.2.2" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.cmml"><mrow id="S4.SS2.SSS3.p6.3.m3.2.2.2.4" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.4.cmml"><mo id="S4.SS2.SSS3.p6.3.m3.2.2.2.4.1" rspace="0.167em" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.4.1.cmml">∇</mo><msub id="S4.SS2.SSS3.p6.3.m3.2.2.2.4.2" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.4.2.cmml"><mi id="S4.SS2.SSS3.p6.3.m3.2.2.2.4.2.2" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.4.2.2.cmml">F</mi><mi id="S4.SS2.SSS3.p6.3.m3.2.2.2.4.2.3" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.4.2.3.cmml">k</mi></msub></mrow><mo id="S4.SS2.SSS3.p6.3.m3.2.2.2.3" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.3.cmml">⁢</mo><mrow id="S4.SS2.SSS3.p6.3.m3.2.2.2.2.2" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.2.3.cmml"><mo id="S4.SS2.SSS3.p6.3.m3.2.2.2.2.2.3" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.2.3.cmml">(</mo><msub id="S4.SS2.SSS3.p6.3.m3.1.1.1.1.1.1" xref="S4.SS2.SSS3.p6.3.m3.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS3.p6.3.m3.1.1.1.1.1.1.2" xref="S4.SS2.SSS3.p6.3.m3.1.1.1.1.1.1.2.cmml">M</mi><mi id="S4.SS2.SSS3.p6.3.m3.1.1.1.1.1.1.3" xref="S4.SS2.SSS3.p6.3.m3.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S4.SS2.SSS3.p6.3.m3.2.2.2.2.2.4" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.2.3.cmml">,</mo><msub id="S4.SS2.SSS3.p6.3.m3.2.2.2.2.2.2" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.2.2.2.cmml"><mi id="S4.SS2.SSS3.p6.3.m3.2.2.2.2.2.2.2" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.2.2.2.2.cmml">ξ</mi><mi id="S4.SS2.SSS3.p6.3.m3.2.2.2.2.2.2.3" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.2.2.2.3.cmml">k</mi></msub><mo id="S4.SS2.SSS3.p6.3.m3.2.2.2.2.2.5" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p6.3.m3.2b"><apply id="S4.SS2.SSS3.p6.3.m3.2.2.cmml" xref="S4.SS2.SSS3.p6.3.m3.2.2"><eq id="S4.SS2.SSS3.p6.3.m3.2.2.3.cmml" xref="S4.SS2.SSS3.p6.3.m3.2.2.3"></eq><ci id="S4.SS2.SSS3.p6.3.m3.2.2.4.cmml" xref="S4.SS2.SSS3.p6.3.m3.2.2.4">𝐻</ci><apply id="S4.SS2.SSS3.p6.3.m3.2.2.2.cmml" xref="S4.SS2.SSS3.p6.3.m3.2.2.2"><times id="S4.SS2.SSS3.p6.3.m3.2.2.2.3.cmml" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.3"></times><apply id="S4.SS2.SSS3.p6.3.m3.2.2.2.4.cmml" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.4"><ci id="S4.SS2.SSS3.p6.3.m3.2.2.2.4.1.cmml" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.4.1">∇</ci><apply id="S4.SS2.SSS3.p6.3.m3.2.2.2.4.2.cmml" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.4.2"><csymbol cd="ambiguous" id="S4.SS2.SSS3.p6.3.m3.2.2.2.4.2.1.cmml" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.4.2">subscript</csymbol><ci id="S4.SS2.SSS3.p6.3.m3.2.2.2.4.2.2.cmml" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.4.2.2">𝐹</ci><ci id="S4.SS2.SSS3.p6.3.m3.2.2.2.4.2.3.cmml" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.4.2.3">𝑘</ci></apply></apply><interval closure="open" id="S4.SS2.SSS3.p6.3.m3.2.2.2.2.3.cmml" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.2.2"><apply id="S4.SS2.SSS3.p6.3.m3.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.p6.3.m3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.p6.3.m3.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.p6.3.m3.1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.p6.3.m3.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS3.p6.3.m3.1.1.1.1.1.1.2">𝑀</ci><ci id="S4.SS2.SSS3.p6.3.m3.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS3.p6.3.m3.1.1.1.1.1.1.3">𝑘</ci></apply><apply id="S4.SS2.SSS3.p6.3.m3.2.2.2.2.2.2.cmml" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS3.p6.3.m3.2.2.2.2.2.2.1.cmml" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS3.p6.3.m3.2.2.2.2.2.2.2.cmml" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.2.2.2.2">𝜉</ci><ci id="S4.SS2.SSS3.p6.3.m3.2.2.2.2.2.2.3.cmml" xref="S4.SS2.SSS3.p6.3.m3.2.2.2.2.2.2.3">𝑘</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p6.3.m3.2c">H=\nabla F_{k}\left(M_{k},\xi_{k}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p6.3.m3.2d">italic_H = ∇ italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="\bar{H}=\nabla F_{k}\left(M_{k}\right)" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p6.4.m4.1"><semantics id="S4.SS2.SSS3.p6.4.m4.1a"><mrow id="S4.SS2.SSS3.p6.4.m4.1.1" xref="S4.SS2.SSS3.p6.4.m4.1.1.cmml"><mover accent="true" id="S4.SS2.SSS3.p6.4.m4.1.1.3" xref="S4.SS2.SSS3.p6.4.m4.1.1.3.cmml"><mi id="S4.SS2.SSS3.p6.4.m4.1.1.3.2" xref="S4.SS2.SSS3.p6.4.m4.1.1.3.2.cmml">H</mi><mo id="S4.SS2.SSS3.p6.4.m4.1.1.3.1" xref="S4.SS2.SSS3.p6.4.m4.1.1.3.1.cmml">¯</mo></mover><mo id="S4.SS2.SSS3.p6.4.m4.1.1.2" xref="S4.SS2.SSS3.p6.4.m4.1.1.2.cmml">=</mo><mrow id="S4.SS2.SSS3.p6.4.m4.1.1.1" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.cmml"><mrow id="S4.SS2.SSS3.p6.4.m4.1.1.1.3" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.3.cmml"><mo id="S4.SS2.SSS3.p6.4.m4.1.1.1.3.1" rspace="0.167em" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.3.1.cmml">∇</mo><msub id="S4.SS2.SSS3.p6.4.m4.1.1.1.3.2" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.3.2.cmml"><mi id="S4.SS2.SSS3.p6.4.m4.1.1.1.3.2.2" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.3.2.2.cmml">F</mi><mi id="S4.SS2.SSS3.p6.4.m4.1.1.1.3.2.3" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.3.2.3.cmml">k</mi></msub></mrow><mo id="S4.SS2.SSS3.p6.4.m4.1.1.1.2" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS3.p6.4.m4.1.1.1.1.1" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS3.p6.4.m4.1.1.1.1.1.2" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.SSS3.p6.4.m4.1.1.1.1.1.1" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS3.p6.4.m4.1.1.1.1.1.1.2" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.1.1.1.2.cmml">M</mi><mi id="S4.SS2.SSS3.p6.4.m4.1.1.1.1.1.1.3" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S4.SS2.SSS3.p6.4.m4.1.1.1.1.1.3" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p6.4.m4.1b"><apply id="S4.SS2.SSS3.p6.4.m4.1.1.cmml" xref="S4.SS2.SSS3.p6.4.m4.1.1"><eq id="S4.SS2.SSS3.p6.4.m4.1.1.2.cmml" xref="S4.SS2.SSS3.p6.4.m4.1.1.2"></eq><apply id="S4.SS2.SSS3.p6.4.m4.1.1.3.cmml" xref="S4.SS2.SSS3.p6.4.m4.1.1.3"><ci id="S4.SS2.SSS3.p6.4.m4.1.1.3.1.cmml" xref="S4.SS2.SSS3.p6.4.m4.1.1.3.1">¯</ci><ci id="S4.SS2.SSS3.p6.4.m4.1.1.3.2.cmml" xref="S4.SS2.SSS3.p6.4.m4.1.1.3.2">𝐻</ci></apply><apply id="S4.SS2.SSS3.p6.4.m4.1.1.1.cmml" xref="S4.SS2.SSS3.p6.4.m4.1.1.1"><times id="S4.SS2.SSS3.p6.4.m4.1.1.1.2.cmml" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.2"></times><apply id="S4.SS2.SSS3.p6.4.m4.1.1.1.3.cmml" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.3"><ci id="S4.SS2.SSS3.p6.4.m4.1.1.1.3.1.cmml" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.3.1">∇</ci><apply id="S4.SS2.SSS3.p6.4.m4.1.1.1.3.2.cmml" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS3.p6.4.m4.1.1.1.3.2.1.cmml" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.3.2">subscript</csymbol><ci id="S4.SS2.SSS3.p6.4.m4.1.1.1.3.2.2.cmml" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.3.2.2">𝐹</ci><ci id="S4.SS2.SSS3.p6.4.m4.1.1.1.3.2.3.cmml" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.3.2.3">𝑘</ci></apply></apply><apply id="S4.SS2.SSS3.p6.4.m4.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.p6.4.m4.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.p6.4.m4.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.1.1.1.2">𝑀</ci><ci id="S4.SS2.SSS3.p6.4.m4.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS3.p6.4.m4.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p6.4.m4.1c">\bar{H}=\nabla F_{k}\left(M_{k}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p6.4.m4.1d">over¯ start_ARG italic_H end_ARG = ∇ italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> Thus, <math alttext="E(H)=\bar{H}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p6.5.m5.1"><semantics id="S4.SS2.SSS3.p6.5.m5.1a"><mrow id="S4.SS2.SSS3.p6.5.m5.1.2" xref="S4.SS2.SSS3.p6.5.m5.1.2.cmml"><mrow id="S4.SS2.SSS3.p6.5.m5.1.2.2" xref="S4.SS2.SSS3.p6.5.m5.1.2.2.cmml"><mi id="S4.SS2.SSS3.p6.5.m5.1.2.2.2" xref="S4.SS2.SSS3.p6.5.m5.1.2.2.2.cmml">E</mi><mo id="S4.SS2.SSS3.p6.5.m5.1.2.2.1" xref="S4.SS2.SSS3.p6.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.p6.5.m5.1.2.2.3.2" xref="S4.SS2.SSS3.p6.5.m5.1.2.2.cmml"><mo id="S4.SS2.SSS3.p6.5.m5.1.2.2.3.2.1" stretchy="false" xref="S4.SS2.SSS3.p6.5.m5.1.2.2.cmml">(</mo><mi id="S4.SS2.SSS3.p6.5.m5.1.1" xref="S4.SS2.SSS3.p6.5.m5.1.1.cmml">H</mi><mo id="S4.SS2.SSS3.p6.5.m5.1.2.2.3.2.2" stretchy="false" xref="S4.SS2.SSS3.p6.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS3.p6.5.m5.1.2.1" xref="S4.SS2.SSS3.p6.5.m5.1.2.1.cmml">=</mo><mover accent="true" id="S4.SS2.SSS3.p6.5.m5.1.2.3" xref="S4.SS2.SSS3.p6.5.m5.1.2.3.cmml"><mi id="S4.SS2.SSS3.p6.5.m5.1.2.3.2" xref="S4.SS2.SSS3.p6.5.m5.1.2.3.2.cmml">H</mi><mo id="S4.SS2.SSS3.p6.5.m5.1.2.3.1" xref="S4.SS2.SSS3.p6.5.m5.1.2.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p6.5.m5.1b"><apply id="S4.SS2.SSS3.p6.5.m5.1.2.cmml" xref="S4.SS2.SSS3.p6.5.m5.1.2"><eq id="S4.SS2.SSS3.p6.5.m5.1.2.1.cmml" xref="S4.SS2.SSS3.p6.5.m5.1.2.1"></eq><apply id="S4.SS2.SSS3.p6.5.m5.1.2.2.cmml" xref="S4.SS2.SSS3.p6.5.m5.1.2.2"><times id="S4.SS2.SSS3.p6.5.m5.1.2.2.1.cmml" xref="S4.SS2.SSS3.p6.5.m5.1.2.2.1"></times><ci id="S4.SS2.SSS3.p6.5.m5.1.2.2.2.cmml" xref="S4.SS2.SSS3.p6.5.m5.1.2.2.2">𝐸</ci><ci id="S4.SS2.SSS3.p6.5.m5.1.1.cmml" xref="S4.SS2.SSS3.p6.5.m5.1.1">𝐻</ci></apply><apply id="S4.SS2.SSS3.p6.5.m5.1.2.3.cmml" xref="S4.SS2.SSS3.p6.5.m5.1.2.3"><ci id="S4.SS2.SSS3.p6.5.m5.1.2.3.1.cmml" xref="S4.SS2.SSS3.p6.5.m5.1.2.3.1">¯</ci><ci id="S4.SS2.SSS3.p6.5.m5.1.2.3.2.cmml" xref="S4.SS2.SSS3.p6.5.m5.1.2.3.2">𝐻</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p6.5.m5.1c">E(H)=\bar{H}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p6.5.m5.1d">italic_E ( italic_H ) = over¯ start_ARG italic_H end_ARG</annotation></semantics></math> and <math alttext="w_{k}=M-\eta_{k}H" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p6.6.m6.1"><semantics id="S4.SS2.SSS3.p6.6.m6.1a"><mrow id="S4.SS2.SSS3.p6.6.m6.1.1" xref="S4.SS2.SSS3.p6.6.m6.1.1.cmml"><msub id="S4.SS2.SSS3.p6.6.m6.1.1.2" xref="S4.SS2.SSS3.p6.6.m6.1.1.2.cmml"><mi id="S4.SS2.SSS3.p6.6.m6.1.1.2.2" xref="S4.SS2.SSS3.p6.6.m6.1.1.2.2.cmml">w</mi><mi id="S4.SS2.SSS3.p6.6.m6.1.1.2.3" xref="S4.SS2.SSS3.p6.6.m6.1.1.2.3.cmml">k</mi></msub><mo id="S4.SS2.SSS3.p6.6.m6.1.1.1" xref="S4.SS2.SSS3.p6.6.m6.1.1.1.cmml">=</mo><mrow id="S4.SS2.SSS3.p6.6.m6.1.1.3" xref="S4.SS2.SSS3.p6.6.m6.1.1.3.cmml"><mi id="S4.SS2.SSS3.p6.6.m6.1.1.3.2" xref="S4.SS2.SSS3.p6.6.m6.1.1.3.2.cmml">M</mi><mo id="S4.SS2.SSS3.p6.6.m6.1.1.3.1" xref="S4.SS2.SSS3.p6.6.m6.1.1.3.1.cmml">−</mo><mrow id="S4.SS2.SSS3.p6.6.m6.1.1.3.3" xref="S4.SS2.SSS3.p6.6.m6.1.1.3.3.cmml"><msub id="S4.SS2.SSS3.p6.6.m6.1.1.3.3.2" xref="S4.SS2.SSS3.p6.6.m6.1.1.3.3.2.cmml"><mi id="S4.SS2.SSS3.p6.6.m6.1.1.3.3.2.2" xref="S4.SS2.SSS3.p6.6.m6.1.1.3.3.2.2.cmml">η</mi><mi id="S4.SS2.SSS3.p6.6.m6.1.1.3.3.2.3" xref="S4.SS2.SSS3.p6.6.m6.1.1.3.3.2.3.cmml">k</mi></msub><mo id="S4.SS2.SSS3.p6.6.m6.1.1.3.3.1" xref="S4.SS2.SSS3.p6.6.m6.1.1.3.3.1.cmml">⁢</mo><mi id="S4.SS2.SSS3.p6.6.m6.1.1.3.3.3" xref="S4.SS2.SSS3.p6.6.m6.1.1.3.3.3.cmml">H</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p6.6.m6.1b"><apply id="S4.SS2.SSS3.p6.6.m6.1.1.cmml" xref="S4.SS2.SSS3.p6.6.m6.1.1"><eq id="S4.SS2.SSS3.p6.6.m6.1.1.1.cmml" xref="S4.SS2.SSS3.p6.6.m6.1.1.1"></eq><apply id="S4.SS2.SSS3.p6.6.m6.1.1.2.cmml" xref="S4.SS2.SSS3.p6.6.m6.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS3.p6.6.m6.1.1.2.1.cmml" xref="S4.SS2.SSS3.p6.6.m6.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS3.p6.6.m6.1.1.2.2.cmml" xref="S4.SS2.SSS3.p6.6.m6.1.1.2.2">𝑤</ci><ci id="S4.SS2.SSS3.p6.6.m6.1.1.2.3.cmml" xref="S4.SS2.SSS3.p6.6.m6.1.1.2.3">𝑘</ci></apply><apply id="S4.SS2.SSS3.p6.6.m6.1.1.3.cmml" xref="S4.SS2.SSS3.p6.6.m6.1.1.3"><minus id="S4.SS2.SSS3.p6.6.m6.1.1.3.1.cmml" xref="S4.SS2.SSS3.p6.6.m6.1.1.3.1"></minus><ci id="S4.SS2.SSS3.p6.6.m6.1.1.3.2.cmml" xref="S4.SS2.SSS3.p6.6.m6.1.1.3.2">𝑀</ci><apply id="S4.SS2.SSS3.p6.6.m6.1.1.3.3.cmml" xref="S4.SS2.SSS3.p6.6.m6.1.1.3.3"><times id="S4.SS2.SSS3.p6.6.m6.1.1.3.3.1.cmml" xref="S4.SS2.SSS3.p6.6.m6.1.1.3.3.1"></times><apply id="S4.SS2.SSS3.p6.6.m6.1.1.3.3.2.cmml" xref="S4.SS2.SSS3.p6.6.m6.1.1.3.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS3.p6.6.m6.1.1.3.3.2.1.cmml" xref="S4.SS2.SSS3.p6.6.m6.1.1.3.3.2">subscript</csymbol><ci id="S4.SS2.SSS3.p6.6.m6.1.1.3.3.2.2.cmml" xref="S4.SS2.SSS3.p6.6.m6.1.1.3.3.2.2">𝜂</ci><ci id="S4.SS2.SSS3.p6.6.m6.1.1.3.3.2.3.cmml" xref="S4.SS2.SSS3.p6.6.m6.1.1.3.3.2.3">𝑘</ci></apply><ci id="S4.SS2.SSS3.p6.6.m6.1.1.3.3.3.cmml" xref="S4.SS2.SSS3.p6.6.m6.1.1.3.3.3">𝐻</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p6.6.m6.1c">w_{k}=M-\eta_{k}H</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p6.6.m6.1d">italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_M - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_H</annotation></semantics></math>, and we get:</p> </div> <div class="ltx_para" id="S4.SS2.SSS3.p7"> <table class="ltx_equation ltx_eqn_table" id="S4.E21"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{gathered}\parallel w_{k}-w^{\ast}\parallel^{2}=\parallel M-\eta_{k}H-w^% {\ast}-\eta_{k}\bar{H}+\eta_{k}\bar{H}\parallel^{2}\\ =\parallel M-\eta_{k}\bar{H}-w^{\ast}+\eta_{k}\bar{H}-\eta_{k}H\parallel^{2}% \end{gathered}" class="ltx_Math" display="block" id="S4.E21.m1.54"><semantics id="S4.E21.m1.54a"><mtable displaystyle="true" id="S4.E21.m1.54.54.6" rowspacing="0pt" xref="S4.E21.m1.51.51.3.cmml"><mtr id="S4.E21.m1.54.54.6a" xref="S4.E21.m1.51.51.3.cmml"><mtd id="S4.E21.m1.54.54.6b" xref="S4.E21.m1.51.51.3.cmml"><mrow id="S4.E21.m1.53.53.5.50.30.30" xref="S4.E21.m1.51.51.3.cmml"><msup id="S4.E21.m1.52.52.4.49.29.29.29" xref="S4.E21.m1.51.51.3.cmml"><mrow id="S4.E21.m1.52.52.4.49.29.29.29.1.1" xref="S4.E21.m1.51.51.3.cmml"><mo id="S4.E21.m1.1.1.1.1.1.1" stretchy="false" xref="S4.E21.m1.51.51.3a.cmml">‖</mo><mrow id="S4.E21.m1.52.52.4.49.29.29.29.1.1.1" xref="S4.E21.m1.51.51.3.cmml"><msub id="S4.E21.m1.52.52.4.49.29.29.29.1.1.1.1" xref="S4.E21.m1.51.51.3.cmml"><mi id="S4.E21.m1.2.2.2.2.2.2" xref="S4.E21.m1.2.2.2.2.2.2.cmml">w</mi><mi id="S4.E21.m1.3.3.3.3.3.3.1" xref="S4.E21.m1.3.3.3.3.3.3.1.cmml">k</mi></msub><mo id="S4.E21.m1.4.4.4.4.4.4" xref="S4.E21.m1.4.4.4.4.4.4.cmml">−</mo><msup id="S4.E21.m1.52.52.4.49.29.29.29.1.1.1.2" xref="S4.E21.m1.51.51.3.cmml"><mi id="S4.E21.m1.5.5.5.5.5.5" xref="S4.E21.m1.5.5.5.5.5.5.cmml">w</mi><mo id="S4.E21.m1.6.6.6.6.6.6.1" xref="S4.E21.m1.6.6.6.6.6.6.1.cmml">∗</mo></msup></mrow><mo id="S4.E21.m1.7.7.7.7.7.7" stretchy="false" xref="S4.E21.m1.51.51.3a.cmml">‖</mo></mrow><mn id="S4.E21.m1.8.8.8.8.8.8.1" xref="S4.E21.m1.8.8.8.8.8.8.1.cmml">2</mn></msup><mo id="S4.E21.m1.9.9.9.9.9.9" xref="S4.E21.m1.9.9.9.9.9.9.cmml">=</mo><msup id="S4.E21.m1.53.53.5.50.30.30.30" xref="S4.E21.m1.51.51.3.cmml"><mrow id="S4.E21.m1.53.53.5.50.30.30.30.1.1" xref="S4.E21.m1.51.51.3.cmml"><mo id="S4.E21.m1.10.10.10.10.10.10" stretchy="false" xref="S4.E21.m1.51.51.3a.cmml">‖</mo><mrow id="S4.E21.m1.53.53.5.50.30.30.30.1.1.1" xref="S4.E21.m1.51.51.3.cmml"><mrow id="S4.E21.m1.53.53.5.50.30.30.30.1.1.1.1" xref="S4.E21.m1.51.51.3.cmml"><mi id="S4.E21.m1.11.11.11.11.11.11" xref="S4.E21.m1.11.11.11.11.11.11.cmml">M</mi><mo id="S4.E21.m1.12.12.12.12.12.12" xref="S4.E21.m1.12.12.12.12.12.12.cmml">−</mo><mrow id="S4.E21.m1.53.53.5.50.30.30.30.1.1.1.1.1" xref="S4.E21.m1.51.51.3.cmml"><msub id="S4.E21.m1.53.53.5.50.30.30.30.1.1.1.1.1.2" xref="S4.E21.m1.51.51.3.cmml"><mi id="S4.E21.m1.13.13.13.13.13.13" xref="S4.E21.m1.13.13.13.13.13.13.cmml">η</mi><mi id="S4.E21.m1.14.14.14.14.14.14.1" xref="S4.E21.m1.14.14.14.14.14.14.1.cmml">k</mi></msub><mo id="S4.E21.m1.53.53.5.50.30.30.30.1.1.1.1.1.1" xref="S4.E21.m1.51.51.3a.cmml">⁢</mo><mi id="S4.E21.m1.15.15.15.15.15.15" xref="S4.E21.m1.15.15.15.15.15.15.cmml">H</mi></mrow><mo id="S4.E21.m1.12.12.12.12.12.12a" xref="S4.E21.m1.12.12.12.12.12.12.cmml">−</mo><msup id="S4.E21.m1.53.53.5.50.30.30.30.1.1.1.1.2" xref="S4.E21.m1.51.51.3.cmml"><mi id="S4.E21.m1.17.17.17.17.17.17" xref="S4.E21.m1.17.17.17.17.17.17.cmml">w</mi><mo id="S4.E21.m1.18.18.18.18.18.18.1" xref="S4.E21.m1.18.18.18.18.18.18.1.cmml">∗</mo></msup><mo id="S4.E21.m1.12.12.12.12.12.12b" xref="S4.E21.m1.12.12.12.12.12.12.cmml">−</mo><mrow id="S4.E21.m1.53.53.5.50.30.30.30.1.1.1.1.3" xref="S4.E21.m1.51.51.3.cmml"><msub id="S4.E21.m1.53.53.5.50.30.30.30.1.1.1.1.3.2" xref="S4.E21.m1.51.51.3.cmml"><mi id="S4.E21.m1.20.20.20.20.20.20" xref="S4.E21.m1.20.20.20.20.20.20.cmml">η</mi><mi id="S4.E21.m1.21.21.21.21.21.21.1" xref="S4.E21.m1.21.21.21.21.21.21.1.cmml">k</mi></msub><mo id="S4.E21.m1.53.53.5.50.30.30.30.1.1.1.1.3.1" xref="S4.E21.m1.51.51.3a.cmml">⁢</mo><mover accent="true" id="S4.E21.m1.22.22.22.22.22.22" xref="S4.E21.m1.22.22.22.22.22.22.cmml"><mi id="S4.E21.m1.22.22.22.22.22.22.2" xref="S4.E21.m1.22.22.22.22.22.22.2.cmml">H</mi><mo id="S4.E21.m1.22.22.22.22.22.22.1" xref="S4.E21.m1.22.22.22.22.22.22.1.cmml">¯</mo></mover></mrow></mrow><mo id="S4.E21.m1.23.23.23.23.23.23" xref="S4.E21.m1.23.23.23.23.23.23.cmml">+</mo><mrow id="S4.E21.m1.53.53.5.50.30.30.30.1.1.1.2" xref="S4.E21.m1.51.51.3.cmml"><msub id="S4.E21.m1.53.53.5.50.30.30.30.1.1.1.2.2" xref="S4.E21.m1.51.51.3.cmml"><mi id="S4.E21.m1.24.24.24.24.24.24" xref="S4.E21.m1.24.24.24.24.24.24.cmml">η</mi><mi id="S4.E21.m1.25.25.25.25.25.25.1" xref="S4.E21.m1.25.25.25.25.25.25.1.cmml">k</mi></msub><mo id="S4.E21.m1.53.53.5.50.30.30.30.1.1.1.2.1" xref="S4.E21.m1.51.51.3a.cmml">⁢</mo><mover accent="true" id="S4.E21.m1.26.26.26.26.26.26" xref="S4.E21.m1.26.26.26.26.26.26.cmml"><mi id="S4.E21.m1.26.26.26.26.26.26.2" xref="S4.E21.m1.26.26.26.26.26.26.2.cmml">H</mi><mo id="S4.E21.m1.26.26.26.26.26.26.1" xref="S4.E21.m1.26.26.26.26.26.26.1.cmml">¯</mo></mover></mrow></mrow><mo id="S4.E21.m1.27.27.27.27.27.27" stretchy="false" xref="S4.E21.m1.51.51.3a.cmml">‖</mo></mrow><mn id="S4.E21.m1.28.28.28.28.28.28.1" xref="S4.E21.m1.28.28.28.28.28.28.1.cmml">2</mn></msup></mrow></mtd></mtr><mtr id="S4.E21.m1.54.54.6c" xref="S4.E21.m1.51.51.3.cmml"><mtd id="S4.E21.m1.54.54.6d" xref="S4.E21.m1.51.51.3.cmml"><mrow id="S4.E21.m1.54.54.6.51.21.21" xref="S4.E21.m1.51.51.3.cmml"><mi id="S4.E21.m1.54.54.6.51.21.21.22" xref="S4.E21.m1.51.51.3a.cmml"></mi><mo id="S4.E21.m1.29.29.29.1.1.1" xref="S4.E21.m1.29.29.29.1.1.1.cmml">=</mo><msup id="S4.E21.m1.54.54.6.51.21.21.21" xref="S4.E21.m1.51.51.3.cmml"><mrow id="S4.E21.m1.54.54.6.51.21.21.21.1.1" xref="S4.E21.m1.51.51.3.cmml"><mo id="S4.E21.m1.30.30.30.2.2.2" stretchy="false" xref="S4.E21.m1.51.51.3a.cmml">‖</mo><mrow id="S4.E21.m1.54.54.6.51.21.21.21.1.1.1" xref="S4.E21.m1.51.51.3.cmml"><mrow id="S4.E21.m1.54.54.6.51.21.21.21.1.1.1.1" xref="S4.E21.m1.51.51.3.cmml"><mrow id="S4.E21.m1.54.54.6.51.21.21.21.1.1.1.1.1" xref="S4.E21.m1.51.51.3.cmml"><mi id="S4.E21.m1.31.31.31.3.3.3" xref="S4.E21.m1.31.31.31.3.3.3.cmml">M</mi><mo id="S4.E21.m1.32.32.32.4.4.4" xref="S4.E21.m1.32.32.32.4.4.4.cmml">−</mo><mrow id="S4.E21.m1.54.54.6.51.21.21.21.1.1.1.1.1.1" xref="S4.E21.m1.51.51.3.cmml"><msub id="S4.E21.m1.54.54.6.51.21.21.21.1.1.1.1.1.1.2" xref="S4.E21.m1.51.51.3.cmml"><mi id="S4.E21.m1.33.33.33.5.5.5" xref="S4.E21.m1.33.33.33.5.5.5.cmml">η</mi><mi id="S4.E21.m1.34.34.34.6.6.6.1" xref="S4.E21.m1.34.34.34.6.6.6.1.cmml">k</mi></msub><mo id="S4.E21.m1.54.54.6.51.21.21.21.1.1.1.1.1.1.1" xref="S4.E21.m1.51.51.3a.cmml">⁢</mo><mover accent="true" id="S4.E21.m1.35.35.35.7.7.7" xref="S4.E21.m1.35.35.35.7.7.7.cmml"><mi id="S4.E21.m1.35.35.35.7.7.7.2" xref="S4.E21.m1.35.35.35.7.7.7.2.cmml">H</mi><mo id="S4.E21.m1.35.35.35.7.7.7.1" xref="S4.E21.m1.35.35.35.7.7.7.1.cmml">¯</mo></mover></mrow><mo id="S4.E21.m1.32.32.32.4.4.4a" xref="S4.E21.m1.32.32.32.4.4.4.cmml">−</mo><msup id="S4.E21.m1.54.54.6.51.21.21.21.1.1.1.1.1.2" xref="S4.E21.m1.51.51.3.cmml"><mi id="S4.E21.m1.37.37.37.9.9.9" xref="S4.E21.m1.37.37.37.9.9.9.cmml">w</mi><mo id="S4.E21.m1.38.38.38.10.10.10.1" xref="S4.E21.m1.38.38.38.10.10.10.1.cmml">∗</mo></msup></mrow><mo id="S4.E21.m1.39.39.39.11.11.11" xref="S4.E21.m1.39.39.39.11.11.11.cmml">+</mo><mrow id="S4.E21.m1.54.54.6.51.21.21.21.1.1.1.1.2" xref="S4.E21.m1.51.51.3.cmml"><msub id="S4.E21.m1.54.54.6.51.21.21.21.1.1.1.1.2.2" xref="S4.E21.m1.51.51.3.cmml"><mi id="S4.E21.m1.40.40.40.12.12.12" xref="S4.E21.m1.40.40.40.12.12.12.cmml">η</mi><mi id="S4.E21.m1.41.41.41.13.13.13.1" xref="S4.E21.m1.41.41.41.13.13.13.1.cmml">k</mi></msub><mo id="S4.E21.m1.54.54.6.51.21.21.21.1.1.1.1.2.1" xref="S4.E21.m1.51.51.3a.cmml">⁢</mo><mover accent="true" id="S4.E21.m1.42.42.42.14.14.14" xref="S4.E21.m1.42.42.42.14.14.14.cmml"><mi id="S4.E21.m1.42.42.42.14.14.14.2" xref="S4.E21.m1.42.42.42.14.14.14.2.cmml">H</mi><mo 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M-\eta_{k}H-w^% {\ast}-\eta_{k}\bar{H}+\eta_{k}\bar{H}\parallel^{2}\\ =\parallel M-\eta_{k}\bar{H}-w^{\ast}+\eta_{k}\bar{H}-\eta_{k}H\parallel^{2}% \end{gathered}</annotation><annotation encoding="application/x-llamapun" id="S4.E21.m1.54d">start_ROW start_CELL ∥ italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = ∥ italic_M - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_H - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT over¯ start_ARG italic_H end_ARG + italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT over¯ start_ARG italic_H end_ARG ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL = ∥ italic_M - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT over¯ start_ARG italic_H end_ARG - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT + italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT over¯ start_ARG italic_H end_ARG - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_H ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(21)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS3.p8"> <p class="ltx_p" id="S4.SS2.SSS3.p8.5">The same applies to the continuous linear, Equation <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S4.E6" title="In IV-B Convergence Analysis ‣ IV Convergence Rate Analysis ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">6</span></a> is equivalent to <math alttext="|M-\eta_{k}\bar{H}-w^{\ast}\|^{2}-2<M-\eta_{k}\bar{H}-w^{\ast},\eta_{k}\bar{H}% 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id="S4.SS2.SSS3.p8.1.m1.3c">|M-\eta_{k}\bar{H}-w^{\ast}\|^{2}-2<M-\eta_{k}\bar{H}-w^{\ast},\eta_{k}\bar{H}% -\eta_{k}H>+\eta^{2}_{k}\parallel\bar{H}\parallel^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p8.1.m1.3d">| italic_M - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT over¯ start_ARG italic_H end_ARG - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 < italic_M - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT over¯ start_ARG italic_H end_ARG - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT over¯ start_ARG italic_H end_ARG - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_H > + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ over¯ start_ARG italic_H end_ARG ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> and we will bond the <math alttext="|M-\eta_{k}\bar{H}-w^{\ast}\|^{2}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p8.2.m2.1"><semantics id="S4.SS2.SSS3.p8.2.m2.1a"><msup id="S4.SS2.SSS3.p8.2.m2.1.1" xref="S4.SS2.SSS3.p8.2.m2.1.1.cmml"><mrow id="S4.SS2.SSS3.p8.2.m2.1.1.1.1" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.2.cmml"><mo id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.2.1.cmml">|</mo><mrow id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.2" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.2.cmml">M</mi><mo id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.1" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.1.cmml">−</mo><mrow id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.cmml"><msub id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.2" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.2.cmml"><mi id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.2.2" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.2.2.cmml">η</mi><mi id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.2.3" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.2.3.cmml">k</mi></msub><mo id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.1" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.1.cmml">⁢</mo><mover accent="true" id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.3" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.3.cmml"><mi id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.3.2" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.3.2.cmml">H</mi><mo id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.3.1" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.3.1.cmml">¯</mo></mover></mrow><mo id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.1a" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.1.cmml">−</mo><msup id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.4" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.4.cmml"><mi id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.4.2" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.4.2.cmml">w</mi><mo id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.4.3" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.4.3.cmml">∗</mo></msup></mrow><mo id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S4.SS2.SSS3.p8.2.m2.1.1.3" xref="S4.SS2.SSS3.p8.2.m2.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p8.2.m2.1b"><apply id="S4.SS2.SSS3.p8.2.m2.1.1.cmml" xref="S4.SS2.SSS3.p8.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.p8.2.m2.1.1.2.cmml" xref="S4.SS2.SSS3.p8.2.m2.1.1">superscript</csymbol><apply id="S4.SS2.SSS3.p8.2.m2.1.1.1.2.cmml" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1"><csymbol cd="latexml" id="S4.SS2.SSS3.p8.2.m2.1.1.1.2.1.cmml" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.2">delimited-|‖</csymbol><apply id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1"><minus id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.1"></minus><ci id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.2">𝑀</ci><apply id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3"><times id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.1.cmml" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.1"></times><apply id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.2.cmml" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.2.1.cmml" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.2">subscript</csymbol><ci id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.2.2.cmml" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.2.2">𝜂</ci><ci id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.2.3.cmml" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.2.3">𝑘</ci></apply><apply id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.3.cmml" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.3"><ci id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.3.1.cmml" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.3.1">¯</ci><ci id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.3.2.cmml" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.3.3.2">𝐻</ci></apply></apply><apply id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.4.cmml" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.4"><csymbol cd="ambiguous" id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.4.1.cmml" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.4">superscript</csymbol><ci id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.4.2.cmml" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.4.2">𝑤</ci><ci id="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.4.3.cmml" xref="S4.SS2.SSS3.p8.2.m2.1.1.1.1.1.4.3">∗</ci></apply></apply></apply><cn id="S4.SS2.SSS3.p8.2.m2.1.1.3.cmml" type="integer" xref="S4.SS2.SSS3.p8.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p8.2.m2.1c">|M-\eta_{k}\bar{H}-w^{\ast}\|^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p8.2.m2.1d">| italic_M - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT over¯ start_ARG italic_H end_ARG - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> first. <math alttext="\|M-\eta_{k}\bar{H}-w^{\ast}\|^{2}=\ \|M-w^{\ast}\|^{2}-2\eta_{k}<M-w^{\ast},% \bar{H}>+\eta^{2}_{t}\|\bar{H}\|^{2}\\ " class="ltx_Math" display="inline" id="S4.SS2.SSS3.p8.3.m3.4"><semantics id="S4.SS2.SSS3.p8.3.m3.4a"><mrow id="S4.SS2.SSS3.p8.3.m3.4.4.2" xref="S4.SS2.SSS3.p8.3.m3.4.4.3.cmml"><mrow id="S4.SS2.SSS3.p8.3.m3.3.3.1.1" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.cmml"><msup id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.cmml"><mrow id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.2.cmml"><mo id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.2.1.cmml">‖</mo><mrow id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.2" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.2.cmml">M</mi><mo id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.1" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.1.cmml">−</mo><mrow id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.3" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.3.cmml"><msub id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.3.2" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.3.2.cmml"><mi id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.3.2.2" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.3.2.2.cmml">η</mi><mi id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.3.2.3" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.3.2.3.cmml">k</mi></msub><mo id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.3.1" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.3.1.cmml">⁢</mo><mover accent="true" id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.3.3" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.3.3.cmml"><mi id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.3.3.2" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.3.3.2.cmml">H</mi><mo id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.3.3.1" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.3.3.1.cmml">¯</mo></mover></mrow><mo id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.1a" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.1.cmml">−</mo><msup id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.4" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.4.cmml"><mi id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.4.2" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.4.2.cmml">w</mi><mo id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.4.3" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.1.4.3.cmml">∗</mo></msup></mrow><mo id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.3" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.1.3.cmml">2</mn></msup><mo id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.4" rspace="0.778em" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.4.cmml">=</mo><mrow id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.2" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.2.cmml"><msup id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.2.1" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.2.1.cmml"><mrow id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.2.1.1.1" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.2.1.1.2.cmml"><mo id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.2.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.2.1.1.2.1.cmml">‖</mo><mrow id="S4.SS2.SSS3.p8.3.m3.3.3.1.1.2.1.1.1.1" xref="S4.SS2.SSS3.p8.3.m3.3.3.1.1.2.1.1.1.1.cmml"><mi 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xref="S4.SS2.SSS3.p8.3.m3.4.4.2.2.2.2.2.2.2">𝜂</ci><cn id="S4.SS2.SSS3.p8.3.m3.4.4.2.2.2.2.2.2.3.cmml" type="integer" xref="S4.SS2.SSS3.p8.3.m3.4.4.2.2.2.2.2.2.3">2</cn></apply><ci id="S4.SS2.SSS3.p8.3.m3.4.4.2.2.2.2.2.3.cmml" xref="S4.SS2.SSS3.p8.3.m3.4.4.2.2.2.2.2.3">𝑡</ci></apply><apply id="S4.SS2.SSS3.p8.3.m3.4.4.2.2.2.2.3.cmml" xref="S4.SS2.SSS3.p8.3.m3.4.4.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS3.p8.3.m3.4.4.2.2.2.2.3.1.cmml" xref="S4.SS2.SSS3.p8.3.m3.4.4.2.2.2.2.3">superscript</csymbol><apply id="S4.SS2.SSS3.p8.3.m3.4.4.2.2.2.2.3.2.1.cmml" xref="S4.SS2.SSS3.p8.3.m3.4.4.2.2.2.2.3.2.2"><csymbol cd="latexml" id="S4.SS2.SSS3.p8.3.m3.4.4.2.2.2.2.3.2.1.1.cmml" xref="S4.SS2.SSS3.p8.3.m3.4.4.2.2.2.2.3.2.2.1">norm</csymbol><apply id="S4.SS2.SSS3.p8.3.m3.2.2.cmml" xref="S4.SS2.SSS3.p8.3.m3.2.2"><ci id="S4.SS2.SSS3.p8.3.m3.2.2.1.cmml" xref="S4.SS2.SSS3.p8.3.m3.2.2.1">¯</ci><ci id="S4.SS2.SSS3.p8.3.m3.2.2.2.cmml" xref="S4.SS2.SSS3.p8.3.m3.2.2.2">𝐻</ci></apply></apply><cn id="S4.SS2.SSS3.p8.3.m3.4.4.2.2.2.2.3.3.cmml" type="integer" xref="S4.SS2.SSS3.p8.3.m3.4.4.2.2.2.2.3.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p8.3.m3.4c">\|M-\eta_{k}\bar{H}-w^{\ast}\|^{2}=\ \|M-w^{\ast}\|^{2}-2\eta_{k}<M-w^{\ast},% \bar{H}>+\eta^{2}_{t}\|\bar{H}\|^{2}\\ </annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p8.3.m3.4d">∥ italic_M - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT over¯ start_ARG italic_H end_ARG - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = ∥ italic_M - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT < italic_M - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , over¯ start_ARG italic_H end_ARG > + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∥ over¯ start_ARG italic_H end_ARG ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> and we will bond the <math alttext="2\eta^{2}_{k}<M-w^{\ast},\bar{H}>" class="ltx_math_unparsed" display="inline" id="S4.SS2.SSS3.p8.4.m4.1"><semantics id="S4.SS2.SSS3.p8.4.m4.1a"><mrow id="S4.SS2.SSS3.p8.4.m4.1b"><mn id="S4.SS2.SSS3.p8.4.m4.1.2">2</mn><msubsup id="S4.SS2.SSS3.p8.4.m4.1.3"><mi id="S4.SS2.SSS3.p8.4.m4.1.3.2.2">η</mi><mi id="S4.SS2.SSS3.p8.4.m4.1.3.3">k</mi><mn id="S4.SS2.SSS3.p8.4.m4.1.3.2.3">2</mn></msubsup><mo id="S4.SS2.SSS3.p8.4.m4.1.4"><</mo><mi id="S4.SS2.SSS3.p8.4.m4.1.5">M</mi><mo id="S4.SS2.SSS3.p8.4.m4.1.6">−</mo><msup id="S4.SS2.SSS3.p8.4.m4.1.7"><mi id="S4.SS2.SSS3.p8.4.m4.1.7.2">w</mi><mo id="S4.SS2.SSS3.p8.4.m4.1.7.3">∗</mo></msup><mo id="S4.SS2.SSS3.p8.4.m4.1.8">,</mo><mover accent="true" id="S4.SS2.SSS3.p8.4.m4.1.1"><mi id="S4.SS2.SSS3.p8.4.m4.1.1.2">H</mi><mo id="S4.SS2.SSS3.p8.4.m4.1.1.1">¯</mo></mover><mo id="S4.SS2.SSS3.p8.4.m4.1.9">></mo></mrow><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p8.4.m4.1c">2\eta^{2}_{k}<M-w^{\ast},\bar{H}></annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p8.4.m4.1d">2 italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT < italic_M - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , over¯ start_ARG italic_H end_ARG ></annotation></semantics></math> and <math alttext="\eta^{2}_{k}\|\bar{H}\|^{2}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p8.5.m5.1"><semantics id="S4.SS2.SSS3.p8.5.m5.1a"><mrow id="S4.SS2.SSS3.p8.5.m5.1.2" xref="S4.SS2.SSS3.p8.5.m5.1.2.cmml"><msubsup id="S4.SS2.SSS3.p8.5.m5.1.2.2" xref="S4.SS2.SSS3.p8.5.m5.1.2.2.cmml"><mi id="S4.SS2.SSS3.p8.5.m5.1.2.2.2.2" xref="S4.SS2.SSS3.p8.5.m5.1.2.2.2.2.cmml">η</mi><mi id="S4.SS2.SSS3.p8.5.m5.1.2.2.3" xref="S4.SS2.SSS3.p8.5.m5.1.2.2.3.cmml">k</mi><mn id="S4.SS2.SSS3.p8.5.m5.1.2.2.2.3" xref="S4.SS2.SSS3.p8.5.m5.1.2.2.2.3.cmml">2</mn></msubsup><mo id="S4.SS2.SSS3.p8.5.m5.1.2.1" xref="S4.SS2.SSS3.p8.5.m5.1.2.1.cmml">⁢</mo><msup id="S4.SS2.SSS3.p8.5.m5.1.2.3" xref="S4.SS2.SSS3.p8.5.m5.1.2.3.cmml"><mrow id="S4.SS2.SSS3.p8.5.m5.1.2.3.2.2" xref="S4.SS2.SSS3.p8.5.m5.1.2.3.2.1.cmml"><mo id="S4.SS2.SSS3.p8.5.m5.1.2.3.2.2.1" stretchy="false" xref="S4.SS2.SSS3.p8.5.m5.1.2.3.2.1.1.cmml">‖</mo><mover accent="true" id="S4.SS2.SSS3.p8.5.m5.1.1" xref="S4.SS2.SSS3.p8.5.m5.1.1.cmml"><mi id="S4.SS2.SSS3.p8.5.m5.1.1.2" xref="S4.SS2.SSS3.p8.5.m5.1.1.2.cmml">H</mi><mo id="S4.SS2.SSS3.p8.5.m5.1.1.1" xref="S4.SS2.SSS3.p8.5.m5.1.1.1.cmml">¯</mo></mover><mo id="S4.SS2.SSS3.p8.5.m5.1.2.3.2.2.2" stretchy="false" xref="S4.SS2.SSS3.p8.5.m5.1.2.3.2.1.1.cmml">‖</mo></mrow><mn id="S4.SS2.SSS3.p8.5.m5.1.2.3.3" xref="S4.SS2.SSS3.p8.5.m5.1.2.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p8.5.m5.1b"><apply id="S4.SS2.SSS3.p8.5.m5.1.2.cmml" 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id="S4.SS2.SSS3.p8.5.m5.1.2.3.2.1.cmml" xref="S4.SS2.SSS3.p8.5.m5.1.2.3.2.2"><csymbol cd="latexml" id="S4.SS2.SSS3.p8.5.m5.1.2.3.2.1.1.cmml" xref="S4.SS2.SSS3.p8.5.m5.1.2.3.2.2.1">norm</csymbol><apply id="S4.SS2.SSS3.p8.5.m5.1.1.cmml" xref="S4.SS2.SSS3.p8.5.m5.1.1"><ci id="S4.SS2.SSS3.p8.5.m5.1.1.1.cmml" xref="S4.SS2.SSS3.p8.5.m5.1.1.1">¯</ci><ci id="S4.SS2.SSS3.p8.5.m5.1.1.2.cmml" xref="S4.SS2.SSS3.p8.5.m5.1.1.2">𝐻</ci></apply></apply><cn id="S4.SS2.SSS3.p8.5.m5.1.2.3.3.cmml" type="integer" xref="S4.SS2.SSS3.p8.5.m5.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p8.5.m5.1c">\eta^{2}_{k}\|\bar{H}\|^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p8.5.m5.1d">italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ over¯ start_ARG italic_H end_ARG ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math></p> </div> <div class="ltx_para" id="S4.SS2.SSS3.p9"> <table class="ltx_equation ltx_eqn_table" id="S4.E22"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{split}2\eta_{k}^{2}\langle M-w^{\ast},\bar{H}\rangle&\leq-2\eta_{k}% \left(F_{k}(M)-F(w^{\ast})\right.\\ &\quad\left.+\frac{\mu}{2}\|M-w^{\ast}\|_{2}^{2}\right)\end{split}" class="ltx_Math" display="block" id="S4.E22.m1.43"><semantics id="S4.E22.m1.43a"><mtable columnspacing="0pt" displaystyle="true" id="S4.E22.m1.43.43.3" rowspacing="0pt" xref="S4.E22.m1.42.42.2.cmml"><mtr id="S4.E22.m1.43.43.3a" xref="S4.E22.m1.42.42.2.cmml"><mtd class="ltx_align_right" columnalign="right" id="S4.E22.m1.43.43.3b" xref="S4.E22.m1.42.42.2.cmml"><mrow id="S4.E22.m1.43.43.3.41.30.13" xref="S4.E22.m1.42.42.2.cmml"><mn id="S4.E22.m1.1.1.1.1.1.1" xref="S4.E22.m1.1.1.1.1.1.1.cmml">2</mn><mo id="S4.E22.m1.43.43.3.41.30.13.14" xref="S4.E22.m1.42.42.2.cmml">⁢</mo><msubsup id="S4.E22.m1.43.43.3.41.30.13.15" xref="S4.E22.m1.42.42.2.cmml"><mi id="S4.E22.m1.2.2.2.2.2.2" xref="S4.E22.m1.2.2.2.2.2.2.cmml">η</mi><mi id="S4.E22.m1.3.3.3.3.3.3.1" xref="S4.E22.m1.3.3.3.3.3.3.1.cmml">k</mi><mn id="S4.E22.m1.4.4.4.4.4.4.1" xref="S4.E22.m1.4.4.4.4.4.4.1.cmml">2</mn></msubsup><mo id="S4.E22.m1.43.43.3.41.30.13.14a" xref="S4.E22.m1.42.42.2.cmml">⁢</mo><mrow id="S4.E22.m1.43.43.3.41.30.13.13.1" xref="S4.E22.m1.42.42.2.cmml"><mo id="S4.E22.m1.5.5.5.5.5.5" stretchy="false" xref="S4.E22.m1.42.42.2.cmml">⟨</mo><mrow id="S4.E22.m1.43.43.3.41.30.13.13.1.1" xref="S4.E22.m1.42.42.2.cmml"><mi id="S4.E22.m1.6.6.6.6.6.6" xref="S4.E22.m1.6.6.6.6.6.6.cmml">M</mi><mo id="S4.E22.m1.7.7.7.7.7.7" xref="S4.E22.m1.7.7.7.7.7.7.cmml">−</mo><msup id="S4.E22.m1.43.43.3.41.30.13.13.1.1.1" xref="S4.E22.m1.42.42.2.cmml"><mi id="S4.E22.m1.8.8.8.8.8.8" xref="S4.E22.m1.8.8.8.8.8.8.cmml">w</mi><mo id="S4.E22.m1.9.9.9.9.9.9.1" xref="S4.E22.m1.9.9.9.9.9.9.1.cmml">∗</mo></msup></mrow><mo id="S4.E22.m1.10.10.10.10.10.10" xref="S4.E22.m1.42.42.2.cmml">,</mo><mover accent="true" id="S4.E22.m1.11.11.11.11.11.11" xref="S4.E22.m1.11.11.11.11.11.11.cmml"><mi id="S4.E22.m1.11.11.11.11.11.11.2" xref="S4.E22.m1.11.11.11.11.11.11.2.cmml">H</mi><mo id="S4.E22.m1.11.11.11.11.11.11.1" xref="S4.E22.m1.11.11.11.11.11.11.1.cmml">¯</mo></mover><mo id="S4.E22.m1.12.12.12.12.12.12" stretchy="false" xref="S4.E22.m1.42.42.2.cmml">⟩</mo></mrow></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.E22.m1.43.43.3c" xref="S4.E22.m1.42.42.2.cmml"><mrow id="S4.E22.m1.29.29.29.29.17" xref="S4.E22.m1.42.42.2.cmml"><mo id="S4.E22.m1.13.13.13.13.1.1" rspace="0em" xref="S4.E22.m1.13.13.13.13.1.1.cmml">≤</mo><mo id="S4.E22.m1.14.14.14.14.2.2" lspace="0em" xref="S4.E22.m1.14.14.14.14.2.2.cmml">−</mo><mn id="S4.E22.m1.15.15.15.15.3.3" xref="S4.E22.m1.15.15.15.15.3.3.cmml">2</mn><msub id="S4.E22.m1.29.29.29.29.17.18" 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italic_M - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , over¯ start_ARG italic_H end_ARG ⟩ end_CELL start_CELL ≤ - 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_M ) - italic_F ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + divide start_ARG italic_μ end_ARG start_ARG 2 end_ARG ∥ italic_M - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(22)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS3.p10"> <table class="ltx_equation ltx_eqn_table" id="S4.E23"> <tbody><tr 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cd="ambiguous" id="S4.E23.m1.4.4.2.1.1.1.3.1.cmml" xref="S4.E23.m1.4.4.2.1.1.1.3">subscript</csymbol><apply id="S4.E23.m1.4.4.2.1.1.1.3.2.cmml" xref="S4.E23.m1.4.4.2.1.1.1.3"><csymbol cd="ambiguous" id="S4.E23.m1.4.4.2.1.1.1.3.2.1.cmml" xref="S4.E23.m1.4.4.2.1.1.1.3">superscript</csymbol><ci id="S4.E23.m1.4.4.2.1.1.1.3.2.2.cmml" xref="S4.E23.m1.4.4.2.1.1.1.3.2.2">𝐹</ci><ci id="S4.E23.m1.4.4.2.1.1.1.3.2.3.cmml" xref="S4.E23.m1.4.4.2.1.1.1.3.2.3">∗</ci></apply><ci id="S4.E23.m1.4.4.2.1.1.1.3.3.cmml" xref="S4.E23.m1.4.4.2.1.1.1.3.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E23.m1.4c">\eta^{2}_{k}\|\nabla F_{k}\left(M\right)\|^{2}\leq 2\eta^{2}_{k}L\left(F_{k}% \left(M\right)-F^{\ast}_{k}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.E23.m1.4d">italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ ∇ italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_M ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ 2 italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_L ( italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_M ) - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(23)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS3.p11"> <p class="ltx_p" id="S4.SS2.SSS3.p11.1">Then, we get</p> <table class="ltx_equation ltx_eqn_table" id="S4.E24"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{split}\|M-\eta_{k}\bar{H}-w^{\ast}\|^{2}\leq&\left(1+\mu\eta_{k}\right)% \|M-w^{\ast}\|^{2}\\ &+2\eta_{k}\left(F_{k}\left(w^{\ast}\right)-F_{k}\left(M\right)\right)\\ &+2\eta_{k}^{2}L\left(F_{k}\left(M\right)-F^{\ast}_{k}\right)\end{split}" class="ltx_Math" display="block" id="S4.E24.m1.71"><semantics id="S4.E24.m1.71a"><mtable columnspacing="0pt" displaystyle="true" id="S4.E24.m1.71.71.10" rowspacing="0pt" xref="S4.E24.m1.66.66.5.cmml"><mtr id="S4.E24.m1.71.71.10a" xref="S4.E24.m1.66.66.5.cmml"><mtd class="ltx_align_right" columnalign="right" id="S4.E24.m1.71.71.10b" xref="S4.E24.m1.66.66.5.cmml"><mrow id="S4.E24.m1.67.67.6.62.27.13" xref="S4.E24.m1.66.66.5.cmml"><msup id="S4.E24.m1.67.67.6.62.27.13.13" xref="S4.E24.m1.66.66.5.cmml"><mrow id="S4.E24.m1.67.67.6.62.27.13.13.1.1" xref="S4.E24.m1.66.66.5.cmml"><mo id="S4.E24.m1.1.1.1.1.1.1" stretchy="false" xref="S4.E24.m1.66.66.5.cmml">‖</mo><mrow id="S4.E24.m1.67.67.6.62.27.13.13.1.1.1" xref="S4.E24.m1.66.66.5.cmml"><mi 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end_POSTSUPERSCRIPT ≤ end_CELL start_CELL ( 1 + italic_μ italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ∥ italic_M - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) - italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_M ) ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_L ( italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_M ) - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(24)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS3.p12"> <p class="ltx_p" id="S4.SS2.SSS3.p12.3">In the same way as the continuous linear, we can get the equation:</p> <table class="ltx_equation ltx_eqn_table" id="S4.E25"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{split}\begin{gathered}A\leq\left(1+\mu\eta_{k}\right)\|M-w^{\ast}\|^{2}% +2\eta_{k}Z+\eta^{2}_{k}L^{2}\|M-w^{\ast}\|^{2}\\ \|w_{k}-w^{\ast}\|^{2}\leq\left(1+\mu\eta_{k}\right)\|M-w^{\ast}\|^{2}\\ +2\eta_{k}Z+\eta^{2}_{k}L^{2}\|M-w^{\ast}\|^{2}+\eta^{2}_{k}\sigma^{2}_{k}\end% {gathered}\end{split}" class="ltx_Math" display="block" id="S4.E25.m1.83"><semantics id="S4.E25.m1.83a"><mtable displaystyle="true" 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xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mi id="S4.E25.m1.29.29.29.29.29.29.29.29.29.29" xref="S4.E25.m1.29.29.29.29.29.29.29.29.29.29.cmml">M</mi><mo id="S4.E25.m1.30.30.30.30.30.30.30.30.30.30" xref="S4.E25.m1.30.30.30.30.30.30.30.30.30.30.cmml">−</mo><msup id="S4.E25.m1.83.83.83.83.83.83.10.85.37.37.37.3.1.1.1.1.1" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mi id="S4.E25.m1.31.31.31.31.31.31.31.31.31.31" xref="S4.E25.m1.31.31.31.31.31.31.31.31.31.31.cmml">w</mi><mo id="S4.E25.m1.32.32.32.32.32.32.32.32.32.32.1" xref="S4.E25.m1.32.32.32.32.32.32.32.32.32.32.1.cmml">∗</mo></msup></mrow><mo id="S4.E25.m1.33.33.33.33.33.33.33.33.33.33" stretchy="false" xref="S4.E25.m1.83.83.83.83.83.83.7a.cmml">‖</mo></mrow><mn id="S4.E25.m1.34.34.34.34.34.34.34.34.34.34.1" xref="S4.E25.m1.34.34.34.34.34.34.34.34.34.34.1.cmml">2</mn></msup></mrow></mrow></mrow></mtd></mtr><mtr id="S4.E25.m1.83.83.83.83.83.83.14c" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mtd id="S4.E25.m1.83.83.83.83.83.83.14d" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mrow id="S4.E25.m1.83.83.83.83.83.83.13.88.26.26" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><msup id="S4.E25.m1.83.83.83.83.83.83.11.86.24.24.24" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mrow id="S4.E25.m1.83.83.83.83.83.83.11.86.24.24.24.1.1" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mo id="S4.E25.m1.35.35.35.35.35.35.35.1.1.1" stretchy="false" xref="S4.E25.m1.83.83.83.83.83.83.7a.cmml">‖</mo><mrow id="S4.E25.m1.83.83.83.83.83.83.11.86.24.24.24.1.1.1" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><msub id="S4.E25.m1.83.83.83.83.83.83.11.86.24.24.24.1.1.1.1" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mi id="S4.E25.m1.36.36.36.36.36.36.36.2.2.2" xref="S4.E25.m1.36.36.36.36.36.36.36.2.2.2.cmml">w</mi><mi id="S4.E25.m1.37.37.37.37.37.37.37.3.3.3.1" xref="S4.E25.m1.37.37.37.37.37.37.37.3.3.3.1.cmml">k</mi></msub><mo id="S4.E25.m1.38.38.38.38.38.38.38.4.4.4" xref="S4.E25.m1.38.38.38.38.38.38.38.4.4.4.cmml">−</mo><msup id="S4.E25.m1.83.83.83.83.83.83.11.86.24.24.24.1.1.1.2" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mi id="S4.E25.m1.39.39.39.39.39.39.39.5.5.5" xref="S4.E25.m1.39.39.39.39.39.39.39.5.5.5.cmml">w</mi><mo id="S4.E25.m1.40.40.40.40.40.40.40.6.6.6.1" xref="S4.E25.m1.40.40.40.40.40.40.40.6.6.6.1.cmml">∗</mo></msup></mrow><mo id="S4.E25.m1.41.41.41.41.41.41.41.7.7.7" stretchy="false" xref="S4.E25.m1.83.83.83.83.83.83.7a.cmml">‖</mo></mrow><mn id="S4.E25.m1.42.42.42.42.42.42.42.8.8.8.1" xref="S4.E25.m1.42.42.42.42.42.42.42.8.8.8.1.cmml">2</mn></msup><mo id="S4.E25.m1.43.43.43.43.43.43.43.9.9.9" xref="S4.E25.m1.43.43.43.43.43.43.43.9.9.9.cmml">≤</mo><mrow id="S4.E25.m1.83.83.83.83.83.83.13.88.26.26.26" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mrow id="S4.E25.m1.83.83.83.83.83.83.12.87.25.25.25.1.1" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mo id="S4.E25.m1.44.44.44.44.44.44.44.10.10.10" xref="S4.E25.m1.83.83.83.83.83.83.7a.cmml">(</mo><mrow id="S4.E25.m1.83.83.83.83.83.83.12.87.25.25.25.1.1.1" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mn id="S4.E25.m1.45.45.45.45.45.45.45.11.11.11" xref="S4.E25.m1.45.45.45.45.45.45.45.11.11.11.cmml">1</mn><mo id="S4.E25.m1.46.46.46.46.46.46.46.12.12.12" xref="S4.E25.m1.46.46.46.46.46.46.46.12.12.12.cmml">+</mo><mrow id="S4.E25.m1.83.83.83.83.83.83.12.87.25.25.25.1.1.1.1" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mi id="S4.E25.m1.47.47.47.47.47.47.47.13.13.13" xref="S4.E25.m1.47.47.47.47.47.47.47.13.13.13.cmml">μ</mi><mo id="S4.E25.m1.83.83.83.83.83.83.12.87.25.25.25.1.1.1.1.1" xref="S4.E25.m1.83.83.83.83.83.83.7a.cmml">⁢</mo><msub id="S4.E25.m1.83.83.83.83.83.83.12.87.25.25.25.1.1.1.1.2" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mi id="S4.E25.m1.48.48.48.48.48.48.48.14.14.14" xref="S4.E25.m1.48.48.48.48.48.48.48.14.14.14.cmml">η</mi><mi id="S4.E25.m1.49.49.49.49.49.49.49.15.15.15.1" xref="S4.E25.m1.49.49.49.49.49.49.49.15.15.15.1.cmml">k</mi></msub></mrow></mrow><mo id="S4.E25.m1.50.50.50.50.50.50.50.16.16.16" xref="S4.E25.m1.83.83.83.83.83.83.7a.cmml">)</mo></mrow><mo id="S4.E25.m1.83.83.83.83.83.83.13.88.26.26.26.3" xref="S4.E25.m1.83.83.83.83.83.83.7a.cmml">⁢</mo><msup id="S4.E25.m1.83.83.83.83.83.83.13.88.26.26.26.2" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mrow id="S4.E25.m1.83.83.83.83.83.83.13.88.26.26.26.2.1.1" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mo id="S4.E25.m1.51.51.51.51.51.51.51.17.17.17" stretchy="false" xref="S4.E25.m1.83.83.83.83.83.83.7a.cmml">‖</mo><mrow id="S4.E25.m1.83.83.83.83.83.83.13.88.26.26.26.2.1.1.1" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mi id="S4.E25.m1.52.52.52.52.52.52.52.18.18.18" xref="S4.E25.m1.52.52.52.52.52.52.52.18.18.18.cmml">M</mi><mo id="S4.E25.m1.53.53.53.53.53.53.53.19.19.19" xref="S4.E25.m1.53.53.53.53.53.53.53.19.19.19.cmml">−</mo><msup id="S4.E25.m1.83.83.83.83.83.83.13.88.26.26.26.2.1.1.1.1" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mi id="S4.E25.m1.54.54.54.54.54.54.54.20.20.20" xref="S4.E25.m1.54.54.54.54.54.54.54.20.20.20.cmml">w</mi><mo id="S4.E25.m1.55.55.55.55.55.55.55.21.21.21.1" xref="S4.E25.m1.55.55.55.55.55.55.55.21.21.21.1.cmml">∗</mo></msup></mrow><mo id="S4.E25.m1.56.56.56.56.56.56.56.22.22.22" stretchy="false" xref="S4.E25.m1.83.83.83.83.83.83.7a.cmml">‖</mo></mrow><mn id="S4.E25.m1.57.57.57.57.57.57.57.23.23.23.1" xref="S4.E25.m1.57.57.57.57.57.57.57.23.23.23.1.cmml">2</mn></msup></mrow></mrow></mtd></mtr><mtr id="S4.E25.m1.83.83.83.83.83.83.14e" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mtd id="S4.E25.m1.83.83.83.83.83.83.14f" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mrow id="S4.E25.m1.83.83.83.83.83.83.14.89.26.26" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mrow id="S4.E25.m1.83.83.83.83.83.83.14.89.26.26.27" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mo id="S4.E25.m1.83.83.83.83.83.83.14.89.26.26.27a" xref="S4.E25.m1.83.83.83.83.83.83.7a.cmml">+</mo><mrow id="S4.E25.m1.83.83.83.83.83.83.14.89.26.26.27.1" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mn id="S4.E25.m1.59.59.59.59.59.59.59.2.2.2" xref="S4.E25.m1.59.59.59.59.59.59.59.2.2.2.cmml">2</mn><mo id="S4.E25.m1.83.83.83.83.83.83.14.89.26.26.27.1.1" xref="S4.E25.m1.83.83.83.83.83.83.7a.cmml">⁢</mo><msub id="S4.E25.m1.83.83.83.83.83.83.14.89.26.26.27.1.2" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mi id="S4.E25.m1.60.60.60.60.60.60.60.3.3.3" xref="S4.E25.m1.60.60.60.60.60.60.60.3.3.3.cmml">η</mi><mi id="S4.E25.m1.61.61.61.61.61.61.61.4.4.4.1" xref="S4.E25.m1.61.61.61.61.61.61.61.4.4.4.1.cmml">k</mi></msub><mo id="S4.E25.m1.83.83.83.83.83.83.14.89.26.26.27.1.1a" xref="S4.E25.m1.83.83.83.83.83.83.7a.cmml">⁢</mo><mi id="S4.E25.m1.62.62.62.62.62.62.62.5.5.5" xref="S4.E25.m1.62.62.62.62.62.62.62.5.5.5.cmml">Z</mi></mrow></mrow><mo id="S4.E25.m1.63.63.63.63.63.63.63.6.6.6" xref="S4.E25.m1.83.83.83.83.83.83.7a.cmml">+</mo><mrow id="S4.E25.m1.83.83.83.83.83.83.14.89.26.26.26" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><msubsup id="S4.E25.m1.83.83.83.83.83.83.14.89.26.26.26.3" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mi id="S4.E25.m1.64.64.64.64.64.64.64.7.7.7" xref="S4.E25.m1.64.64.64.64.64.64.64.7.7.7.cmml">η</mi><mi id="S4.E25.m1.66.66.66.66.66.66.66.9.9.9.1" xref="S4.E25.m1.66.66.66.66.66.66.66.9.9.9.1.cmml">k</mi><mn id="S4.E25.m1.65.65.65.65.65.65.65.8.8.8.1" xref="S4.E25.m1.65.65.65.65.65.65.65.8.8.8.1.cmml">2</mn></msubsup><mo id="S4.E25.m1.83.83.83.83.83.83.14.89.26.26.26.2" xref="S4.E25.m1.83.83.83.83.83.83.7a.cmml">⁢</mo><msup id="S4.E25.m1.83.83.83.83.83.83.14.89.26.26.26.4" xref="S4.E25.m1.83.83.83.83.83.83.7.cmml"><mi id="S4.E25.m1.67.67.67.67.67.67.67.10.10.10" xref="S4.E25.m1.67.67.67.67.67.67.67.10.10.10.cmml">L</mi><mn id="S4.E25.m1.68.68.68.68.68.68.68.11.11.11.1" xref="S4.E25.m1.68.68.68.68.68.68.68.11.11.11.1.cmml">2</mn></msup><mo id="S4.E25.m1.83.83.83.83.83.83.14.89.26.26.26.2a" xref="S4.E25.m1.83.83.83.83.83.83.7a.cmml">⁢</mo><msup id="S4.E25.m1.83.83.83.83.83.83.14.89.26.26.26.1" 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xref="S4.E25.m1.3.3.3.3.3.3.3.3.3.3">superscript</csymbol><ci id="S4.E25.m1.80.80.80.80.80.80.80.23.23.23.cmml" xref="S4.E25.m1.80.80.80.80.80.80.80.23.23.23">𝜎</ci><cn id="S4.E25.m1.81.81.81.81.81.81.81.24.24.24.1.cmml" type="integer" xref="S4.E25.m1.81.81.81.81.81.81.81.24.24.24.1">2</cn></apply><ci id="S4.E25.m1.82.82.82.82.82.82.82.25.25.25.1.cmml" xref="S4.E25.m1.82.82.82.82.82.82.82.25.25.25.1">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E25.m1.83c">\begin{split}\begin{gathered}A\leq\left(1+\mu\eta_{k}\right)\|M-w^{\ast}\|^{2}% +2\eta_{k}Z+\eta^{2}_{k}L^{2}\|M-w^{\ast}\|^{2}\\ \|w_{k}-w^{\ast}\|^{2}\leq\left(1+\mu\eta_{k}\right)\|M-w^{\ast}\|^{2}\\ +2\eta_{k}Z+\eta^{2}_{k}L^{2}\|M-w^{\ast}\|^{2}+\eta^{2}_{k}\sigma^{2}_{k}\end% {gathered}\end{split}</annotation><annotation encoding="application/x-llamapun" id="S4.E25.m1.83d">start_ROW start_CELL start_ROW start_CELL italic_A ≤ ( 1 + italic_μ italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ∥ italic_M - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_Z + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ italic_M - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL ∥ italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ ( 1 + italic_μ italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ∥ italic_M - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL + 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_Z + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ italic_M - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL end_ROW end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(25)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS3.p12.2">By the strong convex property, we can replace the <math alttext="M_{k}=\left(\frac{s_{k-1}w_{k-1}+s_{k-2}w_{k-2}}{\displaystyle\sum^{k-1}_{i=k}% s_{i}}\right)" class="ltx_Math" display="inline" 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start_ARG ∑ start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = italic_k end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG )</annotation></semantics></math> with <math alttext="V_{k}=\left(\frac{s_{k-1}w^{0}+s_{k-2}w^{0}}{\displaystyle\sum^{k-1}_{i=k}s_{i% }}\right)" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p12.2.m2.1"><semantics id="S4.SS2.SSS3.p12.2.m2.1a"><mrow id="S4.SS2.SSS3.p12.2.m2.1.2" xref="S4.SS2.SSS3.p12.2.m2.1.2.cmml"><msub id="S4.SS2.SSS3.p12.2.m2.1.2.2" xref="S4.SS2.SSS3.p12.2.m2.1.2.2.cmml"><mi id="S4.SS2.SSS3.p12.2.m2.1.2.2.2" xref="S4.SS2.SSS3.p12.2.m2.1.2.2.2.cmml">V</mi><mi id="S4.SS2.SSS3.p12.2.m2.1.2.2.3" xref="S4.SS2.SSS3.p12.2.m2.1.2.2.3.cmml">k</mi></msub><mo id="S4.SS2.SSS3.p12.2.m2.1.2.1" xref="S4.SS2.SSS3.p12.2.m2.1.2.1.cmml">=</mo><mrow id="S4.SS2.SSS3.p12.2.m2.1.2.3.2" xref="S4.SS2.SSS3.p12.2.m2.1.1.cmml"><mo id="S4.SS2.SSS3.p12.2.m2.1.2.3.2.1" 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italic_k end_POSTSUBSCRIPT = ( divide start_ARG italic_s start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT italic_w start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT + italic_s start_POSTSUBSCRIPT italic_k - 2 end_POSTSUBSCRIPT italic_w start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT end_ARG start_ARG ∑ start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = italic_k end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG )</annotation></semantics></math></p> </div> <div class="ltx_para" id="S4.SS2.SSS3.p13"> <p class="ltx_p" id="S4.SS2.SSS3.p13.1">Thus, we get the function for aggregate linear:</p> <table class="ltx_equation ltx_eqn_table" id="S4.E26"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{split}E(F(\bar{w}))-F\left(w^{\ast}\right))\leq\frac{L}{2}[\left(1+\mu% \eta_{k}\right)\|V_{k}-w^{\ast}\|^{2}\\ +2\eta_{k}Z+\eta^{2}_{k}L^{2}\|V_{k}-w^{\ast}\|^{2}+\eta^{2}_{k}\sigma^{2}_{k}% ]\end{split}" class="ltx_math_unparsed" display="block" id="S4.E26.m1.59"><semantics id="S4.E26.m1.59a"><mtable displaystyle="true" id="S4.E26.m1.59.59" rowspacing="0pt"><mtr id="S4.E26.m1.59.59a"><mtd class="ltx_align_right" columnalign="right" id="S4.E26.m1.59.59b"><mrow id="S4.E26.m1.32.32.32.32.32"><mi id="S4.E26.m1.1.1.1.1.1.1">E</mi><mrow id="S4.E26.m1.32.32.32.32.32.33"><mo id="S4.E26.m1.2.2.2.2.2.2" stretchy="false">(</mo><mi id="S4.E26.m1.3.3.3.3.3.3">F</mi><mrow id="S4.E26.m1.32.32.32.32.32.33.1"><mo id="S4.E26.m1.4.4.4.4.4.4" stretchy="false">(</mo><mover accent="true" id="S4.E26.m1.5.5.5.5.5.5"><mi id="S4.E26.m1.5.5.5.5.5.5.2">w</mi><mo id="S4.E26.m1.5.5.5.5.5.5.1">¯</mo></mover><mo id="S4.E26.m1.6.6.6.6.6.6" stretchy="false">)</mo></mrow><mo id="S4.E26.m1.7.7.7.7.7.7" stretchy="false">)</mo></mrow><mo id="S4.E26.m1.8.8.8.8.8.8">−</mo><mi id="S4.E26.m1.9.9.9.9.9.9">F</mi><mrow id="S4.E26.m1.32.32.32.32.32.34"><mo id="S4.E26.m1.10.10.10.10.10.10">(</mo><msup id="S4.E26.m1.32.32.32.32.32.34.1"><mi id="S4.E26.m1.11.11.11.11.11.11">w</mi><mo id="S4.E26.m1.12.12.12.12.12.12.1">∗</mo></msup><mo id="S4.E26.m1.13.13.13.13.13.13">)</mo></mrow><mo id="S4.E26.m1.14.14.14.14.14.14" stretchy="false">)</mo><mo id="S4.E26.m1.15.15.15.15.15.15">≤</mo><mfrac id="S4.E26.m1.16.16.16.16.16.16"><mi id="S4.E26.m1.16.16.16.16.16.16.2">L</mi><mn id="S4.E26.m1.16.16.16.16.16.16.3">2</mn></mfrac><mo id="S4.E26.m1.17.17.17.17.17.17" stretchy="false">[</mo><mrow id="S4.E26.m1.32.32.32.32.32.35"><mo id="S4.E26.m1.18.18.18.18.18.18">(</mo><mn id="S4.E26.m1.19.19.19.19.19.19">1</mn><mo id="S4.E26.m1.20.20.20.20.20.20">+</mo><mi id="S4.E26.m1.21.21.21.21.21.21">μ</mi><msub id="S4.E26.m1.32.32.32.32.32.35.1"><mi id="S4.E26.m1.22.22.22.22.22.22">η</mi><mi id="S4.E26.m1.23.23.23.23.23.23.1">k</mi></msub><mo id="S4.E26.m1.24.24.24.24.24.24">)</mo></mrow><mo id="S4.E26.m1.25.25.25.25.25.25" lspace="0em" rspace="0.167em">∥</mo><msub id="S4.E26.m1.32.32.32.32.32.36"><mi id="S4.E26.m1.26.26.26.26.26.26">V</mi><mi id="S4.E26.m1.27.27.27.27.27.27.1">k</mi></msub><mo id="S4.E26.m1.28.28.28.28.28.28">−</mo><msup id="S4.E26.m1.32.32.32.32.32.37"><mi id="S4.E26.m1.29.29.29.29.29.29">w</mi><mo id="S4.E26.m1.30.30.30.30.30.30.1">∗</mo></msup><msup id="S4.E26.m1.32.32.32.32.32.38"><mo id="S4.E26.m1.31.31.31.31.31.31" lspace="0em">∥</mo><mn id="S4.E26.m1.32.32.32.32.32.32.1">2</mn></msup></mrow></mtd></mtr><mtr id="S4.E26.m1.59.59c"><mtd class="ltx_align_right" columnalign="right" id="S4.E26.m1.59.59d"><mrow id="S4.E26.m1.59.59.59.27.27"><mo id="S4.E26.m1.33.33.33.1.1.1">+</mo><mn id="S4.E26.m1.34.34.34.2.2.2">2</mn><msub id="S4.E26.m1.59.59.59.27.27.28"><mi id="S4.E26.m1.35.35.35.3.3.3">η</mi><mi id="S4.E26.m1.36.36.36.4.4.4.1">k</mi></msub><mi id="S4.E26.m1.37.37.37.5.5.5">Z</mi><mo id="S4.E26.m1.38.38.38.6.6.6">+</mo><msubsup id="S4.E26.m1.59.59.59.27.27.29"><mi id="S4.E26.m1.39.39.39.7.7.7">η</mi><mi id="S4.E26.m1.41.41.41.9.9.9.1">k</mi><mn id="S4.E26.m1.40.40.40.8.8.8.1">2</mn></msubsup><msup id="S4.E26.m1.59.59.59.27.27.30"><mi id="S4.E26.m1.42.42.42.10.10.10">L</mi><mn id="S4.E26.m1.43.43.43.11.11.11.1">2</mn></msup><mo id="S4.E26.m1.44.44.44.12.12.12" lspace="0em" rspace="0.167em">∥</mo><msub id="S4.E26.m1.59.59.59.27.27.31"><mi id="S4.E26.m1.45.45.45.13.13.13">V</mi><mi id="S4.E26.m1.46.46.46.14.14.14.1">k</mi></msub><mo id="S4.E26.m1.47.47.47.15.15.15">−</mo><msup id="S4.E26.m1.59.59.59.27.27.32"><mi id="S4.E26.m1.48.48.48.16.16.16">w</mi><mo id="S4.E26.m1.49.49.49.17.17.17.1">∗</mo></msup><msup id="S4.E26.m1.59.59.59.27.27.33"><mo id="S4.E26.m1.50.50.50.18.18.18" lspace="0em" rspace="0em">∥</mo><mn id="S4.E26.m1.51.51.51.19.19.19.1">2</mn></msup><mo id="S4.E26.m1.52.52.52.20.20.20" lspace="0em">+</mo><msubsup id="S4.E26.m1.59.59.59.27.27.34"><mi id="S4.E26.m1.53.53.53.21.21.21">η</mi><mi id="S4.E26.m1.55.55.55.23.23.23.1">k</mi><mn id="S4.E26.m1.54.54.54.22.22.22.1">2</mn></msubsup><msubsup id="S4.E26.m1.59.59.59.27.27.35"><mi id="S4.E26.m1.56.56.56.24.24.24">σ</mi><mi id="S4.E26.m1.58.58.58.26.26.26.1">k</mi><mn id="S4.E26.m1.57.57.57.25.25.25.1">2</mn></msubsup><mo id="S4.E26.m1.59.59.59.27.27.27" stretchy="false">]</mo></mrow></mtd></mtr></mtable><annotation encoding="application/x-tex" id="S4.E26.m1.59b">\begin{split}E(F(\bar{w}))-F\left(w^{\ast}\right))\leq\frac{L}{2}[\left(1+\mu% \eta_{k}\right)\|V_{k}-w^{\ast}\|^{2}\\ +2\eta_{k}Z+\eta^{2}_{k}L^{2}\|V_{k}-w^{\ast}\|^{2}+\eta^{2}_{k}\sigma^{2}_{k}% ]\end{split}</annotation><annotation encoding="application/x-llamapun" id="S4.E26.m1.59c">start_ROW start_CELL italic_E ( italic_F ( over¯ start_ARG italic_w end_ARG ) ) - italic_F ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ) ≤ divide start_ARG italic_L end_ARG start_ARG 2 end_ARG [ ( 1 + italic_μ italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ∥ italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL + 2 italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_Z + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ] end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(26)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS3.p14"> <p class="ltx_p" id="S4.SS2.SSS3.p14.3">We see that all parameters except <math alttext="Z" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p14.1.m1.1"><semantics id="S4.SS2.SSS3.p14.1.m1.1a"><mi id="S4.SS2.SSS3.p14.1.m1.1.1" xref="S4.SS2.SSS3.p14.1.m1.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p14.1.m1.1b"><ci id="S4.SS2.SSS3.p14.1.m1.1.1.cmml" xref="S4.SS2.SSS3.p14.1.m1.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p14.1.m1.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p14.1.m1.1d">italic_Z</annotation></semantics></math> are constants, which means that the right side of the formula is only affected by <math alttext="Z" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p14.2.m2.1"><semantics id="S4.SS2.SSS3.p14.2.m2.1a"><mi id="S4.SS2.SSS3.p14.2.m2.1.1" xref="S4.SS2.SSS3.p14.2.m2.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p14.2.m2.1b"><ci id="S4.SS2.SSS3.p14.2.m2.1.1.cmml" xref="S4.SS2.SSS3.p14.2.m2.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p14.2.m2.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p14.2.m2.1d">italic_Z</annotation></semantics></math>. However, an increase in <math alttext="Z" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p14.3.m3.1"><semantics id="S4.SS2.SSS3.p14.3.m3.1a"><mi id="S4.SS2.SSS3.p14.3.m3.1.1" xref="S4.SS2.SSS3.p14.3.m3.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p14.3.m3.1b"><ci id="S4.SS2.SSS3.p14.3.m3.1.1.cmml" xref="S4.SS2.SSS3.p14.3.m3.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p14.3.m3.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p14.3.m3.1d">italic_Z</annotation></semantics></math> will cause the right side to increase as a whole. This simultaneously affects the value of the left side of the formula, leading to poorer overall convergence of the model.</p> </div> </section> <section class="ltx_subsubsection" id="S4.SS2.SSS4"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection"><span class="ltx_text" id="S4.SS2.SSS4.4.1.1">IV-B</span>4 </span>Aggregate Ring</h4> <div class="ltx_para" id="S4.SS2.SSS4.p1"> <p class="ltx_p" id="S4.SS2.SSS4.p1.1">The aggregate ring is similar to the aggregate linear because they have the same training strategy. A series of interconnected devices form a ring. Each device possesses a unique dataset. There is only one difference. When running to the last device, the last aggregated parameters will be sent back to the first device.</p> </div> <div class="ltx_para" id="S4.SS2.SSS4.p2"> <p class="ltx_p" id="S4.SS2.SSS4.p2.1">This process is mathematically represented by the following equation; when k is at most 2:</p> <table class="ltx_equation ltx_eqn_table" id="S4.E27"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="w_{d}=w_{d-1}-\eta_{d}\nabla F_{d}\left(w_{d-1},\xi_{d}\right)" class="ltx_Math" display="block" id="S4.E27.m1.2"><semantics id="S4.E27.m1.2a"><mrow id="S4.E27.m1.2.2" xref="S4.E27.m1.2.2.cmml"><msub id="S4.E27.m1.2.2.4" xref="S4.E27.m1.2.2.4.cmml"><mi id="S4.E27.m1.2.2.4.2" xref="S4.E27.m1.2.2.4.2.cmml">w</mi><mi id="S4.E27.m1.2.2.4.3" xref="S4.E27.m1.2.2.4.3.cmml">d</mi></msub><mo id="S4.E27.m1.2.2.3" xref="S4.E27.m1.2.2.3.cmml">=</mo><mrow id="S4.E27.m1.2.2.2" xref="S4.E27.m1.2.2.2.cmml"><msub id="S4.E27.m1.2.2.2.4" xref="S4.E27.m1.2.2.2.4.cmml"><mi id="S4.E27.m1.2.2.2.4.2" xref="S4.E27.m1.2.2.2.4.2.cmml">w</mi><mrow id="S4.E27.m1.2.2.2.4.3" xref="S4.E27.m1.2.2.2.4.3.cmml"><mi id="S4.E27.m1.2.2.2.4.3.2" xref="S4.E27.m1.2.2.2.4.3.2.cmml">d</mi><mo id="S4.E27.m1.2.2.2.4.3.1" xref="S4.E27.m1.2.2.2.4.3.1.cmml">−</mo><mn id="S4.E27.m1.2.2.2.4.3.3" xref="S4.E27.m1.2.2.2.4.3.3.cmml">1</mn></mrow></msub><mo id="S4.E27.m1.2.2.2.3" xref="S4.E27.m1.2.2.2.3.cmml">−</mo><mrow id="S4.E27.m1.2.2.2.2" xref="S4.E27.m1.2.2.2.2.cmml"><msub id="S4.E27.m1.2.2.2.2.4" xref="S4.E27.m1.2.2.2.2.4.cmml"><mi id="S4.E27.m1.2.2.2.2.4.2" xref="S4.E27.m1.2.2.2.2.4.2.cmml">η</mi><mi id="S4.E27.m1.2.2.2.2.4.3" xref="S4.E27.m1.2.2.2.2.4.3.cmml">d</mi></msub><mo id="S4.E27.m1.2.2.2.2.3" lspace="0.167em" xref="S4.E27.m1.2.2.2.2.3.cmml">⁢</mo><mrow id="S4.E27.m1.2.2.2.2.5" xref="S4.E27.m1.2.2.2.2.5.cmml"><mo id="S4.E27.m1.2.2.2.2.5.1" rspace="0.167em" 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xref="S4.E27.m1.2.2.2.2.2.3.cmml">,</mo><msub id="S4.E27.m1.2.2.2.2.2.2.2" xref="S4.E27.m1.2.2.2.2.2.2.2.cmml"><mi id="S4.E27.m1.2.2.2.2.2.2.2.2" xref="S4.E27.m1.2.2.2.2.2.2.2.2.cmml">ξ</mi><mi id="S4.E27.m1.2.2.2.2.2.2.2.3" xref="S4.E27.m1.2.2.2.2.2.2.2.3.cmml">d</mi></msub><mo id="S4.E27.m1.2.2.2.2.2.2.5" xref="S4.E27.m1.2.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E27.m1.2b"><apply id="S4.E27.m1.2.2.cmml" xref="S4.E27.m1.2.2"><eq id="S4.E27.m1.2.2.3.cmml" xref="S4.E27.m1.2.2.3"></eq><apply id="S4.E27.m1.2.2.4.cmml" xref="S4.E27.m1.2.2.4"><csymbol cd="ambiguous" id="S4.E27.m1.2.2.4.1.cmml" xref="S4.E27.m1.2.2.4">subscript</csymbol><ci id="S4.E27.m1.2.2.4.2.cmml" xref="S4.E27.m1.2.2.4.2">𝑤</ci><ci id="S4.E27.m1.2.2.4.3.cmml" xref="S4.E27.m1.2.2.4.3">𝑑</ci></apply><apply id="S4.E27.m1.2.2.2.cmml" xref="S4.E27.m1.2.2.2"><minus id="S4.E27.m1.2.2.2.3.cmml" xref="S4.E27.m1.2.2.2.3"></minus><apply id="S4.E27.m1.2.2.2.4.cmml" 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start_POSTSUBSCRIPT italic_d - 1 end_POSTSUBSCRIPT , italic_ξ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(27)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS4.p2.2">when k is larger than 2:</p> <table class="ltx_equation ltx_eqn_table" id="S4.E28"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{split}w_{d}=\left(\frac{\left(s_{d-1}\right)w_{d-1}+\left(s_{d-2}\right% )w_{d-2}}{\displaystyle\sum^{d-1}_{i=1}s_{i}}\right)-\eta_{d}\\ \nabla F_{d}\left(\left(\frac{\left(s_{d-1}\right)w_{d-1}+\left(s_{d-2}\right)% w_{d-2}}{\displaystyle\sum^{d-1}_{i=1}s_{i}}\right),\xi_{d}\right)\\ k=d-\left(t-1\right)N\end{split}" class="ltx_Math" display="block" id="S4.E28.m1.36"><semantics id="S4.E28.m1.36a"><mtable displaystyle="true" id="S4.E28.m1.36.36.6" rowspacing="0pt" xref="S4.E28.m1.33.33.3.cmml"><mtr id="S4.E28.m1.36.36.6a" xref="S4.E28.m1.33.33.3.cmml"><mtd class="ltx_align_right" columnalign="right" id="S4.E28.m1.36.36.6b" xref="S4.E28.m1.33.33.3.cmml"><mrow id="S4.E28.m1.9.9.9.9.9" xref="S4.E28.m1.33.33.3.cmml"><msub id="S4.E28.m1.9.9.9.9.9.10" xref="S4.E28.m1.33.33.3.cmml"><mi id="S4.E28.m1.1.1.1.1.1.1" xref="S4.E28.m1.1.1.1.1.1.1.cmml">w</mi><mi id="S4.E28.m1.2.2.2.2.2.2.1" xref="S4.E28.m1.2.2.2.2.2.2.1.cmml">d</mi></msub><mo id="S4.E28.m1.3.3.3.3.3.3" xref="S4.E28.m1.3.3.3.3.3.3.cmml">=</mo><mrow id="S4.E28.m1.9.9.9.9.9.11" xref="S4.E28.m1.33.33.3.cmml"><mrow id="S4.E28.m1.9.9.9.9.9.11.1" xref="S4.E28.m1.33.33.3.cmml"><mo id="S4.E28.m1.4.4.4.4.4.4" xref="S4.E28.m1.33.33.3a.cmml">(</mo><mfrac id="S4.E28.m1.5.5.5.5.5.5" xref="S4.E28.m1.5.5.5.5.5.5.cmml"><mrow id="S4.E28.m1.5.5.5.5.5.5.2" xref="S4.E28.m1.5.5.5.5.5.5.2.cmml"><mrow id="S4.E28.m1.5.5.5.5.5.5.1.1" xref="S4.E28.m1.5.5.5.5.5.5.1.1.cmml"><mrow id="S4.E28.m1.5.5.5.5.5.5.1.1.1.1" xref="S4.E28.m1.5.5.5.5.5.5.1.1.1.1.1.cmml"><mo id="S4.E28.m1.5.5.5.5.5.5.1.1.1.1.2" xref="S4.E28.m1.5.5.5.5.5.5.1.1.1.1.1.cmml">(</mo><msub id="S4.E28.m1.5.5.5.5.5.5.1.1.1.1.1" xref="S4.E28.m1.5.5.5.5.5.5.1.1.1.1.1.cmml"><mi id="S4.E28.m1.5.5.5.5.5.5.1.1.1.1.1.2" xref="S4.E28.m1.5.5.5.5.5.5.1.1.1.1.1.2.cmml">s</mi><mrow id="S4.E28.m1.5.5.5.5.5.5.1.1.1.1.1.3" xref="S4.E28.m1.5.5.5.5.5.5.1.1.1.1.1.3.cmml"><mi id="S4.E28.m1.5.5.5.5.5.5.1.1.1.1.1.3.2" xref="S4.E28.m1.5.5.5.5.5.5.1.1.1.1.1.3.2.cmml">d</mi><mo id="S4.E28.m1.5.5.5.5.5.5.1.1.1.1.1.3.1" xref="S4.E28.m1.5.5.5.5.5.5.1.1.1.1.1.3.1.cmml">−</mo><mn id="S4.E28.m1.5.5.5.5.5.5.1.1.1.1.1.3.3" xref="S4.E28.m1.5.5.5.5.5.5.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S4.E28.m1.5.5.5.5.5.5.1.1.1.1.3" xref="S4.E28.m1.5.5.5.5.5.5.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.E28.m1.5.5.5.5.5.5.1.1.2" xref="S4.E28.m1.5.5.5.5.5.5.1.1.2.cmml">⁢</mo><msub id="S4.E28.m1.5.5.5.5.5.5.1.1.3" xref="S4.E28.m1.5.5.5.5.5.5.1.1.3.cmml"><mi id="S4.E28.m1.5.5.5.5.5.5.1.1.3.2" xref="S4.E28.m1.5.5.5.5.5.5.1.1.3.2.cmml">w</mi><mrow id="S4.E28.m1.5.5.5.5.5.5.1.1.3.3" xref="S4.E28.m1.5.5.5.5.5.5.1.1.3.3.cmml"><mi id="S4.E28.m1.5.5.5.5.5.5.1.1.3.3.2" xref="S4.E28.m1.5.5.5.5.5.5.1.1.3.3.2.cmml">d</mi><mo id="S4.E28.m1.5.5.5.5.5.5.1.1.3.3.1" xref="S4.E28.m1.5.5.5.5.5.5.1.1.3.3.1.cmml">−</mo><mn id="S4.E28.m1.5.5.5.5.5.5.1.1.3.3.3" xref="S4.E28.m1.5.5.5.5.5.5.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><mo id="S4.E28.m1.5.5.5.5.5.5.2.3" xref="S4.E28.m1.5.5.5.5.5.5.2.3.cmml">+</mo><mrow id="S4.E28.m1.5.5.5.5.5.5.2.2" xref="S4.E28.m1.5.5.5.5.5.5.2.2.cmml"><mrow id="S4.E28.m1.5.5.5.5.5.5.2.2.1.1" xref="S4.E28.m1.5.5.5.5.5.5.2.2.1.1.1.cmml"><mo id="S4.E28.m1.5.5.5.5.5.5.2.2.1.1.2" xref="S4.E28.m1.5.5.5.5.5.5.2.2.1.1.1.cmml">(</mo><msub id="S4.E28.m1.5.5.5.5.5.5.2.2.1.1.1" xref="S4.E28.m1.5.5.5.5.5.5.2.2.1.1.1.cmml"><mi id="S4.E28.m1.5.5.5.5.5.5.2.2.1.1.1.2" xref="S4.E28.m1.5.5.5.5.5.5.2.2.1.1.1.2.cmml">s</mi><mrow id="S4.E28.m1.5.5.5.5.5.5.2.2.1.1.1.3" xref="S4.E28.m1.5.5.5.5.5.5.2.2.1.1.1.3.cmml"><mi id="S4.E28.m1.5.5.5.5.5.5.2.2.1.1.1.3.2" xref="S4.E28.m1.5.5.5.5.5.5.2.2.1.1.1.3.2.cmml">d</mi><mo id="S4.E28.m1.5.5.5.5.5.5.2.2.1.1.1.3.1" xref="S4.E28.m1.5.5.5.5.5.5.2.2.1.1.1.3.1.cmml">−</mo><mn id="S4.E28.m1.5.5.5.5.5.5.2.2.1.1.1.3.3" xref="S4.E28.m1.5.5.5.5.5.5.2.2.1.1.1.3.3.cmml">2</mn></mrow></msub><mo id="S4.E28.m1.5.5.5.5.5.5.2.2.1.1.3" xref="S4.E28.m1.5.5.5.5.5.5.2.2.1.1.1.cmml">)</mo></mrow><mo id="S4.E28.m1.5.5.5.5.5.5.2.2.2" xref="S4.E28.m1.5.5.5.5.5.5.2.2.2.cmml">⁢</mo><msub id="S4.E28.m1.5.5.5.5.5.5.2.2.3" xref="S4.E28.m1.5.5.5.5.5.5.2.2.3.cmml"><mi id="S4.E28.m1.5.5.5.5.5.5.2.2.3.2" xref="S4.E28.m1.5.5.5.5.5.5.2.2.3.2.cmml">w</mi><mrow id="S4.E28.m1.5.5.5.5.5.5.2.2.3.3" xref="S4.E28.m1.5.5.5.5.5.5.2.2.3.3.cmml"><mi id="S4.E28.m1.5.5.5.5.5.5.2.2.3.3.2" xref="S4.E28.m1.5.5.5.5.5.5.2.2.3.3.2.cmml">d</mi><mo id="S4.E28.m1.5.5.5.5.5.5.2.2.3.3.1" xref="S4.E28.m1.5.5.5.5.5.5.2.2.3.3.1.cmml">−</mo><mn id="S4.E28.m1.5.5.5.5.5.5.2.2.3.3.3" xref="S4.E28.m1.5.5.5.5.5.5.2.2.3.3.3.cmml">2</mn></mrow></msub></mrow></mrow><mrow id="S4.E28.m1.5.5.5.5.5.5.4" xref="S4.E28.m1.5.5.5.5.5.5.4.cmml"><mstyle displaystyle="true" id="S4.E28.m1.5.5.5.5.5.5.4.1" xref="S4.E28.m1.5.5.5.5.5.5.4.1.cmml"><munderover id="S4.E28.m1.5.5.5.5.5.5.4.1a" xref="S4.E28.m1.5.5.5.5.5.5.4.1.cmml"><mo id="S4.E28.m1.5.5.5.5.5.5.4.1.2.2" movablelimits="false" xref="S4.E28.m1.5.5.5.5.5.5.4.1.2.2.cmml">∑</mo><mrow id="S4.E28.m1.5.5.5.5.5.5.4.1.3" xref="S4.E28.m1.5.5.5.5.5.5.4.1.3.cmml"><mi id="S4.E28.m1.5.5.5.5.5.5.4.1.3.2" xref="S4.E28.m1.5.5.5.5.5.5.4.1.3.2.cmml">i</mi><mo id="S4.E28.m1.5.5.5.5.5.5.4.1.3.1" xref="S4.E28.m1.5.5.5.5.5.5.4.1.3.1.cmml">=</mo><mn id="S4.E28.m1.5.5.5.5.5.5.4.1.3.3" xref="S4.E28.m1.5.5.5.5.5.5.4.1.3.3.cmml">1</mn></mrow><mrow id="S4.E28.m1.5.5.5.5.5.5.4.1.2.3" xref="S4.E28.m1.5.5.5.5.5.5.4.1.2.3.cmml"><mi id="S4.E28.m1.5.5.5.5.5.5.4.1.2.3.2" xref="S4.E28.m1.5.5.5.5.5.5.4.1.2.3.2.cmml">d</mi><mo id="S4.E28.m1.5.5.5.5.5.5.4.1.2.3.1" xref="S4.E28.m1.5.5.5.5.5.5.4.1.2.3.1.cmml">−</mo><mn id="S4.E28.m1.5.5.5.5.5.5.4.1.2.3.3" xref="S4.E28.m1.5.5.5.5.5.5.4.1.2.3.3.cmml">1</mn></mrow></munderover></mstyle><msub id="S4.E28.m1.5.5.5.5.5.5.4.2" xref="S4.E28.m1.5.5.5.5.5.5.4.2.cmml"><mi id="S4.E28.m1.5.5.5.5.5.5.4.2.2" xref="S4.E28.m1.5.5.5.5.5.5.4.2.2.cmml">s</mi><mi id="S4.E28.m1.5.5.5.5.5.5.4.2.3" xref="S4.E28.m1.5.5.5.5.5.5.4.2.3.cmml">i</mi></msub></mrow></mfrac><mo id="S4.E28.m1.6.6.6.6.6.6" xref="S4.E28.m1.33.33.3a.cmml">)</mo></mrow><mo id="S4.E28.m1.7.7.7.7.7.7" xref="S4.E28.m1.7.7.7.7.7.7.cmml">−</mo><msub id="S4.E28.m1.9.9.9.9.9.11.2" xref="S4.E28.m1.33.33.3.cmml"><mi id="S4.E28.m1.8.8.8.8.8.8" xref="S4.E28.m1.8.8.8.8.8.8.cmml">η</mi><mi id="S4.E28.m1.9.9.9.9.9.9.1" xref="S4.E28.m1.9.9.9.9.9.9.1.cmml">d</mi></msub></mrow></mrow></mtd></mtr><mtr id="S4.E28.m1.36.36.6c" xref="S4.E28.m1.33.33.3.cmml"><mtd class="ltx_align_right" columnalign="right" id="S4.E28.m1.36.36.6d" xref="S4.E28.m1.33.33.3.cmml"><mrow id="S4.E28.m1.35.35.5.32.13.13" xref="S4.E28.m1.33.33.3.cmml"><mrow id="S4.E28.m1.35.35.5.32.13.13.15" xref="S4.E28.m1.33.33.3.cmml"><mo id="S4.E28.m1.10.10.10.1.1.1" rspace="0.167em" xref="S4.E28.m1.10.10.10.1.1.1.cmml">∇</mo><msub id="S4.E28.m1.35.35.5.32.13.13.15.1" xref="S4.E28.m1.33.33.3.cmml"><mi id="S4.E28.m1.11.11.11.2.2.2" xref="S4.E28.m1.11.11.11.2.2.2.cmml">F</mi><mi id="S4.E28.m1.12.12.12.3.3.3.1" xref="S4.E28.m1.12.12.12.3.3.3.1.cmml">d</mi></msub></mrow><mo id="S4.E28.m1.35.35.5.32.13.13.14" xref="S4.E28.m1.33.33.3a.cmml">⁢</mo><mrow id="S4.E28.m1.35.35.5.32.13.13.13.2" xref="S4.E28.m1.33.33.3.cmml"><mo id="S4.E28.m1.13.13.13.4.4.4" xref="S4.E28.m1.33.33.3a.cmml">(</mo><mrow id="S4.E28.m1.34.34.4.31.12.12.12.1.1" xref="S4.E28.m1.33.33.3.cmml"><mo id="S4.E28.m1.14.14.14.5.5.5" xref="S4.E28.m1.33.33.3a.cmml">(</mo><mfrac id="S4.E28.m1.15.15.15.6.6.6" xref="S4.E28.m1.33.33.3.cmml"><mrow id="S4.E28.m1.15.15.15.6.6.6.2" xref="S4.E28.m1.15.15.15.6.6.6.2.cmml"><mrow id="S4.E28.m1.15.15.15.6.6.6.1.1" xref="S4.E28.m1.15.15.15.6.6.6.1.1.cmml"><mrow id="S4.E28.m1.15.15.15.6.6.6.1.1.1.1" xref="S4.E28.m1.15.15.15.6.6.6.1.1.1.1.1.cmml"><mo id="S4.E28.m1.15.15.15.6.6.6.1.1.1.1.2" xref="S4.E28.m1.15.15.15.6.6.6.1.1.1.1.1.cmml">(</mo><msub id="S4.E28.m1.15.15.15.6.6.6.1.1.1.1.1" xref="S4.E28.m1.15.15.15.6.6.6.1.1.1.1.1.cmml"><mi id="S4.E28.m1.15.15.15.6.6.6.1.1.1.1.1.2" xref="S4.E28.m1.15.15.15.6.6.6.1.1.1.1.1.2.cmml">s</mi><mrow id="S4.E28.m1.15.15.15.6.6.6.1.1.1.1.1.3" xref="S4.E28.m1.15.15.15.6.6.6.1.1.1.1.1.3.cmml"><mi id="S4.E28.m1.15.15.15.6.6.6.1.1.1.1.1.3.2" xref="S4.E28.m1.15.15.15.6.6.6.1.1.1.1.1.3.2.cmml">d</mi><mo 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xref="S4.E28.m1.4.4.4.4.4.4"></times><apply id="S4.E28.m1.33.33.3.3.1.1.1.1.cmml" xref="S4.E28.m1.36.36.6"><minus id="S4.E28.m1.27.27.27.7.7.7.cmml" xref="S4.E28.m1.27.27.27.7.7.7"></minus><ci id="S4.E28.m1.26.26.26.6.6.6.cmml" xref="S4.E28.m1.26.26.26.6.6.6">𝑡</ci><cn id="S4.E28.m1.28.28.28.8.8.8.cmml" type="integer" xref="S4.E28.m1.28.28.28.8.8.8">1</cn></apply><ci id="S4.E28.m1.30.30.30.10.10.10.cmml" xref="S4.E28.m1.30.30.30.10.10.10">𝑁</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E28.m1.36c">\begin{split}w_{d}=\left(\frac{\left(s_{d-1}\right)w_{d-1}+\left(s_{d-2}\right% )w_{d-2}}{\displaystyle\sum^{d-1}_{i=1}s_{i}}\right)-\eta_{d}\\ \nabla F_{d}\left(\left(\frac{\left(s_{d-1}\right)w_{d-1}+\left(s_{d-2}\right)% w_{d-2}}{\displaystyle\sum^{d-1}_{i=1}s_{i}}\right),\xi_{d}\right)\\ k=d-\left(t-1\right)N\end{split}</annotation><annotation encoding="application/x-llamapun" id="S4.E28.m1.36d">start_ROW start_CELL italic_w start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = ( divide start_ARG ( italic_s start_POSTSUBSCRIPT italic_d - 1 end_POSTSUBSCRIPT ) italic_w start_POSTSUBSCRIPT italic_d - 1 end_POSTSUBSCRIPT + ( italic_s start_POSTSUBSCRIPT italic_d - 2 end_POSTSUBSCRIPT ) italic_w start_POSTSUBSCRIPT italic_d - 2 end_POSTSUBSCRIPT end_ARG start_ARG ∑ start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ) - italic_η start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL ∇ italic_F start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( ( divide start_ARG ( italic_s start_POSTSUBSCRIPT italic_d - 1 end_POSTSUBSCRIPT ) italic_w start_POSTSUBSCRIPT italic_d - 1 end_POSTSUBSCRIPT + ( italic_s start_POSTSUBSCRIPT italic_d - 2 end_POSTSUBSCRIPT ) italic_w start_POSTSUBSCRIPT italic_d - 2 end_POSTSUBSCRIPT end_ARG start_ARG ∑ start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ) , italic_ξ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL italic_k = italic_d - ( italic_t - 1 ) italic_N end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(28)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS4.p3"> <p class="ltx_p" id="S4.SS2.SSS4.p3.1">And since the aggregate ring have the same training strategy, we get:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S8.EGx4"> <tbody id="S4.Ex11"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle E(F(\bar{w}))-F\left(w^{\ast}\right)" class="ltx_Math" display="inline" id="S4.Ex11.m1.3"><semantics id="S4.Ex11.m1.3a"><mrow id="S4.Ex11.m1.3.3" xref="S4.Ex11.m1.3.3.cmml"><mrow id="S4.Ex11.m1.2.2.1" xref="S4.Ex11.m1.2.2.1.cmml"><mi id="S4.Ex11.m1.2.2.1.3" xref="S4.Ex11.m1.2.2.1.3.cmml">E</mi><mo id="S4.Ex11.m1.2.2.1.2" xref="S4.Ex11.m1.2.2.1.2.cmml">⁢</mo><mrow id="S4.Ex11.m1.2.2.1.1.1" xref="S4.Ex11.m1.2.2.1.1.1.1.cmml"><mo id="S4.Ex11.m1.2.2.1.1.1.2" stretchy="false" xref="S4.Ex11.m1.2.2.1.1.1.1.cmml">(</mo><mrow id="S4.Ex11.m1.2.2.1.1.1.1" xref="S4.Ex11.m1.2.2.1.1.1.1.cmml"><mi id="S4.Ex11.m1.2.2.1.1.1.1.2" xref="S4.Ex11.m1.2.2.1.1.1.1.2.cmml">F</mi><mo id="S4.Ex11.m1.2.2.1.1.1.1.1" xref="S4.Ex11.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.Ex11.m1.2.2.1.1.1.1.3.2" xref="S4.Ex11.m1.1.1.cmml"><mo id="S4.Ex11.m1.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S4.Ex11.m1.1.1.cmml">(</mo><mover accent="true" id="S4.Ex11.m1.1.1" xref="S4.Ex11.m1.1.1.cmml"><mi id="S4.Ex11.m1.1.1.2" xref="S4.Ex11.m1.1.1.2.cmml">w</mi><mo id="S4.Ex11.m1.1.1.1" xref="S4.Ex11.m1.1.1.1.cmml">¯</mo></mover><mo id="S4.Ex11.m1.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S4.Ex11.m1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex11.m1.2.2.1.1.1.3" stretchy="false" xref="S4.Ex11.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex11.m1.3.3.3" xref="S4.Ex11.m1.3.3.3.cmml">−</mo><mrow id="S4.Ex11.m1.3.3.2" xref="S4.Ex11.m1.3.3.2.cmml"><mi id="S4.Ex11.m1.3.3.2.3" xref="S4.Ex11.m1.3.3.2.3.cmml">F</mi><mo id="S4.Ex11.m1.3.3.2.2" xref="S4.Ex11.m1.3.3.2.2.cmml">⁢</mo><mrow id="S4.Ex11.m1.3.3.2.1.1" xref="S4.Ex11.m1.3.3.2.1.1.1.cmml"><mo id="S4.Ex11.m1.3.3.2.1.1.2" xref="S4.Ex11.m1.3.3.2.1.1.1.cmml">(</mo><msup id="S4.Ex11.m1.3.3.2.1.1.1" xref="S4.Ex11.m1.3.3.2.1.1.1.cmml"><mi id="S4.Ex11.m1.3.3.2.1.1.1.2" xref="S4.Ex11.m1.3.3.2.1.1.1.2.cmml">w</mi><mo id="S4.Ex11.m1.3.3.2.1.1.1.3" xref="S4.Ex11.m1.3.3.2.1.1.1.3.cmml">∗</mo></msup><mo id="S4.Ex11.m1.3.3.2.1.1.3" xref="S4.Ex11.m1.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex11.m1.3b"><apply id="S4.Ex11.m1.3.3.cmml" xref="S4.Ex11.m1.3.3"><minus id="S4.Ex11.m1.3.3.3.cmml" xref="S4.Ex11.m1.3.3.3"></minus><apply id="S4.Ex11.m1.2.2.1.cmml" xref="S4.Ex11.m1.2.2.1"><times id="S4.Ex11.m1.2.2.1.2.cmml" xref="S4.Ex11.m1.2.2.1.2"></times><ci id="S4.Ex11.m1.2.2.1.3.cmml" xref="S4.Ex11.m1.2.2.1.3">𝐸</ci><apply id="S4.Ex11.m1.2.2.1.1.1.1.cmml" xref="S4.Ex11.m1.2.2.1.1.1"><times id="S4.Ex11.m1.2.2.1.1.1.1.1.cmml" xref="S4.Ex11.m1.2.2.1.1.1.1.1"></times><ci id="S4.Ex11.m1.2.2.1.1.1.1.2.cmml" xref="S4.Ex11.m1.2.2.1.1.1.1.2">𝐹</ci><apply id="S4.Ex11.m1.1.1.cmml" xref="S4.Ex11.m1.2.2.1.1.1.1.3.2"><ci id="S4.Ex11.m1.1.1.1.cmml" xref="S4.Ex11.m1.1.1.1">¯</ci><ci id="S4.Ex11.m1.1.1.2.cmml" xref="S4.Ex11.m1.1.1.2">𝑤</ci></apply></apply></apply><apply id="S4.Ex11.m1.3.3.2.cmml" xref="S4.Ex11.m1.3.3.2"><times id="S4.Ex11.m1.3.3.2.2.cmml" xref="S4.Ex11.m1.3.3.2.2"></times><ci id="S4.Ex11.m1.3.3.2.3.cmml" xref="S4.Ex11.m1.3.3.2.3">𝐹</ci><apply id="S4.Ex11.m1.3.3.2.1.1.1.cmml" xref="S4.Ex11.m1.3.3.2.1.1"><csymbol cd="ambiguous" id="S4.Ex11.m1.3.3.2.1.1.1.1.cmml" xref="S4.Ex11.m1.3.3.2.1.1">superscript</csymbol><ci id="S4.Ex11.m1.3.3.2.1.1.1.2.cmml" xref="S4.Ex11.m1.3.3.2.1.1.1.2">𝑤</ci><ci id="S4.Ex11.m1.3.3.2.1.1.1.3.cmml" xref="S4.Ex11.m1.3.3.2.1.1.1.3">∗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex11.m1.3c">\displaystyle E(F(\bar{w}))-F\left(w^{\ast}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex11.m1.3d">italic_E ( italic_F ( over¯ start_ARG italic_w end_ARG ) ) - italic_F ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\frac{L}{2}\left[\left(1+\mu\eta_{d}\right)\|V_{d}-w^{\ast}\|% ^{2}\right." class="ltx_math_unparsed" display="inline" id="S4.Ex11.m2.1"><semantics id="S4.Ex11.m2.1a"><mrow id="S4.Ex11.m2.1b"><mo id="S4.Ex11.m2.1.1">≤</mo><mstyle displaystyle="true" id="S4.Ex11.m2.1.2"><mfrac id="S4.Ex11.m2.1.2a"><mi id="S4.Ex11.m2.1.2.2">L</mi><mn id="S4.Ex11.m2.1.2.3">2</mn></mfrac></mstyle><mrow id="S4.Ex11.m2.1.3"><mo id="S4.Ex11.m2.1.3.1">[</mo><mrow id="S4.Ex11.m2.1.3.2"><mo id="S4.Ex11.m2.1.3.2.1">(</mo><mn id="S4.Ex11.m2.1.3.2.2">1</mn><mo id="S4.Ex11.m2.1.3.2.3">+</mo><mi id="S4.Ex11.m2.1.3.2.4">μ</mi><msub id="S4.Ex11.m2.1.3.2.5"><mi id="S4.Ex11.m2.1.3.2.5.2">η</mi><mi id="S4.Ex11.m2.1.3.2.5.3">d</mi></msub><mo id="S4.Ex11.m2.1.3.2.6">)</mo></mrow><mo id="S4.Ex11.m2.1.3.3" lspace="0em" rspace="0.167em">∥</mo><msub id="S4.Ex11.m2.1.3.4"><mi id="S4.Ex11.m2.1.3.4.2">V</mi><mi id="S4.Ex11.m2.1.3.4.3">d</mi></msub><mo id="S4.Ex11.m2.1.3.5">−</mo><msup id="S4.Ex11.m2.1.3.6"><mi id="S4.Ex11.m2.1.3.6.2">w</mi><mo id="S4.Ex11.m2.1.3.6.3">∗</mo></msup><msup id="S4.Ex11.m2.1.3.7"><mo id="S4.Ex11.m2.1.3.7.2" lspace="0em">∥</mo><mn id="S4.Ex11.m2.1.3.7.3">2</mn></msup></mrow></mrow><annotation encoding="application/x-tex" id="S4.Ex11.m2.1c">\displaystyle\leq\frac{L}{2}\left[\left(1+\mu\eta_{d}\right)\|V_{d}-w^{\ast}\|% ^{2}\right.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex11.m2.1d">≤ divide start_ARG italic_L end_ARG start_ARG 2 end_ARG [ ( 1 + italic_μ italic_η start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) ∥ italic_V start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.E29"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\quad\left.+2\eta_{d}Z+\eta^{2}_{d}L^{2}\|V_{d}-w^{\ast}\|^{2}+% \eta^{2}_{d}\sigma^{2}_{d}\right]" class="ltx_math_unparsed" display="inline" id="S4.E29.m1.1"><semantics id="S4.E29.m1.1a"><mrow id="S4.E29.m1.1b"><mo id="S4.E29.m1.1.1">+</mo><mn id="S4.E29.m1.1.2">2</mn><msub id="S4.E29.m1.1.3"><mi id="S4.E29.m1.1.3.2">η</mi><mi id="S4.E29.m1.1.3.3">d</mi></msub><mi id="S4.E29.m1.1.4">Z</mi><mo id="S4.E29.m1.1.5">+</mo><msubsup id="S4.E29.m1.1.6"><mi id="S4.E29.m1.1.6.2.2">η</mi><mi id="S4.E29.m1.1.6.3">d</mi><mn id="S4.E29.m1.1.6.2.3">2</mn></msubsup><msup id="S4.E29.m1.1.7"><mi id="S4.E29.m1.1.7.2">L</mi><mn id="S4.E29.m1.1.7.3">2</mn></msup><mo id="S4.E29.m1.1.8" lspace="0em" rspace="0.167em">∥</mo><msub id="S4.E29.m1.1.9"><mi id="S4.E29.m1.1.9.2">V</mi><mi id="S4.E29.m1.1.9.3">d</mi></msub><mo id="S4.E29.m1.1.10">−</mo><msup id="S4.E29.m1.1.11"><mi id="S4.E29.m1.1.11.2">w</mi><mo id="S4.E29.m1.1.11.3">∗</mo></msup><msup id="S4.E29.m1.1.12"><mo id="S4.E29.m1.1.12.2" lspace="0em" rspace="0em">∥</mo><mn id="S4.E29.m1.1.12.3">2</mn></msup><mo id="S4.E29.m1.1.13" lspace="0em">+</mo><msubsup id="S4.E29.m1.1.14"><mi id="S4.E29.m1.1.14.2.2">η</mi><mi id="S4.E29.m1.1.14.3">d</mi><mn id="S4.E29.m1.1.14.2.3">2</mn></msubsup><msubsup id="S4.E29.m1.1.15"><mi id="S4.E29.m1.1.15.2.2">σ</mi><mi id="S4.E29.m1.1.15.3">d</mi><mn id="S4.E29.m1.1.15.2.3">2</mn></msubsup><mo id="S4.E29.m1.1.16">]</mo></mrow><annotation encoding="application/x-tex" id="S4.E29.m1.1c">\displaystyle\quad\left.+2\eta_{d}Z+\eta^{2}_{d}L^{2}\|V_{d}-w^{\ast}\|^{2}+% \eta^{2}_{d}\sigma^{2}_{d}\right]</annotation><annotation encoding="application/x-llamapun" id="S4.E29.m1.1d">+ 2 italic_η start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT italic_Z + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ italic_V start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(29)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS4.p4"> <p class="ltx_p" id="S4.SS2.SSS4.p4.1">Since A_ring and A_linear use the same aggregation strategy, their mathematical convergence analysis formulas are similar. Therefore, the derivation process is omitted. By observing the formula, we can see that as <math alttext="Z" class="ltx_Math" display="inline" id="S4.SS2.SSS4.p4.1.m1.1"><semantics id="S4.SS2.SSS4.p4.1.m1.1a"><mi id="S4.SS2.SSS4.p4.1.m1.1.1" xref="S4.SS2.SSS4.p4.1.m1.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS4.p4.1.m1.1b"><ci id="S4.SS2.SSS4.p4.1.m1.1.1.cmml" xref="S4.SS2.SSS4.p4.1.m1.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS4.p4.1.m1.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS4.p4.1.m1.1d">italic_Z</annotation></semantics></math> (the degree of non-IID) increases, the value on the left side becomes larger, and the gap between the optimal solution and the actual value on the right side also increases, causing the model to become more divergent.</p> </div> </section> <section class="ltx_subsubsection" id="S4.SS2.SSS5"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection"><span class="ltx_text" id="S4.SS2.SSS5.4.1.1">IV-B</span>5 </span>Aggregate Star</h4> <div class="ltx_para" id="S4.SS2.SSS5.p1"> <p class="ltx_p" id="S4.SS2.SSS5.p1.1">The aggregate star is similar to CFL<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib9" title="">9</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib7" title="">7</a>]</cite>, however, the center device no longer only plays the role of a server, instead, it can participate in the training of local models. First, select a central device as the center of the whole deployment. After that, the initial parameters are issued by the central device to each device, and then each device (including the central device) is trained based on the initial parameters. After that, the central device receives the parameters of all devices for aggregation. This process is repeated iteratively.</p> </div> <div class="ltx_para" id="S4.SS2.SSS5.p2"> <p class="ltx_p" id="S4.SS2.SSS5.p2.1">This process is mathematically represented by the following equation:</p> <table class="ltx_equation ltx_eqn_table" id="S4.E30"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="w_{k}=\sum^{N}_{i=1}\left(\frac{s_{i}w_{i}}{\displaystyle\sum^{n}_{j=1}s_{j}}% \right)-\eta_{k}\Delta F_{k}\left(\sum^{N}_{i=1}\left(\frac{s_{i}w_{i}}{% \displaystyle\sum^{n}_{j=1}s_{j}}\right),\xi_{k}\right)" class="ltx_Math" display="block" id="S4.E30.m1.4"><semantics id="S4.E30.m1.4a"><mrow id="S4.E30.m1.4.4" xref="S4.E30.m1.4.4.cmml"><msub id="S4.E30.m1.4.4.4" xref="S4.E30.m1.4.4.4.cmml"><mi id="S4.E30.m1.4.4.4.2" xref="S4.E30.m1.4.4.4.2.cmml">w</mi><mi id="S4.E30.m1.4.4.4.3" xref="S4.E30.m1.4.4.4.3.cmml">k</mi></msub><mo id="S4.E30.m1.4.4.3" 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italic_j = 1 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG ) - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_Δ italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( ∑ start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT ( divide start_ARG italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG ∑ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG ) , italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(30)</span></td> </tr></tbody> </table> </div> <div 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id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.3" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.3.cmml">≤</mo><mrow id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.cmml"><mfrac id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.3" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.3.cmml"><mrow id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.3.2" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.3.2.cmml"><mn id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.3.2.2" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.3.2.2.cmml">2</mn><mo id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.3.2.1" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.3.2.1.cmml">⁢</mo><mi id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.3.2.3" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.3.2.3.cmml">κ</mi></mrow><mrow id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.3.3" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.3.3.cmml"><mi id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.3.3.2" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.3.3.2.cmml">γ</mi><mo id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.3.3.1" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.3.3.1.cmml">+</mo><mi id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.3.3.3" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.3.3.3.cmml">T</mi></mrow></mfrac><mo id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.2" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.2.cmml">⁢</mo><mrow id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.cmml"><mo id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.2" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.cmml"><mfrac id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.3" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.3.cmml"><mi id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.3.2" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.3.2.cmml">B</mi><mi id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.3.3" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.3.3.cmml">μ</mi></mfrac><mo id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.2" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.2.cmml">+</mo><mrow id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.cmml"><mn id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.3" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.3.cmml">2</mn><mo id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.2" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.2.cmml">⁢</mo><mi id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.4" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.4.cmml">L</mi><mo id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.2a" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.2.cmml">⁢</mo><msup id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.cmml"><mrow id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.1.1" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.1.2.cmml"><mo id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.1.1.1" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.cmml"><msup id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.2" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.2.cmml"><mi id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.2.2" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.2.2.cmml">w</mi><mn id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.2.3" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.2.3.cmml">0</mn></msup><mo id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.cmml">−</mo><msup id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.3" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.3.cmml"><mi id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.3.2" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.3.2.cmml">w</mi><mo id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.3.3" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.3.3.cmml">∗</mo></msup></mrow><mo id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.3" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.1.1.3.cmml">2</mn></msup></mrow></mrow><mo id="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.3" xref="S4.SS2.SSS5.p3.1.m1.1.1.1.1.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.SS2.SSS5.p3.1.m1.2.2.2.3" xref="S4.SS2.SSS5.p3.1.m1.2.2.3a.cmml">,</mo><mrow id="S4.SS2.SSS5.p3.1.m1.2.2.2.2" xref="S4.SS2.SSS5.p3.1.m1.2.2.2.2.cmml"><mi id="S4.SS2.SSS5.p3.1.m1.2.2.2.2.3" xref="S4.SS2.SSS5.p3.1.m1.2.2.2.2.3.cmml">B</mi><mo id="S4.SS2.SSS5.p3.1.m1.2.2.2.2.2" xref="S4.SS2.SSS5.p3.1.m1.2.2.2.2.2.cmml">=</mo><mrow id="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1" xref="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.cmml"><mrow id="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3" xref="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.cmml"><mstyle displaystyle="true" id="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.1" xref="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.1.cmml"><munderover id="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.1a" xref="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.1.cmml"><mo id="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.1.2.2" movablelimits="false" xref="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.1.2.2.cmml">∑</mo><mrow id="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.1.2.3" xref="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.1.2.3.cmml"><mi id="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.1.2.3.2" xref="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.1.2.3.2.cmml">k</mi><mo id="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.1.2.3.1" xref="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.1.2.3.1.cmml">=</mo><mn id="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.1.2.3.3" xref="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.1.2.3.3.cmml">1</mn></mrow><mi id="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.1.3" xref="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.1.3.cmml">N</mi></munderover></mstyle><mrow id="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.2" xref="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.2.cmml"><msubsup id="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.2.2" xref="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.2.2.cmml"><mi id="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.2.2.2.2" xref="S4.SS2.SSS5.p3.1.m1.2.2.2.2.1.3.2.2.2.2.cmml">p</mi><mi 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id="S4.SS2.SSS5.p3.1.m1.2c">\mathbb{E}[F(w_{T})]-F^{*}\leq\frac{2\kappa}{\gamma+T}\left(\frac{B}{\mu}+2L\|% w^{0}-w^{*}\|^{2}\right),B=\displaystyle\sum_{k=1}^{N}p_{k}^{2}\sigma_{k}^{2}+% 6LZ+8(E-1)^{2}G^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS5.p3.1.m1.2d">blackboard_E [ italic_F ( italic_w start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ) ] - italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ≤ divide start_ARG 2 italic_κ end_ARG start_ARG italic_γ + italic_T end_ARG ( divide start_ARG italic_B end_ARG start_ARG italic_μ end_ARG + 2 italic_L ∥ italic_w start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT - italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) , italic_B = ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 6 italic_L italic_Z + 8 ( italic_E - 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="T" class="ltx_Math" display="inline" id="S4.SS2.SSS5.p3.2.m2.1"><semantics id="S4.SS2.SSS5.p3.2.m2.1a"><mi id="S4.SS2.SSS5.p3.2.m2.1.1" xref="S4.SS2.SSS5.p3.2.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS5.p3.2.m2.1b"><ci id="S4.SS2.SSS5.p3.2.m2.1.1.cmml" xref="S4.SS2.SSS5.p3.2.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS5.p3.2.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS5.p3.2.m2.1d">italic_T</annotation></semantics></math> is the number of iterations.</p> </div> <div class="ltx_para" id="S4.SS2.SSS5.p4"> <p class="ltx_p" id="S4.SS2.SSS5.p4.3">We see that all parameters except <math alttext="Z" class="ltx_Math" display="inline" id="S4.SS2.SSS5.p4.1.m1.1"><semantics id="S4.SS2.SSS5.p4.1.m1.1a"><mi id="S4.SS2.SSS5.p4.1.m1.1.1" xref="S4.SS2.SSS5.p4.1.m1.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS5.p4.1.m1.1b"><ci id="S4.SS2.SSS5.p4.1.m1.1.1.cmml" xref="S4.SS2.SSS5.p4.1.m1.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS5.p4.1.m1.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS5.p4.1.m1.1d">italic_Z</annotation></semantics></math> are constants, which means that the right side of the formula is only affected by <math alttext="Z" class="ltx_Math" display="inline" id="S4.SS2.SSS5.p4.2.m2.1"><semantics id="S4.SS2.SSS5.p4.2.m2.1a"><mi id="S4.SS2.SSS5.p4.2.m2.1.1" xref="S4.SS2.SSS5.p4.2.m2.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS5.p4.2.m2.1b"><ci id="S4.SS2.SSS5.p4.2.m2.1.1.cmml" xref="S4.SS2.SSS5.p4.2.m2.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS5.p4.2.m2.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS5.p4.2.m2.1d">italic_Z</annotation></semantics></math>. However, an increase in <math alttext="Z" class="ltx_Math" display="inline" id="S4.SS2.SSS5.p4.3.m3.1"><semantics id="S4.SS2.SSS5.p4.3.m3.1a"><mi id="S4.SS2.SSS5.p4.3.m3.1.1" xref="S4.SS2.SSS5.p4.3.m3.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS5.p4.3.m3.1b"><ci id="S4.SS2.SSS5.p4.3.m3.1.1.cmml" xref="S4.SS2.SSS5.p4.3.m3.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS5.p4.3.m3.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS5.p4.3.m3.1d">italic_Z</annotation></semantics></math> will cause the right side to increase as a whole. This simultaneously affects the value of the left side of the formula, leading to poorer overall convergence of the model.</p> </div> </section> <section class="ltx_subsubsection" id="S4.SS2.SSS6"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection"><span class="ltx_text" id="S4.SS2.SSS6.4.1.1">IV-B</span>6 </span>Aggregate Mesh</h4> <div class="ltx_para" id="S4.SS2.SSS6.p1"> <p class="ltx_p" id="S4.SS2.SSS6.p1.1">Unlike the aggregation star, each device can play the role of the central device and can be trained for local models. First of all, each device has preset parameters. Each device is trained locally according to the parameters and then sent to other devices. Each device aggregates parameters separately, and the aggregated parameters are recorded as the initial parameters of the next round. The next round of training is carried out based on the initial parameters. The training process is the same as the aggregate star.</p> </div> <div class="ltx_para ltx_noindent" id="S4.SS2.SSS6.p2"> <svg class="ltx_picture" height="235.17" id="S4.SS2.SSS6.p2.pic1" overflow="visible" version="1.1" width="600"><g fill="#000000" stroke="#000000" stroke-width="0.4pt" transform="translate(0,235.17) matrix(1 0 0 -1 0 0)"><g fill="#757677" fill-opacity="1.0"><path d="M 0 3.94 L 0 231.24 C 0 233.41 1.76 235.17 3.94 235.17 L 596.06 235.17 C 598.24 235.17 600 233.41 600 231.24 L 600 3.94 C 600 1.76 598.24 0 596.06 0 L 3.94 0 C 1.76 0 0 1.76 0 3.94 Z" style="stroke:none"></path></g><g fill="#DEE0E3" fill-opacity="1.0"><path d="M 8.3 3.94 L 8.3 231.24 C 8.3 233.41 10.06 235.17 12.24 235.17 L 596.06 235.17 C 598.24 235.17 600 233.41 600 231.24 L 600 3.94 C 600 1.76 598.24 0 596.06 0 L 12.24 0 C 10.06 0 8.3 1.76 8.3 3.94 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 27.99 11.81)"><foreignobject color="#000000" height="211.55" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="552.33"> <span class="ltx_inline-block ltx_minipage ltx_align_bottom" id="S4.SS2.SSS6.p2.pic1.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9" style="width:399.2pt;"> <span class="ltx_p" id="S4.SS2.SSS6.p2.pic1.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9"><span class="ltx_text ltx_font_bold" id="S4.SS2.SSS6.p2.pic1.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.1">Key Takeaways:</span> In general, non-IID data distribution has adverse impact on the convergence rate irrespective of topologies and training strategies. However, different topologies converge differently. In continuous linear and ring-based DFL, the performance depends on <math alttext="Z" class="ltx_Math" display="inline" id="S4.SS2.SSS6.p2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S4.SS2.SSS6.p2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mi id="S4.SS2.SSS6.p2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S4.SS2.SSS6.p2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS6.p2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="S4.SS2.SSS6.p2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S4.SS2.SSS6.p2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS6.p2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS6.p2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_Z</annotation></semantics></math> (representing the degree of non-IID distribution). As <math alttext="Z" class="ltx_Math" display="inline" id="S4.SS2.SSS6.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1"><semantics id="S4.SS2.SSS6.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1a"><mi id="S4.SS2.SSS6.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1.1" xref="S4.SS2.SSS6.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS6.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1b"><ci id="S4.SS2.SSS6.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1.1.cmml" xref="S4.SS2.SSS6.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS6.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS6.p2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.m2.1d">italic_Z</annotation></semantics></math> increases, the convergence of the model worsens. When <math alttext="Z=0" class="ltx_Math" display="inline" id="S4.SS2.SSS6.p2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.m3.1"><semantics id="S4.SS2.SSS6.p2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.m3.1a"><mrow id="S4.SS2.SSS6.p2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.m3.1.1" xref="S4.SS2.SSS6.p2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.m3.1.1.cmml"><mi id="S4.SS2.SSS6.p2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.m3.1.1.2" xref="S4.SS2.SSS6.p2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.m3.1.1.2.cmml">Z</mi><mo id="S4.SS2.SSS6.p2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.m3.1.1.1" xref="S4.SS2.SSS6.p2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.m3.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS6.p2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.m3.1.1.3" xref="S4.SS2.SSS6.p2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS6.p2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.m3.1b"><apply id="S4.SS2.SSS6.p2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.m3.1.1.cmml" xref="S4.SS2.SSS6.p2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.m3.1.1"><eq id="S4.SS2.SSS6.p2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.m3.1.1.1.cmml" xref="S4.SS2.SSS6.p2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.m3.1.1.1"></eq><ci id="S4.SS2.SSS6.p2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.m3.1.1.2.cmml" xref="S4.SS2.SSS6.p2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.m3.1.1.2">𝑍</ci><cn id="S4.SS2.SSS6.p2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.m3.1.1.3.cmml" type="integer" xref="S4.SS2.SSS6.p2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS6.p2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.m3.1c">Z=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS6.p2.pic1.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.m3.1d">italic_Z = 0</annotation></semantics></math>, the gap between the optimal and the actual solution becomes a constant value, indicating that the model has converged. In aggregate linear and ring structures, the performance depends on two variables, <math alttext="V_{k}" class="ltx_Math" display="inline" id="S4.SS2.SSS6.p2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.m4.1"><semantics id="S4.SS2.SSS6.p2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.m4.1a"><msub id="S4.SS2.SSS6.p2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.m4.1.1" xref="S4.SS2.SSS6.p2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.m4.1.1.cmml"><mi id="S4.SS2.SSS6.p2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.m4.1.1.2" xref="S4.SS2.SSS6.p2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.m4.1.1.2.cmml">V</mi><mi id="S4.SS2.SSS6.p2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.m4.1.1.3" xref="S4.SS2.SSS6.p2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.m4.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS6.p2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.m4.1b"><apply id="S4.SS2.SSS6.p2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.m4.1.1.cmml" xref="S4.SS2.SSS6.p2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS6.p2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.m4.1.1.1.cmml" xref="S4.SS2.SSS6.p2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.m4.1.1">subscript</csymbol><ci id="S4.SS2.SSS6.p2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.m4.1.1.2.cmml" xref="S4.SS2.SSS6.p2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.m4.1.1.2">𝑉</ci><ci id="S4.SS2.SSS6.p2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.m4.1.1.3.cmml" xref="S4.SS2.SSS6.p2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.m4.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS6.p2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.m4.1c">V_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS6.p2.pic1.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.4.m4.1d">italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="Z" class="ltx_Math" display="inline" id="S4.SS2.SSS6.p2.pic1.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.m5.1"><semantics id="S4.SS2.SSS6.p2.pic1.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.m5.1a"><mi id="S4.SS2.SSS6.p2.pic1.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.m5.1.1" xref="S4.SS2.SSS6.p2.pic1.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.m5.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS6.p2.pic1.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.m5.1b"><ci id="S4.SS2.SSS6.p2.pic1.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.m5.1.1.cmml" xref="S4.SS2.SSS6.p2.pic1.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.m5.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS6.p2.pic1.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.m5.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS6.p2.pic1.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.m5.1d">italic_Z</annotation></semantics></math>. Here, <math alttext="V_{k}" class="ltx_Math" display="inline" id="S4.SS2.SSS6.p2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.m6.1"><semantics id="S4.SS2.SSS6.p2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.m6.1a"><msub id="S4.SS2.SSS6.p2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.m6.1.1" xref="S4.SS2.SSS6.p2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.m6.1.1.cmml"><mi id="S4.SS2.SSS6.p2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.m6.1.1.2" xref="S4.SS2.SSS6.p2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.m6.1.1.2.cmml">V</mi><mi id="S4.SS2.SSS6.p2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.m6.1.1.3" xref="S4.SS2.SSS6.p2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.m6.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS6.p2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.m6.1b"><apply id="S4.SS2.SSS6.p2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.m6.1.1.cmml" xref="S4.SS2.SSS6.p2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS6.p2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.m6.1.1.1.cmml" xref="S4.SS2.SSS6.p2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.m6.1.1">subscript</csymbol><ci id="S4.SS2.SSS6.p2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.m6.1.1.2.cmml" xref="S4.SS2.SSS6.p2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.m6.1.1.2">𝑉</ci><ci id="S4.SS2.SSS6.p2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.m6.1.1.3.cmml" xref="S4.SS2.SSS6.p2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.m6.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS6.p2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.m6.1c">V_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS6.p2.pic1.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.6.m6.1d">italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> represents the ratio of the sum of the samples of the current and the previous devices to the total number of samples. If this ratio is 1, aggregate and continuous structures are equivalent; however, when the ratio is greater or less than 1, continuous structures converges better than the aggregate and vice versa, respectively. Finally, in star and mesh structures, performance depends not only on <math alttext="Z" class="ltx_Math" display="inline" id="S4.SS2.SSS6.p2.pic1.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.m7.1"><semantics id="S4.SS2.SSS6.p2.pic1.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.m7.1a"><mi id="S4.SS2.SSS6.p2.pic1.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.m7.1.1" xref="S4.SS2.SSS6.p2.pic1.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.m7.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS6.p2.pic1.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.m7.1b"><ci id="S4.SS2.SSS6.p2.pic1.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.m7.1.1.cmml" xref="S4.SS2.SSS6.p2.pic1.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.m7.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS6.p2.pic1.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.m7.1c">Z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS6.p2.pic1.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.m7.1d">italic_Z</annotation></semantics></math> but also on <math alttext="T" class="ltx_Math" display="inline" id="S4.SS2.SSS6.p2.pic1.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.m8.1"><semantics id="S4.SS2.SSS6.p2.pic1.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.m8.1a"><mi id="S4.SS2.SSS6.p2.pic1.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.m8.1.1" xref="S4.SS2.SSS6.p2.pic1.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.m8.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS6.p2.pic1.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.m8.1b"><ci id="S4.SS2.SSS6.p2.pic1.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.m8.1.1.cmml" xref="S4.SS2.SSS6.p2.pic1.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.m8.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS6.p2.pic1.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.m8.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS6.p2.pic1.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.m8.1d">italic_T</annotation></semantics></math> (the number of rounds), i.e., the greater the number of rounds <math alttext="T" class="ltx_Math" display="inline" id="S4.SS2.SSS6.p2.pic1.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.m9.1"><semantics id="S4.SS2.SSS6.p2.pic1.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.m9.1a"><mi id="S4.SS2.SSS6.p2.pic1.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.m9.1.1" xref="S4.SS2.SSS6.p2.pic1.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.m9.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS6.p2.pic1.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.m9.1b"><ci id="S4.SS2.SSS6.p2.pic1.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.m9.1.1.cmml" xref="S4.SS2.SSS6.p2.pic1.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.m9.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS6.p2.pic1.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.m9.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS6.p2.pic1.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.m9.1d">italic_T</annotation></semantics></math>, the better the convergence.</span> </span></foreignobject></g></g></svg> </div> </section> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">V </span><span class="ltx_text ltx_font_smallcaps" id="S5.1.1">Implementation Details</span> </h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">This section presents our evaluation process, including implementation and evaluation details, model and data selection, and non-IID data generation.</p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.1">We evaluate the performance of all models in DFL deployment over a NVIDIA Quadro RTX 8000 GPU. This device is equipped with 48GB of memory, driver version 535.86.05, and CUDA version 12.2. We simulate five different devices on this server, each capable of training models and sending parameters to each other. The models are implemented using Python 3.12.1, NumPy version 1.26.4, scikit-learn version 1.5.0 and PyTorch version 2.3.0+cu121. Finally, we implement MiniGPT-4 using Python 3.10 and PyTorch version 2.3.0+cu121 following <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib15" title="">15</a>]</cite>. Unlike traditional and deep learning models, due to the enormous size of MiniGPT-4, its training is divided into two parts: the first part trains the connections between the vision and language models, and the second part involves fine-tuning the model. In this experiment, we only trained the fine-tuning part.</p> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.1"><span class="ltx_text ltx_font_bold" id="S5.p3.1.1">Model and Data Selection:</span> To ensure that our research adapts to multiple fields, we have chosen three different types of learning models: traditional, deep learning, and large language models. The loss functions of traditional models are often convex, while those for deep learning and large language models are nearly convex. The models and their datasets are listed in Table <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S5.T2" title="TABLE II ‣ V Implementation Details ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">II</span></a>.</p> </div> <figure class="ltx_table" id="S5.T2"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S5.T2.2.1.1" style="font-size:90%;">TABLE II</span>: </span><span class="ltx_text" id="S5.T2.3.2" style="font-size:90%;">Model & Dataset Table</span></figcaption> <table class="ltx_tabular ltx_centering ltx_align_middle" id="S5.T2.4"> <tr class="ltx_tr" id="S5.T2.4.1"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S5.T2.4.1.1"><span class="ltx_text ltx_font_bold" id="S5.T2.4.1.1.1">Model</span></td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S5.T2.4.1.2"><span class="ltx_text ltx_font_bold" id="S5.T2.4.1.2.1">Dataset</span></td> </tr> <tr class="ltx_tr" id="S5.T2.4.2"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S5.T2.4.2.1">SVM</td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S5.T2.4.2.2">Breast Cancer Dataset</td> </tr> <tr class="ltx_tr" id="S5.T2.4.3"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S5.T2.4.3.1">Logistic Regression</td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S5.T2.4.3.2">Breast Cancer Dataset</td> </tr> <tr class="ltx_tr" id="S5.T2.4.4"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S5.T2.4.4.1">ResNet-18</td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S5.T2.4.4.2">MNIST dataset</td> </tr> <tr class="ltx_tr" id="S5.T2.4.5"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_r ltx_border_t" id="S5.T2.4.5.1">DistilBERT</td> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S5.T2.4.5.2">TREC_6 dataset</td> </tr> <tr class="ltx_tr" id="S5.T2.4.6"> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="S5.T2.4.6.1">MiniGPT-4</td> <td class="ltx_td ltx_align_left ltx_border_b ltx_border_r ltx_border_t" id="S5.T2.4.6.2">cc sub dataset</td> </tr> </table> </figure> <div class="ltx_para" id="S5.p4"> <p class="ltx_p" id="S5.p4.1">We choose Support Vector Machines (SVM) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib16" title="">16</a>]</cite> and Logistic Regression <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib17" title="">17</a>]</cite> as they are the most widely adopted baseline models both in classifications and regressions. In the case of deep learning models, we select ResNet-18 <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib18" title="">18</a>]</cite> for vision and DistilBERT <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib19" title="">19</a>]</cite> for NLP tasks. ResNet-18 has a relatively shallow structure, low computational cost, and is suitable for experiments in resource-constrained environments while still providing strong image feature extraction capabilities. Similarly, DistilBERT is a lighter variant of the BERT model suitable for edge deployments. Finally, we use MiniGPT-4 <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib15" title="">15</a>]</cite> as the LLM representative to test in the edge environment. We deploy Stochastic Gradient Descent (SGD) as the optimizer during the training of these chosen models.</p> </div> <div class="ltx_para" id="S5.p5"> <p class="ltx_p" id="S5.p5.1"><span class="ltx_text ltx_font_bold" id="S5.p5.1.1">Non-IID Data Distribution:</span> In our experiment, we consider the most prevalent form of NON-IID data —label skew —as this is commonly observed in vertical Federated Learning models <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib11" title="">11</a>]</cite>. Label skew is defined as the variation in the distribution of dataset labels across different clients. In the following, we present the process of generating multiple levels using approaches presented in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib20" title="">20</a>]</cite>.</p> </div> <div class="ltx_para" id="S5.p6"> <p class="ltx_p" id="S5.p6.6">SVM and logistic regression perform binary classifications on the Breast Cancer Dataset, containing only two labels (true vs. false); thus, we employ three levels of non-IID data. To determine these levels, we used KL divergence as the metric <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib20" title="">20</a>]</cite>. We first set the positive and negative distribution of data labels. For example, if <math alttext="[0.1,0.3,0.5,0.7,0.9]" class="ltx_Math" display="inline" id="S5.p6.1.m1.5"><semantics id="S5.p6.1.m1.5a"><mrow id="S5.p6.1.m1.5.6.2" xref="S5.p6.1.m1.5.6.1.cmml"><mo id="S5.p6.1.m1.5.6.2.1" stretchy="false" xref="S5.p6.1.m1.5.6.1.cmml">[</mo><mn id="S5.p6.1.m1.1.1" xref="S5.p6.1.m1.1.1.cmml">0.1</mn><mo id="S5.p6.1.m1.5.6.2.2" xref="S5.p6.1.m1.5.6.1.cmml">,</mo><mn id="S5.p6.1.m1.2.2" xref="S5.p6.1.m1.2.2.cmml">0.3</mn><mo id="S5.p6.1.m1.5.6.2.3" xref="S5.p6.1.m1.5.6.1.cmml">,</mo><mn id="S5.p6.1.m1.3.3" xref="S5.p6.1.m1.3.3.cmml">0.5</mn><mo id="S5.p6.1.m1.5.6.2.4" xref="S5.p6.1.m1.5.6.1.cmml">,</mo><mn id="S5.p6.1.m1.4.4" xref="S5.p6.1.m1.4.4.cmml">0.7</mn><mo id="S5.p6.1.m1.5.6.2.5" xref="S5.p6.1.m1.5.6.1.cmml">,</mo><mn id="S5.p6.1.m1.5.5" xref="S5.p6.1.m1.5.5.cmml">0.9</mn><mo id="S5.p6.1.m1.5.6.2.6" stretchy="false" xref="S5.p6.1.m1.5.6.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p6.1.m1.5b"><list id="S5.p6.1.m1.5.6.1.cmml" xref="S5.p6.1.m1.5.6.2"><cn id="S5.p6.1.m1.1.1.cmml" type="float" xref="S5.p6.1.m1.1.1">0.1</cn><cn id="S5.p6.1.m1.2.2.cmml" type="float" xref="S5.p6.1.m1.2.2">0.3</cn><cn id="S5.p6.1.m1.3.3.cmml" type="float" xref="S5.p6.1.m1.3.3">0.5</cn><cn id="S5.p6.1.m1.4.4.cmml" type="float" xref="S5.p6.1.m1.4.4">0.7</cn><cn id="S5.p6.1.m1.5.5.cmml" type="float" xref="S5.p6.1.m1.5.5">0.9</cn></list></annotation-xml><annotation encoding="application/x-tex" id="S5.p6.1.m1.5c">[0.1,0.3,0.5,0.7,0.9]</annotation><annotation encoding="application/x-llamapun" id="S5.p6.1.m1.5d">[ 0.1 , 0.3 , 0.5 , 0.7 , 0.9 ]</annotation></semantics></math> and <math alttext="[0.9,0.7,0.5,0.3,0.1]" class="ltx_Math" display="inline" id="S5.p6.2.m2.5"><semantics id="S5.p6.2.m2.5a"><mrow id="S5.p6.2.m2.5.6.2" xref="S5.p6.2.m2.5.6.1.cmml"><mo id="S5.p6.2.m2.5.6.2.1" stretchy="false" xref="S5.p6.2.m2.5.6.1.cmml">[</mo><mn id="S5.p6.2.m2.1.1" xref="S5.p6.2.m2.1.1.cmml">0.9</mn><mo id="S5.p6.2.m2.5.6.2.2" xref="S5.p6.2.m2.5.6.1.cmml">,</mo><mn id="S5.p6.2.m2.2.2" xref="S5.p6.2.m2.2.2.cmml">0.7</mn><mo id="S5.p6.2.m2.5.6.2.3" xref="S5.p6.2.m2.5.6.1.cmml">,</mo><mn id="S5.p6.2.m2.3.3" xref="S5.p6.2.m2.3.3.cmml">0.5</mn><mo id="S5.p6.2.m2.5.6.2.4" xref="S5.p6.2.m2.5.6.1.cmml">,</mo><mn id="S5.p6.2.m2.4.4" xref="S5.p6.2.m2.4.4.cmml">0.3</mn><mo id="S5.p6.2.m2.5.6.2.5" xref="S5.p6.2.m2.5.6.1.cmml">,</mo><mn id="S5.p6.2.m2.5.5" xref="S5.p6.2.m2.5.5.cmml">0.1</mn><mo id="S5.p6.2.m2.5.6.2.6" stretchy="false" xref="S5.p6.2.m2.5.6.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p6.2.m2.5b"><list id="S5.p6.2.m2.5.6.1.cmml" xref="S5.p6.2.m2.5.6.2"><cn id="S5.p6.2.m2.1.1.cmml" type="float" xref="S5.p6.2.m2.1.1">0.9</cn><cn id="S5.p6.2.m2.2.2.cmml" type="float" xref="S5.p6.2.m2.2.2">0.7</cn><cn id="S5.p6.2.m2.3.3.cmml" type="float" xref="S5.p6.2.m2.3.3">0.5</cn><cn id="S5.p6.2.m2.4.4.cmml" type="float" xref="S5.p6.2.m2.4.4">0.3</cn><cn id="S5.p6.2.m2.5.5.cmml" type="float" xref="S5.p6.2.m2.5.5">0.1</cn></list></annotation-xml><annotation encoding="application/x-tex" id="S5.p6.2.m2.5c">[0.9,0.7,0.5,0.3,0.1]</annotation><annotation encoding="application/x-llamapun" id="S5.p6.2.m2.5d">[ 0.9 , 0.7 , 0.5 , 0.3 , 0.1 ]</annotation></semantics></math> are the positive and negative labels across five devices, respectively, then there will be 10% of the positive and 90% of the negative labels from the complete dataset on the first device. Similarly, the second device will have 30% positive labels and 70% negative labels. We consider three different non-IID distributions: <math alttext="[0.5,0.6,0.7,0.8,0.9]" class="ltx_Math" display="inline" id="S5.p6.3.m3.5"><semantics id="S5.p6.3.m3.5a"><mrow id="S5.p6.3.m3.5.6.2" xref="S5.p6.3.m3.5.6.1.cmml"><mo id="S5.p6.3.m3.5.6.2.1" stretchy="false" xref="S5.p6.3.m3.5.6.1.cmml">[</mo><mn id="S5.p6.3.m3.1.1" xref="S5.p6.3.m3.1.1.cmml">0.5</mn><mo id="S5.p6.3.m3.5.6.2.2" xref="S5.p6.3.m3.5.6.1.cmml">,</mo><mn id="S5.p6.3.m3.2.2" xref="S5.p6.3.m3.2.2.cmml">0.6</mn><mo id="S5.p6.3.m3.5.6.2.3" xref="S5.p6.3.m3.5.6.1.cmml">,</mo><mn id="S5.p6.3.m3.3.3" xref="S5.p6.3.m3.3.3.cmml">0.7</mn><mo id="S5.p6.3.m3.5.6.2.4" xref="S5.p6.3.m3.5.6.1.cmml">,</mo><mn id="S5.p6.3.m3.4.4" xref="S5.p6.3.m3.4.4.cmml">0.8</mn><mo id="S5.p6.3.m3.5.6.2.5" xref="S5.p6.3.m3.5.6.1.cmml">,</mo><mn id="S5.p6.3.m3.5.5" xref="S5.p6.3.m3.5.5.cmml">0.9</mn><mo id="S5.p6.3.m3.5.6.2.6" stretchy="false" xref="S5.p6.3.m3.5.6.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p6.3.m3.5b"><list id="S5.p6.3.m3.5.6.1.cmml" xref="S5.p6.3.m3.5.6.2"><cn id="S5.p6.3.m3.1.1.cmml" type="float" xref="S5.p6.3.m3.1.1">0.5</cn><cn id="S5.p6.3.m3.2.2.cmml" type="float" xref="S5.p6.3.m3.2.2">0.6</cn><cn id="S5.p6.3.m3.3.3.cmml" type="float" xref="S5.p6.3.m3.3.3">0.7</cn><cn id="S5.p6.3.m3.4.4.cmml" type="float" xref="S5.p6.3.m3.4.4">0.8</cn><cn id="S5.p6.3.m3.5.5.cmml" type="float" xref="S5.p6.3.m3.5.5">0.9</cn></list></annotation-xml><annotation encoding="application/x-tex" id="S5.p6.3.m3.5c">[0.5,0.6,0.7,0.8,0.9]</annotation><annotation encoding="application/x-llamapun" id="S5.p6.3.m3.5d">[ 0.5 , 0.6 , 0.7 , 0.8 , 0.9 ]</annotation></semantics></math>, and <math alttext="[0.1,0.3,0.5,0.7,0.9]" class="ltx_Math" display="inline" id="S5.p6.4.m4.5"><semantics id="S5.p6.4.m4.5a"><mrow id="S5.p6.4.m4.5.6.2" xref="S5.p6.4.m4.5.6.1.cmml"><mo id="S5.p6.4.m4.5.6.2.1" stretchy="false" xref="S5.p6.4.m4.5.6.1.cmml">[</mo><mn id="S5.p6.4.m4.1.1" xref="S5.p6.4.m4.1.1.cmml">0.1</mn><mo id="S5.p6.4.m4.5.6.2.2" xref="S5.p6.4.m4.5.6.1.cmml">,</mo><mn id="S5.p6.4.m4.2.2" xref="S5.p6.4.m4.2.2.cmml">0.3</mn><mo id="S5.p6.4.m4.5.6.2.3" xref="S5.p6.4.m4.5.6.1.cmml">,</mo><mn id="S5.p6.4.m4.3.3" xref="S5.p6.4.m4.3.3.cmml">0.5</mn><mo id="S5.p6.4.m4.5.6.2.4" xref="S5.p6.4.m4.5.6.1.cmml">,</mo><mn id="S5.p6.4.m4.4.4" xref="S5.p6.4.m4.4.4.cmml">0.7</mn><mo id="S5.p6.4.m4.5.6.2.5" xref="S5.p6.4.m4.5.6.1.cmml">,</mo><mn id="S5.p6.4.m4.5.5" xref="S5.p6.4.m4.5.5.cmml">0.9</mn><mo id="S5.p6.4.m4.5.6.2.6" stretchy="false" xref="S5.p6.4.m4.5.6.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p6.4.m4.5b"><list id="S5.p6.4.m4.5.6.1.cmml" xref="S5.p6.4.m4.5.6.2"><cn id="S5.p6.4.m4.1.1.cmml" type="float" xref="S5.p6.4.m4.1.1">0.1</cn><cn id="S5.p6.4.m4.2.2.cmml" type="float" xref="S5.p6.4.m4.2.2">0.3</cn><cn id="S5.p6.4.m4.3.3.cmml" type="float" xref="S5.p6.4.m4.3.3">0.5</cn><cn id="S5.p6.4.m4.4.4.cmml" type="float" xref="S5.p6.4.m4.4.4">0.7</cn><cn id="S5.p6.4.m4.5.5.cmml" type="float" xref="S5.p6.4.m4.5.5">0.9</cn></list></annotation-xml><annotation encoding="application/x-tex" id="S5.p6.4.m4.5c">[0.1,0.3,0.5,0.7,0.9]</annotation><annotation encoding="application/x-llamapun" id="S5.p6.4.m4.5d">[ 0.1 , 0.3 , 0.5 , 0.7 , 0.9 ]</annotation></semantics></math>,<math alttext="[1,0,0.7,1,0]" class="ltx_Math" display="inline" id="S5.p6.5.m5.5"><semantics id="S5.p6.5.m5.5a"><mrow id="S5.p6.5.m5.5.6.2" xref="S5.p6.5.m5.5.6.1.cmml"><mo id="S5.p6.5.m5.5.6.2.1" stretchy="false" xref="S5.p6.5.m5.5.6.1.cmml">[</mo><mn id="S5.p6.5.m5.1.1" xref="S5.p6.5.m5.1.1.cmml">1</mn><mo id="S5.p6.5.m5.5.6.2.2" xref="S5.p6.5.m5.5.6.1.cmml">,</mo><mn id="S5.p6.5.m5.2.2" xref="S5.p6.5.m5.2.2.cmml">0</mn><mo id="S5.p6.5.m5.5.6.2.3" xref="S5.p6.5.m5.5.6.1.cmml">,</mo><mn id="S5.p6.5.m5.3.3" xref="S5.p6.5.m5.3.3.cmml">0.7</mn><mo id="S5.p6.5.m5.5.6.2.4" xref="S5.p6.5.m5.5.6.1.cmml">,</mo><mn id="S5.p6.5.m5.4.4" xref="S5.p6.5.m5.4.4.cmml">1</mn><mo id="S5.p6.5.m5.5.6.2.5" xref="S5.p6.5.m5.5.6.1.cmml">,</mo><mn id="S5.p6.5.m5.5.5" xref="S5.p6.5.m5.5.5.cmml">0</mn><mo id="S5.p6.5.m5.5.6.2.6" stretchy="false" xref="S5.p6.5.m5.5.6.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p6.5.m5.5b"><list id="S5.p6.5.m5.5.6.1.cmml" xref="S5.p6.5.m5.5.6.2"><cn id="S5.p6.5.m5.1.1.cmml" type="integer" xref="S5.p6.5.m5.1.1">1</cn><cn id="S5.p6.5.m5.2.2.cmml" type="integer" xref="S5.p6.5.m5.2.2">0</cn><cn id="S5.p6.5.m5.3.3.cmml" type="float" xref="S5.p6.5.m5.3.3">0.7</cn><cn id="S5.p6.5.m5.4.4.cmml" type="integer" xref="S5.p6.5.m5.4.4">1</cn><cn id="S5.p6.5.m5.5.5.cmml" type="integer" xref="S5.p6.5.m5.5.5">0</cn></list></annotation-xml><annotation encoding="application/x-tex" id="S5.p6.5.m5.5c">[1,0,0.7,1,0]</annotation><annotation encoding="application/x-llamapun" id="S5.p6.5.m5.5d">[ 1 , 0 , 0.7 , 1 , 0 ]</annotation></semantics></math>. We compare these distributions with the complete data distribution (<math alttext="[1,1,1,1,1]" class="ltx_Math" display="inline" id="S5.p6.6.m6.5"><semantics id="S5.p6.6.m6.5a"><mrow id="S5.p6.6.m6.5.6.2" xref="S5.p6.6.m6.5.6.1.cmml"><mo id="S5.p6.6.m6.5.6.2.1" stretchy="false" xref="S5.p6.6.m6.5.6.1.cmml">[</mo><mn id="S5.p6.6.m6.1.1" xref="S5.p6.6.m6.1.1.cmml">1</mn><mo id="S5.p6.6.m6.5.6.2.2" xref="S5.p6.6.m6.5.6.1.cmml">,</mo><mn id="S5.p6.6.m6.2.2" xref="S5.p6.6.m6.2.2.cmml">1</mn><mo id="S5.p6.6.m6.5.6.2.3" xref="S5.p6.6.m6.5.6.1.cmml">,</mo><mn id="S5.p6.6.m6.3.3" xref="S5.p6.6.m6.3.3.cmml">1</mn><mo id="S5.p6.6.m6.5.6.2.4" xref="S5.p6.6.m6.5.6.1.cmml">,</mo><mn id="S5.p6.6.m6.4.4" xref="S5.p6.6.m6.4.4.cmml">1</mn><mo id="S5.p6.6.m6.5.6.2.5" xref="S5.p6.6.m6.5.6.1.cmml">,</mo><mn id="S5.p6.6.m6.5.5" xref="S5.p6.6.m6.5.5.cmml">1</mn><mo id="S5.p6.6.m6.5.6.2.6" stretchy="false" xref="S5.p6.6.m6.5.6.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p6.6.m6.5b"><list id="S5.p6.6.m6.5.6.1.cmml" xref="S5.p6.6.m6.5.6.2"><cn id="S5.p6.6.m6.1.1.cmml" type="integer" xref="S5.p6.6.m6.1.1">1</cn><cn id="S5.p6.6.m6.2.2.cmml" type="integer" xref="S5.p6.6.m6.2.2">1</cn><cn id="S5.p6.6.m6.3.3.cmml" type="integer" xref="S5.p6.6.m6.3.3">1</cn><cn id="S5.p6.6.m6.4.4.cmml" type="integer" xref="S5.p6.6.m6.4.4">1</cn><cn id="S5.p6.6.m6.5.5.cmml" type="integer" xref="S5.p6.6.m6.5.5">1</cn></list></annotation-xml><annotation encoding="application/x-tex" id="S5.p6.6.m6.5c">[1,1,1,1,1]</annotation><annotation encoding="application/x-llamapun" id="S5.p6.6.m6.5d">[ 1 , 1 , 1 , 1 , 1 ]</annotation></semantics></math>) and calculate the KL divergence, obtaining values of 0.0206,0.18013 and 0.52371 which represent three different levels. This method ensures that each device has all labels, but the label distribution varies between devices.</p> </div> <div class="ltx_para" id="S5.p7"> <p class="ltx_p" id="S5.p7.1">In the case of multi-label data used in deep learning, we consider two scenarios: 1) each device has all labels, but the label distribution differs between devices, and 2) each device does not have all labels, and the label distribution differs between devices. In total, our experiment has five different levels of non-IID settings. Levels 1 and 2 are the same as in the traditional models, whereas the remaining levels do not have the same levels. Specifically, devices in Level 3 have 90% of the labels from the complete dataset, Level 4 has 70% of the labels, and Level 5 has 50% of the labels. The cc_sub dataset is inherently a non-IID dataset composed of images and text and thus does not need any processing. Each image and its associated text in this dataset differ from others, naturally exhibiting non-IID characteristics <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib15" title="">15</a>]</cite>.</p> </div> <div class="ltx_para" id="S5.p8"> <p class="ltx_p" id="S5.p8.2"><span class="ltx_text ltx_font_bold" id="S5.p8.2.1">Evaluation Scheme and Metrics:</span> We have defined a baseline model, referred to as <math alttext="F^{*}" class="ltx_Math" display="inline" id="S5.p8.1.m1.1"><semantics id="S5.p8.1.m1.1a"><msup id="S5.p8.1.m1.1.1" xref="S5.p8.1.m1.1.1.cmml"><mi id="S5.p8.1.m1.1.1.2" xref="S5.p8.1.m1.1.1.2.cmml">F</mi><mo id="S5.p8.1.m1.1.1.3" xref="S5.p8.1.m1.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S5.p8.1.m1.1b"><apply id="S5.p8.1.m1.1.1.cmml" xref="S5.p8.1.m1.1.1"><csymbol cd="ambiguous" id="S5.p8.1.m1.1.1.1.cmml" xref="S5.p8.1.m1.1.1">superscript</csymbol><ci id="S5.p8.1.m1.1.1.2.cmml" xref="S5.p8.1.m1.1.1.2">𝐹</ci><times id="S5.p8.1.m1.1.1.3.cmml" xref="S5.p8.1.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p8.1.m1.1c">F^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.p8.1.m1.1d">italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, representing the state with the minimum loss value. Mathematical analysis has confirmed that <math alttext="F^{*}" class="ltx_Math" display="inline" id="S5.p8.2.m2.1"><semantics id="S5.p8.2.m2.1a"><msup id="S5.p8.2.m2.1.1" xref="S5.p8.2.m2.1.1.cmml"><mi id="S5.p8.2.m2.1.1.2" xref="S5.p8.2.m2.1.1.2.cmml">F</mi><mo id="S5.p8.2.m2.1.1.3" xref="S5.p8.2.m2.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S5.p8.2.m2.1b"><apply id="S5.p8.2.m2.1.1.cmml" xref="S5.p8.2.m2.1.1"><csymbol cd="ambiguous" id="S5.p8.2.m2.1.1.1.cmml" xref="S5.p8.2.m2.1.1">superscript</csymbol><ci id="S5.p8.2.m2.1.1.2.cmml" xref="S5.p8.2.m2.1.1.2">𝐹</ci><times id="S5.p8.2.m2.1.1.3.cmml" xref="S5.p8.2.m2.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p8.2.m2.1c">F^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.p8.2.m2.1d">italic_F start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> constitutes the optimal solution under ideal conditions. Such perfect conditions presume the existence of a machine with unlimited computational power and memory capable of processing the entire dataset. In essence, the baseline can be regarded as the optimal model outcome obtained under ideal circumstances. However, the real deployment differs from the ideal one, i.e., we can measure how much a DFL-based model training differs from the baseline.</p> </div> <div class="ltx_para" id="S5.p9"> <p class="ltx_p" id="S5.p9.1">Our first evaluation verifies whether all six DFL deployments support convergence for the chosen models. Thus, we set each device with the complete dataset and the same hyperparameters as the baseline model. In the second evaluation, we consider non-IID data across devices, i.e., the models operate with the same hyperparameter as the baseline models but with different degrees of non-IID datasets. The convergence is measured by examining loss curves. Specifically, when the loss curve becomes gradually flat or the training and validation loss curves intersect, we consider the models converged. We also measure the model accuracies as defined below.</p> </div> <div class="ltx_para" id="S5.p10"> <p class="ltx_p" id="S5.p10.1">We use binary classification for traditional machine learning datasets and multi-class classification for deep learning datasets. The binary classification F1 score is given by </p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="F1=\frac{2\cdot\text{Precision}\cdot\text{Recall}}{\text{Precision}+\text{% Recall}}," class="ltx_Math" display="block" id="S5.Ex12.m1.1"><semantics id="S5.Ex12.m1.1a"><mrow id="S5.Ex12.m1.1.1.1" xref="S5.Ex12.m1.1.1.1.1.cmml"><mrow id="S5.Ex12.m1.1.1.1.1" xref="S5.Ex12.m1.1.1.1.1.cmml"><mrow id="S5.Ex12.m1.1.1.1.1.2" xref="S5.Ex12.m1.1.1.1.1.2.cmml"><mi id="S5.Ex12.m1.1.1.1.1.2.2" xref="S5.Ex12.m1.1.1.1.1.2.2.cmml">F</mi><mo id="S5.Ex12.m1.1.1.1.1.2.1" xref="S5.Ex12.m1.1.1.1.1.2.1.cmml">⁢</mo><mn id="S5.Ex12.m1.1.1.1.1.2.3" xref="S5.Ex12.m1.1.1.1.1.2.3.cmml">1</mn></mrow><mo id="S5.Ex12.m1.1.1.1.1.1" xref="S5.Ex12.m1.1.1.1.1.1.cmml">=</mo><mfrac id="S5.Ex12.m1.1.1.1.1.3" xref="S5.Ex12.m1.1.1.1.1.3.cmml"><mrow id="S5.Ex12.m1.1.1.1.1.3.2" xref="S5.Ex12.m1.1.1.1.1.3.2.cmml"><mn id="S5.Ex12.m1.1.1.1.1.3.2.2" xref="S5.Ex12.m1.1.1.1.1.3.2.2.cmml">2</mn><mo id="S5.Ex12.m1.1.1.1.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S5.Ex12.m1.1.1.1.1.3.2.1.cmml">⋅</mo><mtext id="S5.Ex12.m1.1.1.1.1.3.2.3" xref="S5.Ex12.m1.1.1.1.1.3.2.3a.cmml">Precision</mtext><mo id="S5.Ex12.m1.1.1.1.1.3.2.1a" lspace="0.222em" rspace="0.222em" xref="S5.Ex12.m1.1.1.1.1.3.2.1.cmml">⋅</mo><mtext id="S5.Ex12.m1.1.1.1.1.3.2.4" xref="S5.Ex12.m1.1.1.1.1.3.2.4a.cmml">Recall</mtext></mrow><mrow id="S5.Ex12.m1.1.1.1.1.3.3" xref="S5.Ex12.m1.1.1.1.1.3.3.cmml"><mtext id="S5.Ex12.m1.1.1.1.1.3.3.2" xref="S5.Ex12.m1.1.1.1.1.3.3.2a.cmml">Precision</mtext><mo id="S5.Ex12.m1.1.1.1.1.3.3.1" xref="S5.Ex12.m1.1.1.1.1.3.3.1.cmml">+</mo><mtext id="S5.Ex12.m1.1.1.1.1.3.3.3" xref="S5.Ex12.m1.1.1.1.1.3.3.3a.cmml">Recall</mtext></mrow></mfrac></mrow><mo id="S5.Ex12.m1.1.1.1.2" xref="S5.Ex12.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex12.m1.1b"><apply id="S5.Ex12.m1.1.1.1.1.cmml" xref="S5.Ex12.m1.1.1.1"><eq id="S5.Ex12.m1.1.1.1.1.1.cmml" xref="S5.Ex12.m1.1.1.1.1.1"></eq><apply id="S5.Ex12.m1.1.1.1.1.2.cmml" xref="S5.Ex12.m1.1.1.1.1.2"><times id="S5.Ex12.m1.1.1.1.1.2.1.cmml" xref="S5.Ex12.m1.1.1.1.1.2.1"></times><ci id="S5.Ex12.m1.1.1.1.1.2.2.cmml" xref="S5.Ex12.m1.1.1.1.1.2.2">𝐹</ci><cn id="S5.Ex12.m1.1.1.1.1.2.3.cmml" type="integer" xref="S5.Ex12.m1.1.1.1.1.2.3">1</cn></apply><apply id="S5.Ex12.m1.1.1.1.1.3.cmml" xref="S5.Ex12.m1.1.1.1.1.3"><divide id="S5.Ex12.m1.1.1.1.1.3.1.cmml" xref="S5.Ex12.m1.1.1.1.1.3"></divide><apply id="S5.Ex12.m1.1.1.1.1.3.2.cmml" xref="S5.Ex12.m1.1.1.1.1.3.2"><ci id="S5.Ex12.m1.1.1.1.1.3.2.1.cmml" xref="S5.Ex12.m1.1.1.1.1.3.2.1">⋅</ci><cn id="S5.Ex12.m1.1.1.1.1.3.2.2.cmml" type="integer" xref="S5.Ex12.m1.1.1.1.1.3.2.2">2</cn><ci id="S5.Ex12.m1.1.1.1.1.3.2.3a.cmml" xref="S5.Ex12.m1.1.1.1.1.3.2.3"><mtext id="S5.Ex12.m1.1.1.1.1.3.2.3.cmml" xref="S5.Ex12.m1.1.1.1.1.3.2.3">Precision</mtext></ci><ci id="S5.Ex12.m1.1.1.1.1.3.2.4a.cmml" xref="S5.Ex12.m1.1.1.1.1.3.2.4"><mtext id="S5.Ex12.m1.1.1.1.1.3.2.4.cmml" xref="S5.Ex12.m1.1.1.1.1.3.2.4">Recall</mtext></ci></apply><apply id="S5.Ex12.m1.1.1.1.1.3.3.cmml" xref="S5.Ex12.m1.1.1.1.1.3.3"><plus id="S5.Ex12.m1.1.1.1.1.3.3.1.cmml" xref="S5.Ex12.m1.1.1.1.1.3.3.1"></plus><ci id="S5.Ex12.m1.1.1.1.1.3.3.2a.cmml" xref="S5.Ex12.m1.1.1.1.1.3.3.2"><mtext id="S5.Ex12.m1.1.1.1.1.3.3.2.cmml" xref="S5.Ex12.m1.1.1.1.1.3.3.2">Precision</mtext></ci><ci id="S5.Ex12.m1.1.1.1.1.3.3.3a.cmml" xref="S5.Ex12.m1.1.1.1.1.3.3.3"><mtext id="S5.Ex12.m1.1.1.1.1.3.3.3.cmml" xref="S5.Ex12.m1.1.1.1.1.3.3.3">Recall</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex12.m1.1c">F1=\frac{2\cdot\text{Precision}\cdot\text{Recall}}{\text{Precision}+\text{% Recall}},</annotation><annotation encoding="application/x-llamapun" id="S5.Ex12.m1.1d">italic_F 1 = divide start_ARG 2 ⋅ Precision ⋅ Recall end_ARG start_ARG Precision + Recall end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.p10.2">while the multi-class F1 score is calculated using macro-averaging as</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex13"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="F1_{\text{macro}}=\frac{1}{N}\sum_{i=1}^{N}F1_{i}." class="ltx_Math" display="block" id="S5.Ex13.m1.1"><semantics id="S5.Ex13.m1.1a"><mrow id="S5.Ex13.m1.1.1.1" xref="S5.Ex13.m1.1.1.1.1.cmml"><mrow id="S5.Ex13.m1.1.1.1.1" xref="S5.Ex13.m1.1.1.1.1.cmml"><mrow id="S5.Ex13.m1.1.1.1.1.2" xref="S5.Ex13.m1.1.1.1.1.2.cmml"><mi id="S5.Ex13.m1.1.1.1.1.2.2" xref="S5.Ex13.m1.1.1.1.1.2.2.cmml">F</mi><mo id="S5.Ex13.m1.1.1.1.1.2.1" xref="S5.Ex13.m1.1.1.1.1.2.1.cmml">⁢</mo><msub id="S5.Ex13.m1.1.1.1.1.2.3" xref="S5.Ex13.m1.1.1.1.1.2.3.cmml"><mn id="S5.Ex13.m1.1.1.1.1.2.3.2" xref="S5.Ex13.m1.1.1.1.1.2.3.2.cmml">1</mn><mtext id="S5.Ex13.m1.1.1.1.1.2.3.3" xref="S5.Ex13.m1.1.1.1.1.2.3.3a.cmml">macro</mtext></msub></mrow><mo id="S5.Ex13.m1.1.1.1.1.1" xref="S5.Ex13.m1.1.1.1.1.1.cmml">=</mo><mrow id="S5.Ex13.m1.1.1.1.1.3" xref="S5.Ex13.m1.1.1.1.1.3.cmml"><mfrac id="S5.Ex13.m1.1.1.1.1.3.2" xref="S5.Ex13.m1.1.1.1.1.3.2.cmml"><mn id="S5.Ex13.m1.1.1.1.1.3.2.2" xref="S5.Ex13.m1.1.1.1.1.3.2.2.cmml">1</mn><mi id="S5.Ex13.m1.1.1.1.1.3.2.3" xref="S5.Ex13.m1.1.1.1.1.3.2.3.cmml">N</mi></mfrac><mo id="S5.Ex13.m1.1.1.1.1.3.1" xref="S5.Ex13.m1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S5.Ex13.m1.1.1.1.1.3.3" xref="S5.Ex13.m1.1.1.1.1.3.3.cmml"><munderover id="S5.Ex13.m1.1.1.1.1.3.3.1" xref="S5.Ex13.m1.1.1.1.1.3.3.1.cmml"><mo id="S5.Ex13.m1.1.1.1.1.3.3.1.2.2" movablelimits="false" 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id="S5.Ex13.m1.1.1.1.1.3.3.1.2.2.cmml" xref="S5.Ex13.m1.1.1.1.1.3.3.1.2.2"></sum><apply id="S5.Ex13.m1.1.1.1.1.3.3.1.2.3.cmml" xref="S5.Ex13.m1.1.1.1.1.3.3.1.2.3"><eq id="S5.Ex13.m1.1.1.1.1.3.3.1.2.3.1.cmml" xref="S5.Ex13.m1.1.1.1.1.3.3.1.2.3.1"></eq><ci id="S5.Ex13.m1.1.1.1.1.3.3.1.2.3.2.cmml" xref="S5.Ex13.m1.1.1.1.1.3.3.1.2.3.2">𝑖</ci><cn id="S5.Ex13.m1.1.1.1.1.3.3.1.2.3.3.cmml" type="integer" xref="S5.Ex13.m1.1.1.1.1.3.3.1.2.3.3">1</cn></apply></apply><ci id="S5.Ex13.m1.1.1.1.1.3.3.1.3.cmml" xref="S5.Ex13.m1.1.1.1.1.3.3.1.3">𝑁</ci></apply><apply id="S5.Ex13.m1.1.1.1.1.3.3.2.cmml" xref="S5.Ex13.m1.1.1.1.1.3.3.2"><times id="S5.Ex13.m1.1.1.1.1.3.3.2.1.cmml" xref="S5.Ex13.m1.1.1.1.1.3.3.2.1"></times><ci id="S5.Ex13.m1.1.1.1.1.3.3.2.2.cmml" xref="S5.Ex13.m1.1.1.1.1.3.3.2.2">𝐹</ci><apply id="S5.Ex13.m1.1.1.1.1.3.3.2.3.cmml" xref="S5.Ex13.m1.1.1.1.1.3.3.2.3"><csymbol cd="ambiguous" id="S5.Ex13.m1.1.1.1.1.3.3.2.3.1.cmml" xref="S5.Ex13.m1.1.1.1.1.3.3.2.3">subscript</csymbol><cn id="S5.Ex13.m1.1.1.1.1.3.3.2.3.2.cmml" type="integer" xref="S5.Ex13.m1.1.1.1.1.3.3.2.3.2">1</cn><ci id="S5.Ex13.m1.1.1.1.1.3.3.2.3.3.cmml" xref="S5.Ex13.m1.1.1.1.1.3.3.2.3.3">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex13.m1.1c">F1_{\text{macro}}=\frac{1}{N}\sum_{i=1}^{N}F1_{i}.</annotation><annotation encoding="application/x-llamapun" id="S5.Ex13.m1.1d">italic_F 1 start_POSTSUBSCRIPT macro end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_N end_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_F 1 start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.p10.3">This approach comprehensively measures the model’s performance across different classes. For multi-class tasks, the macro-averaged F1 score reflects the balanced performance of each class, avoiding bias towards classes with larger data volumes.</p> </div> <div class="ltx_para" id="S5.p11"> <p class="ltx_p" id="S5.p11.1">To test the LLM accuracy as described in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#bib.bib15" title="">15</a>]</cite>, we ask the model four different questions in the form of multiple-choice questions:</p> <ul class="ltx_itemize" id="S5.I1"> <li class="ltx_item" id="S5.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I1.i1.p1"> <p class="ltx_p" id="S5.I1.i1.p1.1">Help me draft a professional advertisement for this.</p> </div> </li> <li class="ltx_item" id="S5.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I1.i2.p1"> <p class="ltx_p" id="S5.I1.i2.p1.1">Can you craft a beautiful poem about this image?</p> </div> </li> <li class="ltx_item" id="S5.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I1.i3.p1"> <p class="ltx_p" id="S5.I1.i3.p1.1">Explain why this meme is funny.</p> </div> </li> <li class="ltx_item" id="S5.I1.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I1.i4.p1"> <p class="ltx_p" id="S5.I1.i4.p1.1">How should I make something like this?</p> </div> </li> </ul> <p class="ltx_p" id="S5.p11.2">We provide the model with an image and then ask these questions to see if there are any obvious errors in the responses, such as mentioning objects not present in the image. In the case of error, the score is 0; otherwise, it is 1.</p> </div> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">VI </span><span class="ltx_text ltx_font_smallcaps" id="S6.1.1">Baseline and DFL Evaluations</span> </h2> <div class="ltx_para" id="S6.p1"> <p class="ltx_p" id="S6.p1.1">This section first presents our evaluation process along with the baseline results.</p> </div> <div class="ltx_para" id="S6.p2"> <p class="ltx_p" id="S6.p2.1">The baseline evaluation refers to training a model in a single machine with adequate resources without relying on distributed training. This assessment offers the optimal set of hyperparameters and the number of epochs for each model. Thus, we can assess how closely the performance of DFL deployments matches with the baseline one using the same hyperparameters. Specifically, we use 80%, 10%, and 10% split as training, validation, and test, respectively. The list of hyperparameters is presented in Table <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S6.T3" title="TABLE III ‣ VI Baseline and DFL Evaluations ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">III</span></a>. We report the learning rate (LR), batch size, number of epochs to converge, and total epochs for each model. The trained model is tested to check their performance in terms of F1 score except for the MiniGPT4. The F1 score of the SVM and Regression model is 0.95 and 0.94, respectively. ReNet18 and DistilBert offer an F1 score of 0.99 and 0.97, respectively. In the case of MiniGPT4, we usually measure its accuracy and loss as we must ask the minigpt4 questions and check the correctness of the response. Its accuracy is 0.90.</p> </div> <figure class="ltx_table" id="S6.T3"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S6.T3.2.1.1" style="font-size:90%;">TABLE III</span>: </span><span class="ltx_text" id="S6.T3.3.2" style="font-size:90%;"> Baseline trained models and their Hyperparameters. </span></figcaption> <p class="ltx_p ltx_align_center" id="S6.T3.4"><span class="ltx_text" id="S6.T3.4.1"> <span class="ltx_inline-block ltx_transformed_outer" id="S6.T3.4.1.1" style="width:286.9pt;height:338.4pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(0.0pt,0.0pt) scale(1,1) ;"> <span class="ltx_p" id="S6.T3.4.1.1.1"><span class="ltx_text" id="S6.T3.4.1.1.1.1"> <span class="ltx_tabular ltx_align_middle" id="S6.T3.4.1.1.1.1.1"> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.1"> <span class="ltx_td ltx_align_left ltx_border_l ltx_border_r ltx_border_t" id="S6.T3.4.1.1.1.1.1.1.1">Models</span> <span class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S6.T3.4.1.1.1.1.1.1.2">Hyperparameters</span> <span class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S6.T3.4.1.1.1.1.1.1.3"><span class="ltx_text" id="S6.T3.4.1.1.1.1.1.1.3.1"></span><span class="ltx_text" id="S6.T3.4.1.1.1.1.1.1.3.2"> <span class="ltx_tabular ltx_align_middle" id="S6.T3.4.1.1.1.1.1.1.3.2.1"> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.1.3.2.1.1"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S6.T3.4.1.1.1.1.1.1.3.2.1.1.1">coverage/total</span></span> </span></span><span class="ltx_text" id="S6.T3.4.1.1.1.1.1.1.3.3"></span></span></span> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.2"> <span class="ltx_td ltx_align_left ltx_border_l ltx_border_r ltx_border_t" id="S6.T3.4.1.1.1.1.1.2.1">SVM</span> <span class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S6.T3.4.1.1.1.1.1.2.2"><span class="ltx_text" id="S6.T3.4.1.1.1.1.1.2.2.1"></span><span class="ltx_text" id="S6.T3.4.1.1.1.1.1.2.2.2"> <span class="ltx_tabular ltx_align_middle" id="S6.T3.4.1.1.1.1.1.2.2.2.1"> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.2.2.2.1.1"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S6.T3.4.1.1.1.1.1.2.2.2.1.1.1">lr: 0.00001</span></span> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.2.2.2.1.2"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S6.T3.4.1.1.1.1.1.2.2.2.1.2.1">batch size: 1</span></span> </span></span><span class="ltx_text" id="S6.T3.4.1.1.1.1.1.2.2.3"></span></span> <span class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S6.T3.4.1.1.1.1.1.2.3">60/500</span></span> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.3"> <span class="ltx_td ltx_align_left ltx_border_l ltx_border_r ltx_border_t" id="S6.T3.4.1.1.1.1.1.3.1"><span class="ltx_text" id="S6.T3.4.1.1.1.1.1.3.1.1"></span><span class="ltx_text" id="S6.T3.4.1.1.1.1.1.3.1.2"> <span class="ltx_tabular ltx_align_middle" id="S6.T3.4.1.1.1.1.1.3.1.2.1"> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.3.1.2.1.1"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S6.T3.4.1.1.1.1.1.3.1.2.1.1.1">Logistic</span></span> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.3.1.2.1.2"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S6.T3.4.1.1.1.1.1.3.1.2.1.2.1">Regression</span></span> </span></span><span class="ltx_text" id="S6.T3.4.1.1.1.1.1.3.1.3"></span></span> <span class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S6.T3.4.1.1.1.1.1.3.2"><span class="ltx_text" id="S6.T3.4.1.1.1.1.1.3.2.1"></span><span class="ltx_text" id="S6.T3.4.1.1.1.1.1.3.2.2"> <span class="ltx_tabular ltx_align_middle" id="S6.T3.4.1.1.1.1.1.3.2.2.1"> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.3.2.2.1.1"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S6.T3.4.1.1.1.1.1.3.2.2.1.1.1">lr: 0.00001</span></span> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.3.2.2.1.2"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S6.T3.4.1.1.1.1.1.3.2.2.1.2.1">batch size: 1</span></span> </span></span><span class="ltx_text" id="S6.T3.4.1.1.1.1.1.3.2.3"></span></span> <span class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S6.T3.4.1.1.1.1.1.3.3">230/1000</span></span> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.4"> <span class="ltx_td ltx_align_left ltx_border_l ltx_border_r ltx_border_t" id="S6.T3.4.1.1.1.1.1.4.1">ReNet18</span> <span class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S6.T3.4.1.1.1.1.1.4.2"><span class="ltx_text" id="S6.T3.4.1.1.1.1.1.4.2.1"></span><span class="ltx_text" id="S6.T3.4.1.1.1.1.1.4.2.2"> <span class="ltx_tabular ltx_align_middle" id="S6.T3.4.1.1.1.1.1.4.2.2.1"> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.4.2.2.1.1"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S6.T3.4.1.1.1.1.1.4.2.2.1.1.1">lr: 0.001</span></span> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.4.2.2.1.2"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S6.T3.4.1.1.1.1.1.4.2.2.1.2.1">batch size: 64</span></span> </span></span><span class="ltx_text" id="S6.T3.4.1.1.1.1.1.4.2.3"></span></span> <span class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S6.T3.4.1.1.1.1.1.4.3">16/100</span></span> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.5"> <span class="ltx_td ltx_align_left ltx_border_l ltx_border_r ltx_border_t" id="S6.T3.4.1.1.1.1.1.5.1">DistilBert</span> <span class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S6.T3.4.1.1.1.1.1.5.2"><span class="ltx_text" id="S6.T3.4.1.1.1.1.1.5.2.1"></span><span class="ltx_text" id="S6.T3.4.1.1.1.1.1.5.2.2"> <span class="ltx_tabular ltx_align_middle" id="S6.T3.4.1.1.1.1.1.5.2.2.1"> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.5.2.2.1.1"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S6.T3.4.1.1.1.1.1.5.2.2.1.1.1">lr: 0.00001</span></span> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.5.2.2.1.2"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S6.T3.4.1.1.1.1.1.5.2.2.1.2.1">batch size: 16</span></span> </span></span><span class="ltx_text" id="S6.T3.4.1.1.1.1.1.5.2.3"></span></span> <span class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S6.T3.4.1.1.1.1.1.5.3">3/20</span></span> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.6"> <span class="ltx_td ltx_align_left ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="S6.T3.4.1.1.1.1.1.6.1">MiniGPT-4</span> <span class="ltx_td ltx_align_left ltx_border_b ltx_border_r ltx_border_t" id="S6.T3.4.1.1.1.1.1.6.2"><span class="ltx_text" id="S6.T3.4.1.1.1.1.1.6.2.1"></span><span class="ltx_text" id="S6.T3.4.1.1.1.1.1.6.2.2"> <span class="ltx_tabular ltx_align_middle" id="S6.T3.4.1.1.1.1.1.6.2.2.1"> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.6.2.2.1.1"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S6.T3.4.1.1.1.1.1.6.2.2.1.1.1">Initial lr: 3e-5</span></span> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.6.2.2.1.2"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S6.T3.4.1.1.1.1.1.6.2.2.1.2.1">Minimum lr: 1e-5</span></span> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.6.2.2.1.3"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S6.T3.4.1.1.1.1.1.6.2.2.1.3.1">Warmup lr: 1e-6</span></span> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.6.2.2.1.4"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S6.T3.4.1.1.1.1.1.6.2.2.1.4.1">Weight decay: 0.05</span></span> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.6.2.2.1.5"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S6.T3.4.1.1.1.1.1.6.2.2.1.5.1">Iterating: 200 times</span></span> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.6.2.2.1.6"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S6.T3.4.1.1.1.1.1.6.2.2.1.6.1">Warmup step count: 200</span></span> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.6.2.2.1.7"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S6.T3.4.1.1.1.1.1.6.2.2.1.7.1">batch size: 12</span></span> <span class="ltx_tr" id="S6.T3.4.1.1.1.1.1.6.2.2.1.8"> <span class="ltx_td ltx_nopad_r ltx_align_left" id="S6.T3.4.1.1.1.1.1.6.2.2.1.8.1">Image size: 224</span></span> </span></span><span class="ltx_text" id="S6.T3.4.1.1.1.1.1.6.2.3"></span></span> <span class="ltx_td ltx_align_left ltx_border_b ltx_border_r ltx_border_t" id="S6.T3.4.1.1.1.1.1.6.3">45/150</span></span> </span></span></span> </span></span></span></p> </figure> <div class="ltx_para" id="S6.p3"> <p class="ltx_p" id="S6.p3.1"><span class="ltx_text ltx_font_bold" id="S6.p3.1.1">DFL Evaluation:</span> In this evaluation, we consider five identical devices (we start the training with the first device) to train a model collaboratively using the same set of hyperparameters used in the baseline evaluation. Thus, the total number of epochs in all DFL configurations is consistent with the baseline. In linear topologies, the number of epochs in each client is equal to the total number of epochs divided by the number of clients. For example, SVM training takes 500 epochs in the baseline; thus, all the participating devices must use at most 500 epochs to complete the training together and be consistent with the baseline. We allocate an equal number of epochs to each device, e.g., 100 epochs for each of the five participating devices when using SVM. 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For instance, each device is assigned 50 epochs in SVM to be consistent with the total of 500 epochs. Finally, star and mesh topologies use concurrent training; thus, the assigned number of epochs to each device is equal to the total number of epochs divided by the number of rounds. Thus, each device takes 100 epochs as the number of rounds is five. This fixed-epoch strategy helps maintain consistency and fairness across different experiments. By standardizing the number of epochs, we can more accurately compare the performance of different DFL structures under the same conditions. We measure the F1 score as the performance metric in the above setup and evaluations; in particular, in linear and ring topologies, it is the average across five devices, whereas it is the average of multiple rounds in star and mesh topologies. In the following sections, we present the impact of different topologies and the degree of non-IId data on the performance of the five chosen models.</p> </div> </section> <section class="ltx_section" id="S7"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">VII </span><span class="ltx_text ltx_font_smallcaps" id="S7.1.1">Evaluation Results</span> </h2> <div class="ltx_para" id="S7.p1"> <p class="ltx_p" id="S7.p1.1">This section presents the results of all models with six different topological deployments of DFL. We present evaluation results with IID and non-IID data and the final takeaway messages. To better understand and avoid confusion, we need to define two concepts: <span class="ltx_text ltx_font_italic" id="S7.p1.1.1">Transmission Round</span> for ring topology and <span class="ltx_text ltx_font_italic" id="S7.p1.1.2">Aggregation Round</span> for star topology. A <span class="ltx_text ltx_font_italic" id="S7.p1.1.3">transmission round</span> refers to the process where all devices sequentially pass the model parameters and complete one training cycle. An <span class="ltx_text ltx_font_italic" id="S7.p1.1.4">aggregation round</span> refers to the process where the central server and all clients complete one round of parameter aggregation.</p> </div> <section class="ltx_subsection" id="S7.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S7.SS1.4.1.1">VII-A</span> </span><span class="ltx_text ltx_font_italic" id="S7.SS1.5.2">Impact of Topologies on the Convergence Rate</span> </h3> <div class="ltx_para" id="S7.SS1.p1"> <p class="ltx_p" id="S7.SS1.p1.1">The impact of training strategy in linear and ring topologies is different. In the former one, each client only trains a model once irrespective of the training strategy (continuous or aggregated). Thus, the number of epochs in each client is equal to the total number of epochs over the number of clients. On the other hand, the ring topology has two rounds of training; thus, each device gets more than one training chance. Finally, the star and mesh topologies involve concurrent training. In the following, we present the convergence rate of these topologies with IID data.</p> </div> <div class="ltx_para" id="S7.SS1.p2"> <p class="ltx_p" id="S7.SS1.p2.1"><span class="ltx_text ltx_font_bold" id="S7.SS1.p2.1.1">Traditional models:</span> Table <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S7.T4" title="TABLE IV ‣ VII-A Impact of Topologies on the Convergence Rate ‣ VII Evaluation Results ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">IV</span></a> presents the convergence of SVM with 500 training epochs, following the baseline performance. The model converges after 83 and 92 epochs on the first device in the continuous and aggregate linear topologies, respectively. Consequently, subsequent devices converge as they inherit the converged model from the previous device and use their own IID data. In the case of ring topology, the model does not converge in the first round. It is observed in the baseline that SVM needs 60 epochs to converge, whereas the first device in the first round gets 50 epochs. Thus, it needs additional epochs during the second round while receiving parameters from the last device of the previous round. Finally, since each device operates concurrently, they exhibit similar convergence behavior in star and mesh topologies.</p> </div> <div class="ltx_para" id="S7.SS1.p3"> <p class="ltx_p" id="S7.SS1.p3.1">The convergence behaviour of logistic regression with 1000 epochs is presented in Table <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S7.T5" title="TABLE V ‣ VII-A Impact of Topologies on the Convergence Rate ‣ VII Evaluation Results ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">V</span></a>. It exhibits similar performance as in SVM. The model converges in the second device in linear topology, as it needs 230 epochs to converge. However, ring topology resolves this issue, as the first device converges in the second round. Finally, the convergence behaviour of star and mesh topologies is similar to SVM with a higher number of epochs, as the model needs that to converge.</p> </div> <figure class="ltx_table" id="S7.T4"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S7.T4.2.1.1" style="font-size:90%;">TABLE IV</span>: </span><span class="ltx_text" id="S7.T4.3.2" style="font-size:90%;">Convergence rate of SVM.</span></figcaption> <table class="ltx_tabular ltx_centering ltx_align_middle" id="S7.T4.4"> <tr class="ltx_tr" id="S7.T4.4.1"> <td class="ltx_td ltx_align_left ltx_border_tt" id="S7.T4.4.1.1"><span class="ltx_text ltx_font_bold" id="S7.T4.4.1.1.1">Topology</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T4.4.1.2"><span class="ltx_text ltx_font_bold" id="S7.T4.4.1.2.1">Client1</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T4.4.1.3"><span class="ltx_text ltx_font_bold" id="S7.T4.4.1.3.1">Client2</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T4.4.1.4"><span class="ltx_text ltx_font_bold" id="S7.T4.4.1.4.1">Client3</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T4.4.1.5"><span class="ltx_text ltx_font_bold" id="S7.T4.4.1.5.1">Client4</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T4.4.1.6"><span class="ltx_text ltx_font_bold" id="S7.T4.4.1.6.1">Client5</span></td> </tr> <tr class="ltx_tr" id="S7.T4.4.2"> <td class="ltx_td ltx_align_left ltx_border_t" id="S7.T4.4.2.1">C_linear</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T4.4.2.2">83</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T4.4.2.3">100</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T4.4.2.4">200</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T4.4.2.5">300</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T4.4.2.6">400</td> </tr> <tr class="ltx_tr" id="S7.T4.4.3"> <td class="ltx_td ltx_align_left" id="S7.T4.4.3.1">C_ring</td> <td class="ltx_td ltx_align_center" id="S7.T4.4.3.2">250</td> <td class="ltx_td ltx_align_center" id="S7.T4.4.3.3">80</td> <td class="ltx_td ltx_align_center" id="S7.T4.4.3.4">100</td> <td class="ltx_td ltx_align_center" id="S7.T4.4.3.5">150</td> <td class="ltx_td ltx_align_center" id="S7.T4.4.3.6">200</td> </tr> <tr class="ltx_tr" id="S7.T4.4.4"> <td class="ltx_td ltx_align_left" id="S7.T4.4.4.1">A_linear</td> <td class="ltx_td ltx_align_center" id="S7.T4.4.4.2">92</td> <td class="ltx_td ltx_align_center" id="S7.T4.4.4.3">100</td> <td class="ltx_td ltx_align_center" id="S7.T4.4.4.4">200</td> <td class="ltx_td ltx_align_center" id="S7.T4.4.4.5">300</td> <td class="ltx_td ltx_align_center" id="S7.T4.4.4.6">400</td> </tr> <tr class="ltx_tr" id="S7.T4.4.5"> <td class="ltx_td ltx_align_left" id="S7.T4.4.5.1">A_ring</td> <td class="ltx_td ltx_align_center" id="S7.T4.4.5.2">250</td> <td class="ltx_td ltx_align_center" id="S7.T4.4.5.3">95</td> <td class="ltx_td ltx_align_center" id="S7.T4.4.5.4">100</td> <td class="ltx_td ltx_align_center" id="S7.T4.4.5.5">150</td> <td class="ltx_td ltx_align_center" id="S7.T4.4.5.6">200</td> </tr> <tr class="ltx_tr" id="S7.T4.4.6"> <td class="ltx_td ltx_align_left" id="S7.T4.4.6.1">Star</td> <td class="ltx_td ltx_align_center" id="S7.T4.4.6.2">97</td> <td class="ltx_td ltx_align_center" id="S7.T4.4.6.3">99</td> <td class="ltx_td ltx_align_center" id="S7.T4.4.6.4">93</td> <td class="ltx_td ltx_align_center" id="S7.T4.4.6.5">94</td> <td class="ltx_td ltx_align_center" id="S7.T4.4.6.6">95</td> </tr> <tr class="ltx_tr" id="S7.T4.4.7"> <td class="ltx_td ltx_align_left ltx_border_bb" id="S7.T4.4.7.1">Mesh</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T4.4.7.2">96</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T4.4.7.3">91</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T4.4.7.4">92</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T4.4.7.5">98</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T4.4.7.6">99</td> </tr> </table> </figure> <figure class="ltx_table" id="S7.T5"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S7.T5.2.1.1" style="font-size:90%;">TABLE V</span>: </span><span class="ltx_text" id="S7.T5.3.2" style="font-size:90%;">Convergence rate of Logical Regression.</span></figcaption> <table class="ltx_tabular ltx_centering ltx_align_middle" id="S7.T5.4"> <tr class="ltx_tr" id="S7.T5.4.1"> <td class="ltx_td ltx_align_left ltx_border_tt" id="S7.T5.4.1.1"><span class="ltx_text ltx_font_bold" id="S7.T5.4.1.1.1">Topology</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T5.4.1.2"><span class="ltx_text ltx_font_bold" id="S7.T5.4.1.2.1">Client1</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T5.4.1.3"><span class="ltx_text ltx_font_bold" id="S7.T5.4.1.3.1">Client2</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T5.4.1.4"><span class="ltx_text ltx_font_bold" id="S7.T5.4.1.4.1">Client3</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T5.4.1.5"><span class="ltx_text ltx_font_bold" id="S7.T5.4.1.5.1">Client4</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T5.4.1.6"><span class="ltx_text ltx_font_bold" id="S7.T5.4.1.6.1">Client5</span></td> </tr> <tr class="ltx_tr" id="S7.T5.4.2"> <td class="ltx_td ltx_align_left ltx_border_t" id="S7.T5.4.2.1">C_linear</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T5.4.2.2">NC</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T5.4.2.3">230</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T5.4.2.4">400</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T5.4.2.5">600</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T5.4.2.6">800</td> </tr> <tr class="ltx_tr" id="S7.T5.4.3"> <td class="ltx_td ltx_align_left" id="S7.T5.4.3.1">C_ring</td> <td class="ltx_td ltx_align_center" id="S7.T5.4.3.2">500</td> <td class="ltx_td ltx_align_center" id="S7.T5.4.3.3">233</td> <td class="ltx_td ltx_align_center" id="S7.T5.4.3.4">300</td> <td class="ltx_td ltx_align_center" id="S7.T5.4.3.5">400</td> <td class="ltx_td ltx_align_center" id="S7.T5.4.3.6">500</td> </tr> <tr class="ltx_tr" id="S7.T5.4.4"> <td class="ltx_td ltx_align_left" id="S7.T5.4.4.1">A_linear</td> <td class="ltx_td ltx_align_center" id="S7.T5.4.4.2">NC</td> <td class="ltx_td ltx_align_center" id="S7.T5.4.4.3">250</td> <td class="ltx_td ltx_align_center" id="S7.T5.4.4.4">400</td> <td class="ltx_td ltx_align_center" id="S7.T5.4.4.5">600</td> <td class="ltx_td ltx_align_center" id="S7.T5.4.4.6">800</td> </tr> <tr class="ltx_tr" id="S7.T5.4.5"> <td class="ltx_td ltx_align_left" id="S7.T5.4.5.1">A_ring</td> <td class="ltx_td ltx_align_center" id="S7.T5.4.5.2">500</td> <td class="ltx_td ltx_align_center" id="S7.T5.4.5.3">250</td> <td class="ltx_td ltx_align_center" id="S7.T5.4.5.4">300</td> <td class="ltx_td ltx_align_center" id="S7.T5.4.5.5">400</td> <td class="ltx_td ltx_align_center" id="S7.T5.4.5.6">500</td> </tr> <tr class="ltx_tr" id="S7.T5.4.6"> <td class="ltx_td ltx_align_left" id="S7.T5.4.6.1">Star</td> <td class="ltx_td ltx_align_center" id="S7.T5.4.6.2">400</td> <td class="ltx_td ltx_align_center" id="S7.T5.4.6.3">401</td> <td class="ltx_td ltx_align_center" id="S7.T5.4.6.4">403</td> <td class="ltx_td ltx_align_center" id="S7.T5.4.6.5">400</td> <td class="ltx_td ltx_align_center" id="S7.T5.4.6.6">402</td> </tr> <tr class="ltx_tr" id="S7.T5.4.7"> <td class="ltx_td ltx_align_left ltx_border_bb" id="S7.T5.4.7.1">Mesh</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T5.4.7.2">400</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T5.4.7.3">401</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T5.4.7.4">402</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T5.4.7.5">400</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T5.4.7.6">402</td> </tr> </table> </figure> <div class="ltx_para" id="S7.SS1.p4"> <p class="ltx_p" id="S7.SS1.p4.1"><span class="ltx_text ltx_font_bold" id="S7.SS1.p4.1.1">Deep neural network models:</span> Similar to traditional models, we fixed the total number of epochs in deep learning, keeping it consistent with the baseline. Table <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S7.T6" title="TABLE VI ‣ VII-A Impact of Topologies on the Convergence Rate ‣ VII Evaluation Results ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">VI</span></a> and Table <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S7.T7" title="TABLE VII ‣ VII-A Impact of Topologies on the Convergence Rate ‣ VII Evaluation Results ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">VII</span></a> present the convergence rate of ResNet and DistilBERT, where the trend is similar to the traditional models. Furthermore, we observe that, since ResNet and DistilBERT are not convex models, unlike traditional models, each device typically has its unique convergence time. This is because when parameters are passed from one device to the next, the loss exhibits fluctuations over 1 to 5 epochs. We speculate that this could be due to the deep structure of models like ResNet, where residual connections introduce non-linear updates during training on different devices, causing such fluctuations. Additionally, we observed that the star and mesh topologies exhibited similar performance to traditional models, as they involve concurrent training. Their results are more stable and do not show fluctuations in loss over several epochs.</p> </div> <figure class="ltx_table" id="S7.T6"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S7.T6.2.1.1" style="font-size:90%;">TABLE VI</span>: </span><span class="ltx_text" id="S7.T6.3.2" style="font-size:90%;"> Convergence rate of ResNet. </span></figcaption> <table class="ltx_tabular ltx_centering ltx_align_middle" id="S7.T6.4"> <tr class="ltx_tr" id="S7.T6.4.1"> <td class="ltx_td ltx_align_left ltx_border_tt" id="S7.T6.4.1.1"><span class="ltx_text ltx_font_bold" id="S7.T6.4.1.1.1">Topology</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T6.4.1.2"><span class="ltx_text ltx_font_bold" id="S7.T6.4.1.2.1">Client1</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T6.4.1.3"><span class="ltx_text ltx_font_bold" id="S7.T6.4.1.3.1">Client2</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T6.4.1.4"><span class="ltx_text ltx_font_bold" id="S7.T6.4.1.4.1">Client3</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T6.4.1.5"><span class="ltx_text ltx_font_bold" id="S7.T6.4.1.5.1">Client4</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T6.4.1.6"><span class="ltx_text ltx_font_bold" id="S7.T6.4.1.6.1">Client5</span></td> </tr> <tr class="ltx_tr" id="S7.T6.4.2"> <td class="ltx_td ltx_align_left ltx_border_t" id="S7.T6.4.2.1">C_linear</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T6.4.2.2">18</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T6.4.2.3">27</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T6.4.2.4">57</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T6.4.2.5">75</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T6.4.2.6">87</td> </tr> <tr class="ltx_tr" id="S7.T6.4.3"> <td class="ltx_td ltx_align_left" id="S7.T6.4.3.1">C_ring</td> <td class="ltx_td ltx_align_center" id="S7.T6.4.3.2">55</td> <td class="ltx_td ltx_align_center" id="S7.T6.4.3.3">17</td> <td class="ltx_td ltx_align_center" id="S7.T6.4.3.4">25</td> <td class="ltx_td ltx_align_center" id="S7.T6.4.3.5">35</td> <td class="ltx_td ltx_align_center" id="S7.T6.4.3.6">46</td> </tr> <tr class="ltx_tr" id="S7.T6.4.4"> <td class="ltx_td ltx_align_left" id="S7.T6.4.4.1">A_linear</td> <td class="ltx_td ltx_align_center" id="S7.T6.4.4.2">19</td> <td class="ltx_td ltx_align_center" id="S7.T6.4.4.3">25</td> <td class="ltx_td ltx_align_center" id="S7.T6.4.4.4">55</td> <td class="ltx_td ltx_align_center" id="S7.T6.4.4.5">78</td> <td class="ltx_td ltx_align_center" id="S7.T6.4.4.6">90</td> </tr> <tr class="ltx_tr" id="S7.T6.4.5"> <td class="ltx_td ltx_align_left" id="S7.T6.4.5.1">A_ring</td> <td class="ltx_td ltx_align_center" id="S7.T6.4.5.2">56</td> <td class="ltx_td ltx_align_center" id="S7.T6.4.5.3">18</td> <td class="ltx_td ltx_align_center" id="S7.T6.4.5.4">25</td> <td class="ltx_td ltx_align_center" id="S7.T6.4.5.5">36</td> <td class="ltx_td ltx_align_center" id="S7.T6.4.5.6">47</td> </tr> <tr class="ltx_tr" id="S7.T6.4.6"> <td class="ltx_td ltx_align_left" id="S7.T6.4.6.1">Star</td> <td class="ltx_td ltx_align_center" id="S7.T6.4.6.2">16</td> <td class="ltx_td ltx_align_center" id="S7.T6.4.6.3">25</td> <td class="ltx_td ltx_align_center" id="S7.T6.4.6.4">53</td> <td class="ltx_td ltx_align_center" id="S7.T6.4.6.5">77</td> <td class="ltx_td ltx_align_center" id="S7.T6.4.6.6">92</td> </tr> <tr class="ltx_tr" id="S7.T6.4.7"> <td class="ltx_td ltx_align_left ltx_border_bb" id="S7.T6.4.7.1">Mesh</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T6.4.7.2">16</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T6.4.7.3">25</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T6.4.7.4">53</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T6.4.7.5">77</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T6.4.7.6">92</td> </tr> </table> </figure> <figure class="ltx_table" id="S7.T7"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S7.T7.2.1.1" style="font-size:90%;">TABLE VII</span>: </span><span class="ltx_text" id="S7.T7.3.2" style="font-size:90%;"> Convergence rate of DistilBERT. </span></figcaption> <table class="ltx_tabular ltx_centering ltx_align_middle" id="S7.T7.4"> <tr class="ltx_tr" id="S7.T7.4.1"> <td class="ltx_td ltx_align_left ltx_border_tt" id="S7.T7.4.1.1"><span class="ltx_text ltx_font_bold" id="S7.T7.4.1.1.1">Topology</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T7.4.1.2"><span class="ltx_text ltx_font_bold" id="S7.T7.4.1.2.1">Client1</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T7.4.1.3"><span class="ltx_text ltx_font_bold" id="S7.T7.4.1.3.1">Client2</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T7.4.1.4"><span class="ltx_text ltx_font_bold" id="S7.T7.4.1.4.1">Client3</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T7.4.1.5"><span class="ltx_text ltx_font_bold" id="S7.T7.4.1.5.1">Client4</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T7.4.1.6"><span class="ltx_text ltx_font_bold" id="S7.T7.4.1.6.1">Client5</span></td> </tr> <tr class="ltx_tr" id="S7.T7.4.2"> <td class="ltx_td ltx_align_left ltx_border_t" id="S7.T7.4.2.1">C_linear</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T7.4.2.2">4</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T7.4.2.3">8</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T7.4.2.4">11</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T7.4.2.5">16</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T7.4.2.6">19</td> </tr> <tr class="ltx_tr" id="S7.T7.4.3"> <td class="ltx_td ltx_align_left" id="S7.T7.4.3.1">C_ring</td> <td class="ltx_td ltx_align_center" id="S7.T7.4.3.2">12</td> <td class="ltx_td ltx_align_center" id="S7.T7.4.3.3">4</td> <td class="ltx_td ltx_align_center" id="S7.T7.4.3.4">16</td> <td class="ltx_td ltx_align_center" id="S7.T7.4.3.5">18</td> <td class="ltx_td ltx_align_center" id="S7.T7.4.3.6">20</td> </tr> <tr class="ltx_tr" id="S7.T7.4.4"> <td class="ltx_td ltx_align_left" id="S7.T7.4.4.1">A_linear</td> <td class="ltx_td ltx_align_center" id="S7.T7.4.4.2">3</td> <td class="ltx_td ltx_align_center" id="S7.T7.4.4.3">7</td> <td class="ltx_td ltx_align_center" id="S7.T7.4.4.4">12</td> <td class="ltx_td ltx_align_center" id="S7.T7.4.4.5">15</td> <td class="ltx_td ltx_align_center" id="S7.T7.4.4.6">20</td> </tr> <tr class="ltx_tr" id="S7.T7.4.5"> <td class="ltx_td ltx_align_left" id="S7.T7.4.5.1">A_ring</td> <td class="ltx_td ltx_align_center" id="S7.T7.4.5.2">11</td> <td class="ltx_td ltx_align_center" id="S7.T7.4.5.3">4</td> <td class="ltx_td ltx_align_center" id="S7.T7.4.5.4">15</td> <td class="ltx_td ltx_align_center" id="S7.T7.4.5.5">18</td> <td class="ltx_td ltx_align_center" id="S7.T7.4.5.6">20</td> </tr> <tr class="ltx_tr" id="S7.T7.4.6"> <td class="ltx_td ltx_align_left" id="S7.T7.4.6.1">Star</td> <td class="ltx_td ltx_align_center" id="S7.T7.4.6.2">4</td> <td class="ltx_td ltx_align_center" id="S7.T7.4.6.3">8</td> <td class="ltx_td ltx_align_center" id="S7.T7.4.6.4">11</td> <td class="ltx_td ltx_align_center" id="S7.T7.4.6.5">15</td> <td class="ltx_td ltx_align_center" id="S7.T7.4.6.6">19</td> </tr> <tr class="ltx_tr" id="S7.T7.4.7"> <td class="ltx_td ltx_align_left ltx_border_bb" id="S7.T7.4.7.1">Mesh</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T7.4.7.2">4</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T7.4.7.3">8</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T7.4.7.4">11</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T7.4.7.5">15</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T7.4.7.6">19</td> </tr> </table> </figure> <figure class="ltx_table" id="S7.T8"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S7.T8.2.1.1" style="font-size:90%;">TABLE VIII</span>: </span><span class="ltx_text" id="S7.T8.3.2" style="font-size:90%;">Convergence rate of MiniGPT.</span></figcaption> <table class="ltx_tabular ltx_centering ltx_align_middle" id="S7.T8.4"> <tr class="ltx_tr" id="S7.T8.4.1"> <td class="ltx_td ltx_align_left ltx_border_tt" id="S7.T8.4.1.1"><span class="ltx_text ltx_font_bold" id="S7.T8.4.1.1.1">Topology</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T8.4.1.2"><span class="ltx_text ltx_font_bold" id="S7.T8.4.1.2.1">Client1</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T8.4.1.3"><span class="ltx_text ltx_font_bold" id="S7.T8.4.1.3.1">Client2</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T8.4.1.4"><span class="ltx_text ltx_font_bold" id="S7.T8.4.1.4.1">Client3</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T8.4.1.5"><span class="ltx_text ltx_font_bold" id="S7.T8.4.1.5.1">Client4</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T8.4.1.6"><span class="ltx_text ltx_font_bold" id="S7.T8.4.1.6.1">Client5</span></td> </tr> <tr class="ltx_tr" id="S7.T8.4.2"> <td class="ltx_td ltx_align_left ltx_border_t" id="S7.T8.4.2.1">C_linear</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T8.4.2.2">NC</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T8.4.2.3">45</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T8.4.2.4">62</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T8.4.2.5">93</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T8.4.2.6">123</td> </tr> <tr class="ltx_tr" id="S7.T8.4.3"> <td class="ltx_td ltx_align_left" id="S7.T8.4.3.1">C_ring</td> <td class="ltx_td ltx_align_center" id="S7.T8.4.3.2">65</td> <td class="ltx_td ltx_align_center" id="S7.T8.4.3.3">26</td> <td class="ltx_td ltx_align_center" id="S7.T8.4.3.4">30</td> <td class="ltx_td ltx_align_center" id="S7.T8.4.3.5">45</td> <td class="ltx_td ltx_align_center" id="S7.T8.4.3.6">60</td> </tr> <tr class="ltx_tr" id="S7.T8.4.4"> <td class="ltx_td ltx_align_left" id="S7.T8.4.4.1">A_linear</td> <td class="ltx_td ltx_align_center" id="S7.T8.4.4.2">NC</td> <td class="ltx_td ltx_align_center" id="S7.T8.4.4.3">39</td> <td class="ltx_td ltx_align_center" id="S7.T8.4.4.4">63</td> <td class="ltx_td ltx_align_center" id="S7.T8.4.4.5">91</td> <td class="ltx_td ltx_align_center" id="S7.T8.4.4.6">121</td> </tr> <tr class="ltx_tr" id="S7.T8.4.5"> <td class="ltx_td ltx_align_left" id="S7.T8.4.5.1">A_ring</td> <td class="ltx_td ltx_align_center" id="S7.T8.4.5.2">65</td> <td class="ltx_td ltx_align_center" id="S7.T8.4.5.3">24</td> <td class="ltx_td ltx_align_center" id="S7.T8.4.5.4">30</td> <td class="ltx_td ltx_align_center" id="S7.T8.4.5.5">45</td> <td class="ltx_td ltx_align_center" id="S7.T8.4.5.6">60</td> </tr> <tr class="ltx_tr" id="S7.T8.4.6"> <td class="ltx_td ltx_align_left" id="S7.T8.4.6.1">Star</td> <td class="ltx_td ltx_align_center" id="S7.T8.4.6.2">40</td> <td class="ltx_td ltx_align_center" id="S7.T8.4.6.3">40</td> <td class="ltx_td ltx_align_center" id="S7.T8.4.6.4">41</td> <td class="ltx_td ltx_align_center" id="S7.T8.4.6.5">40</td> <td class="ltx_td ltx_align_center" id="S7.T8.4.6.6">42</td> </tr> <tr class="ltx_tr" id="S7.T8.4.7"> <td class="ltx_td ltx_align_left ltx_border_bb" id="S7.T8.4.7.1">Mesh</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T8.4.7.2">40</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T8.4.7.3">41</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T8.4.7.4">40</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T8.4.7.5">41</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T8.4.7.6">40</td> </tr> </table> </figure> <div class="ltx_para" id="S7.SS1.p5"> <p class="ltx_p" id="S7.SS1.p5.1"><span class="ltx_text ltx_font_bold" id="S7.SS1.p5.1.1">Large language models:</span> Table <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S7.T8" title="TABLE VIII ‣ VII-A Impact of Topologies on the Convergence Rate ‣ VII Evaluation Results ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">VIII</span></a> presents the convergence rate of MiniGPT. Like deep learning models, MiniGPT loss exhibits fluctuations when parameters are passed from one device to another, as the model is not convex. The behaviour across topologies is expected and aligned with the previous trend. Specifically, the model converges consistently over star and mesh topologies.</p> </div> <figure class="ltx_table" id="S7.T9"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S7.T9.2.1.1" style="font-size:90%;">TABLE IX</span>: </span><span class="ltx_text" id="S7.T9.3.2" style="font-size:90%;">The average F1 score on different DFL deployments.</span></figcaption> <p class="ltx_p ltx_align_center" id="S7.T9.4"><span class="ltx_text" id="S7.T9.4.1"> <span class="ltx_inline-block ltx_transformed_outer" id="S7.T9.4.1.1" style="width:249.0pt;height:123.1pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(0.0pt,0.0pt) scale(1,1) ;"> <span class="ltx_p" id="S7.T9.4.1.1.1"><span class="ltx_text" id="S7.T9.4.1.1.1.1"> <span class="ltx_tabular ltx_align_middle" id="S7.T9.4.1.1.1.1.1"> <span class="ltx_tr" id="S7.T9.4.1.1.1.1.1.1"> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_l ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.1.1"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.1.1.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.1.1.1.1" style="width:34.1pt;"><span class="ltx_text ltx_font_bold" id="S7.T9.4.1.1.1.1.1.1.1.1.1.1">Model</span></span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.1.2"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.1.2.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.1.2.1.1" style="width:25.6pt;"><span class="ltx_text ltx_font_bold" id="S7.T9.4.1.1.1.1.1.1.2.1.1.1">C_linear</span></span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.1.3"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.1.3.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.1.3.1.1" style="width:22.8pt;"><span class="ltx_text ltx_font_bold" id="S7.T9.4.1.1.1.1.1.1.3.1.1.1">C_ring</span></span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.1.4"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.1.4.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.1.4.1.1" style="width:25.6pt;"><span class="ltx_text ltx_font_bold" id="S7.T9.4.1.1.1.1.1.1.4.1.1.1">A_linear</span></span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.1.5"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.1.5.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.1.5.1.1" style="width:22.8pt;"><span class="ltx_text ltx_font_bold" id="S7.T9.4.1.1.1.1.1.1.5.1.1.1">A_ring</span></span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.1.6"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.1.6.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.1.6.1.1" style="width:17.1pt;"><span class="ltx_text ltx_font_bold" id="S7.T9.4.1.1.1.1.1.1.6.1.1.1">Star</span></span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.1.7"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.1.7.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.1.7.1.1" style="width:17.1pt;"><span class="ltx_text ltx_font_bold" id="S7.T9.4.1.1.1.1.1.1.7.1.1.1">Mesh</span></span> </span></span></span> <span class="ltx_tr" id="S7.T9.4.1.1.1.1.1.2"> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_l ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.2.1"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.2.1.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.2.1.1.1" style="width:34.1pt;">SVM</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.2.2"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.2.2.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.2.2.1.1" style="width:25.6pt;">0.964</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.2.3"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.2.3.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.2.3.1.1" style="width:22.8pt;">0.960</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.2.4"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.2.4.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.2.4.1.1" style="width:25.6pt;">0.957</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.2.5"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.2.5.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.2.5.1.1" style="width:22.8pt;">0.957</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.2.6"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.2.6.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.2.6.1.1" style="width:17.1pt;">0.959</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.2.7"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.2.7.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.2.7.1.1" style="width:17.1pt;">0.960</span> </span></span></span> <span class="ltx_tr" id="S7.T9.4.1.1.1.1.1.3"> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_l ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.3.1"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.3.1.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.3.1.1.1" style="width:34.1pt;">Logistic</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.3.2"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.3.2.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.3.2.1.1" style="width:25.6pt;">0.938</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.3.3"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.3.3.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.3.3.1.1" style="width:22.8pt;">0.946</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.3.4"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.3.4.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.3.4.1.1" style="width:25.6pt;">0.937</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.3.5"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.3.5.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.3.5.1.1" style="width:22.8pt;">0.940</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.3.6"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.3.6.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.3.6.1.1" style="width:17.1pt;">0.944</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.3.7"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.3.7.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.3.7.1.1" style="width:17.1pt;">0.944</span> </span></span></span> <span class="ltx_tr" id="S7.T9.4.1.1.1.1.1.4"> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_l ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.4.1"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.4.1.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.4.1.1.1" style="width:34.1pt;">ResNet</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.4.2"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.4.2.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.4.2.1.1" style="width:25.6pt;">0.987</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.4.3"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.4.3.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.4.3.1.1" style="width:22.8pt;">0.979</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.4.4"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.4.4.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.4.4.1.1" style="width:25.6pt;">0.985</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.4.5"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.4.5.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.4.5.1.1" style="width:22.8pt;">0.981</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.4.6"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.4.6.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.4.6.1.1" style="width:17.1pt;">0.985</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.4.7"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.4.7.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.4.7.1.1" style="width:17.1pt;">0.969</span> </span></span></span> <span class="ltx_tr" id="S7.T9.4.1.1.1.1.1.5"> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.5.1"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.5.1.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.5.1.1.1" style="width:34.1pt;">DistilBERT</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_b ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.5.2"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.5.2.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.5.2.1.1" style="width:25.6pt;">0.938</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_b ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.5.3"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.5.3.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.5.3.1.1" style="width:22.8pt;">0.946</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_b ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.5.4"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.5.4.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.5.4.1.1" style="width:25.6pt;">0.937</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_b ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.5.5"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.5.5.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.5.5.1.1" style="width:22.8pt;">0.940</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_b ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.5.6"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.5.6.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.5.6.1.1" style="width:17.1pt;">0.944</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_b ltx_border_r ltx_border_t" id="S7.T9.4.1.1.1.1.1.5.7"> <span class="ltx_inline-block ltx_align_top" id="S7.T9.4.1.1.1.1.1.5.7.1"> <span class="ltx_p" id="S7.T9.4.1.1.1.1.1.5.7.1.1" style="width:17.1pt;">0.944</span> </span></span></span> </span></span></span> </span></span></span></p> </figure> <div class="ltx_para" id="S7.SS1.p6"> <p class="ltx_p" id="S7.SS1.p6.4"><span class="ltx_text ltx_font_bold" id="S7.SS1.p6.4.1">F1 Score:</span> We present the average F1 score of all models over all topologies in Table <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S7.T9" title="TABLE IX ‣ VII-A Impact of Topologies on the Convergence Rate ‣ VII Evaluation Results ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">IX</span></a>. The results indicate that once a model converges, it offers a good score proportional to its convergence rate. Note that we can only measure the accuracy of MiniGPT, which is 0.9 in C_linear, star, and mesh topologies and 0.85 in C_ring, A_linear, and A_ring topologies. Overall, once a model converges in a topology, its performance matches the baseline. The comparison shows that the F1 scores under different topologies are almost identical to the baseline. This indicates that the impact of various topologies on model performance is minimal, and the effectiveness of different deployment methods remains largely consistent. Specifically, the F1 scores of SVM and Logistic Regression under different topologies are almost identical to the baseline, with maximum differences of <math alttext="\textpm 0.014" class="ltx_Math" display="inline" id="S7.SS1.p6.1.m1.1"><semantics id="S7.SS1.p6.1.m1.1a"><mrow id="S7.SS1.p6.1.m1.1.1" xref="S7.SS1.p6.1.m1.1.1.cmml"><mi id="S7.SS1.p6.1.m1.1.1.2" mathvariant="normal" xref="S7.SS1.p6.1.m1.1.1.2.cmml">±</mi><mo id="S7.SS1.p6.1.m1.1.1.1" xref="S7.SS1.p6.1.m1.1.1.1.cmml">⁢</mo><mn id="S7.SS1.p6.1.m1.1.1.3" xref="S7.SS1.p6.1.m1.1.1.3.cmml">0.014</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p6.1.m1.1b"><apply id="S7.SS1.p6.1.m1.1.1.cmml" xref="S7.SS1.p6.1.m1.1.1"><times id="S7.SS1.p6.1.m1.1.1.1.cmml" xref="S7.SS1.p6.1.m1.1.1.1"></times><ci id="S7.SS1.p6.1.m1.1.1.2.cmml" xref="S7.SS1.p6.1.m1.1.1.2">±</ci><cn id="S7.SS1.p6.1.m1.1.1.3.cmml" type="float" xref="S7.SS1.p6.1.m1.1.1.3">0.014</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p6.1.m1.1c">\textpm 0.014</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p6.1.m1.1d">± 0.014</annotation></semantics></math> and <math alttext="\textpm 0.006" class="ltx_Math" display="inline" id="S7.SS1.p6.2.m2.1"><semantics id="S7.SS1.p6.2.m2.1a"><mrow id="S7.SS1.p6.2.m2.1.1" xref="S7.SS1.p6.2.m2.1.1.cmml"><mi id="S7.SS1.p6.2.m2.1.1.2" mathvariant="normal" xref="S7.SS1.p6.2.m2.1.1.2.cmml">±</mi><mo id="S7.SS1.p6.2.m2.1.1.1" xref="S7.SS1.p6.2.m2.1.1.1.cmml">⁢</mo><mn id="S7.SS1.p6.2.m2.1.1.3" xref="S7.SS1.p6.2.m2.1.1.3.cmml">0.006</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p6.2.m2.1b"><apply id="S7.SS1.p6.2.m2.1.1.cmml" xref="S7.SS1.p6.2.m2.1.1"><times id="S7.SS1.p6.2.m2.1.1.1.cmml" xref="S7.SS1.p6.2.m2.1.1.1"></times><ci id="S7.SS1.p6.2.m2.1.1.2.cmml" xref="S7.SS1.p6.2.m2.1.1.2">±</ci><cn id="S7.SS1.p6.2.m2.1.1.3.cmml" type="float" xref="S7.SS1.p6.2.m2.1.1.3">0.006</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p6.2.m2.1c">\textpm 0.006</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p6.2.m2.1d">± 0.006</annotation></semantics></math>, respectively. ResNet shows a maximum difference of <math alttext="0.021" class="ltx_Math" display="inline" id="S7.SS1.p6.3.m3.1"><semantics id="S7.SS1.p6.3.m3.1a"><mn id="S7.SS1.p6.3.m3.1.1" xref="S7.SS1.p6.3.m3.1.1.cmml">0.021</mn><annotation-xml encoding="MathML-Content" id="S7.SS1.p6.3.m3.1b"><cn id="S7.SS1.p6.3.m3.1.1.cmml" type="float" xref="S7.SS1.p6.3.m3.1.1">0.021</cn></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p6.3.m3.1c">0.021</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p6.3.m3.1d">0.021</annotation></semantics></math>, while DistilBERT exhibits a maximum difference of <math alttext="0.033" class="ltx_Math" display="inline" id="S7.SS1.p6.4.m4.1"><semantics id="S7.SS1.p6.4.m4.1a"><mn id="S7.SS1.p6.4.m4.1.1" xref="S7.SS1.p6.4.m4.1.1.cmml">0.033</mn><annotation-xml encoding="MathML-Content" id="S7.SS1.p6.4.m4.1b"><cn id="S7.SS1.p6.4.m4.1.1.cmml" type="float" xref="S7.SS1.p6.4.m4.1.1">0.033</cn></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p6.4.m4.1c">0.033</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p6.4.m4.1d">0.033</annotation></semantics></math> within acceptable limits. Furthermore, MiniGPT-4’s baseline accuracy is 0.9, while Experiment 1 achieves an accuracy of either 0.9 or 0.85, also very close to the baseline. These results suggest that the combinations of different models and topologies have a negligible effect on performance, maintaining a high level of consistency.</p> </div> <div class="ltx_para ltx_noindent" id="S7.SS1.p7"> <svg class="ltx_picture" height="135.55" id="S7.SS1.p7.pic1" overflow="visible" version="1.1" width="600"><g fill="#000000" stroke="#000000" stroke-width="0.4pt" transform="translate(0,135.55) matrix(1 0 0 -1 0 0)"><g fill="#757677" fill-opacity="1.0"><path d="M 0 3.94 L 0 131.61 C 0 133.79 1.76 135.55 3.94 135.55 L 596.06 135.55 C 598.24 135.55 600 133.79 600 131.61 L 600 3.94 C 600 1.76 598.24 0 596.06 0 L 3.94 0 C 1.76 0 0 1.76 0 3.94 Z" style="stroke:none"></path></g><g fill="#DEE0E3" fill-opacity="1.0"><path d="M 8.3 3.94 L 8.3 131.61 C 8.3 133.79 10.06 135.55 12.24 135.55 L 596.06 135.55 C 598.24 135.55 600 133.79 600 131.61 L 600 3.94 C 600 1.76 598.24 0 596.06 0 L 12.24 0 C 10.06 0 8.3 1.76 8.3 3.94 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 27.99 11.81)"><foreignobject color="#000000" height="111.93" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="552.33"> <span class="ltx_inline-block ltx_minipage ltx_align_bottom" id="S7.SS1.p7.pic1.1.1.1.1.1" style="width:399.2pt;"> <span class="ltx_p" id="S7.SS1.p7.pic1.1.1.1.1.1.1"><span class="ltx_text ltx_font_bold" id="S7.SS1.p7.pic1.1.1.1.1.1.1.1">Key Takeaways:</span> We observe that whether the local model is a traditional model, a deep learning model, or an LLM, it successfully converges, and the performance matches the baseline with IID data. However, different topologies require varying computations (epochs), e.g., star and mesh offer the most stable training due to their concurrent training nature. Linear and ring, on the other hand, require adequate computation in each device for model convergence. Finally, deep neural network models may suffer from fluctuations in loss functions while parameters move from one device to another due to their non-convex nature.</span> </span></foreignobject></g></g></svg> </div> </section> <section class="ltx_subsection" id="S7.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S7.SS2.4.1.1">VII-B</span> </span><span class="ltx_text ltx_font_italic" id="S7.SS2.5.2">Impact of Non-IID data Distribution</span> </h3> <div class="ltx_para" id="S7.SS2.p1"> <p class="ltx_p" id="S7.SS2.p1.4">This evaluation aims to investigate the impact of the degree of non-IID data on the performance of DFL with the same hyperparameters as in the baseline model. We evaluate multiple degrees of nonIID data as stated in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S5" title="V Implementation Details ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">V</span></a>, specifically <math alttext="[0.5,0.6,0.7,0.8,0.9]" class="ltx_Math" display="inline" id="S7.SS2.p1.1.m1.5"><semantics id="S7.SS2.p1.1.m1.5a"><mrow id="S7.SS2.p1.1.m1.5.6.2" xref="S7.SS2.p1.1.m1.5.6.1.cmml"><mo id="S7.SS2.p1.1.m1.5.6.2.1" stretchy="false" xref="S7.SS2.p1.1.m1.5.6.1.cmml">[</mo><mn id="S7.SS2.p1.1.m1.1.1" xref="S7.SS2.p1.1.m1.1.1.cmml">0.5</mn><mo id="S7.SS2.p1.1.m1.5.6.2.2" xref="S7.SS2.p1.1.m1.5.6.1.cmml">,</mo><mn id="S7.SS2.p1.1.m1.2.2" xref="S7.SS2.p1.1.m1.2.2.cmml">0.6</mn><mo id="S7.SS2.p1.1.m1.5.6.2.3" xref="S7.SS2.p1.1.m1.5.6.1.cmml">,</mo><mn id="S7.SS2.p1.1.m1.3.3" xref="S7.SS2.p1.1.m1.3.3.cmml">0.7</mn><mo id="S7.SS2.p1.1.m1.5.6.2.4" xref="S7.SS2.p1.1.m1.5.6.1.cmml">,</mo><mn 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We discuss the evaluations results from all three distributions while include only the Level 2 non-IID distribution (<math alttext="[0.1,0.3,0.5,0.7,0.9]" class="ltx_Math" display="inline" id="S7.SS2.p1.4.m4.5"><semantics id="S7.SS2.p1.4.m4.5a"><mrow id="S7.SS2.p1.4.m4.5.6.2" xref="S7.SS2.p1.4.m4.5.6.1.cmml"><mo id="S7.SS2.p1.4.m4.5.6.2.1" stretchy="false" xref="S7.SS2.p1.4.m4.5.6.1.cmml">[</mo><mn id="S7.SS2.p1.4.m4.1.1" xref="S7.SS2.p1.4.m4.1.1.cmml">0.1</mn><mo id="S7.SS2.p1.4.m4.5.6.2.2" xref="S7.SS2.p1.4.m4.5.6.1.cmml">,</mo><mn id="S7.SS2.p1.4.m4.2.2" xref="S7.SS2.p1.4.m4.2.2.cmml">0.3</mn><mo id="S7.SS2.p1.4.m4.5.6.2.3" xref="S7.SS2.p1.4.m4.5.6.1.cmml">,</mo><mn id="S7.SS2.p1.4.m4.3.3" xref="S7.SS2.p1.4.m4.3.3.cmml">0.5</mn><mo id="S7.SS2.p1.4.m4.5.6.2.4" xref="S7.SS2.p1.4.m4.5.6.1.cmml">,</mo><mn id="S7.SS2.p1.4.m4.4.4" xref="S7.SS2.p1.4.m4.4.4.cmml">0.7</mn><mo id="S7.SS2.p1.4.m4.5.6.2.5" xref="S7.SS2.p1.4.m4.5.6.1.cmml">,</mo><mn id="S7.SS2.p1.4.m4.5.5" xref="S7.SS2.p1.4.m4.5.5.cmml">0.9</mn><mo id="S7.SS2.p1.4.m4.5.6.2.6" stretchy="false" xref="S7.SS2.p1.4.m4.5.6.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS2.p1.4.m4.5b"><list id="S7.SS2.p1.4.m4.5.6.1.cmml" xref="S7.SS2.p1.4.m4.5.6.2"><cn id="S7.SS2.p1.4.m4.1.1.cmml" type="float" xref="S7.SS2.p1.4.m4.1.1">0.1</cn><cn id="S7.SS2.p1.4.m4.2.2.cmml" type="float" xref="S7.SS2.p1.4.m4.2.2">0.3</cn><cn id="S7.SS2.p1.4.m4.3.3.cmml" type="float" xref="S7.SS2.p1.4.m4.3.3">0.5</cn><cn id="S7.SS2.p1.4.m4.4.4.cmml" type="float" xref="S7.SS2.p1.4.m4.4.4">0.7</cn><cn id="S7.SS2.p1.4.m4.5.5.cmml" type="float" xref="S7.SS2.p1.4.m4.5.5">0.9</cn></list></annotation-xml><annotation encoding="application/x-tex" id="S7.SS2.p1.4.m4.5c">[0.1,0.3,0.5,0.7,0.9]</annotation><annotation encoding="application/x-llamapun" id="S7.SS2.p1.4.m4.5d">[ 0.1 , 0.3 , 0.5 , 0.7 , 0.9 ]</annotation></semantics></math>) evaluation outcome due to space constraints. Also, we do not evaluate the performance of MiniGPT-4 on non-IID data because its data is inherently non-IID, i.e., different images are associated with different texts with a non-IID distribution.</p> </div> <div class="ltx_para" id="S7.SS2.p2"> <p class="ltx_p" id="S7.SS2.p2.1"><span class="ltx_text ltx_font_bold" id="S7.SS2.p2.1.1">Traditional models:</span> Table <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S7.T10" title="TABLE X ‣ VII-B Impact of Non-IID data Distribution ‣ VII Evaluation Results ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">X</span></a> presents the convergence rate of the SVM with the Level 2 distribution. In the case of linear and ring topologies, irrespective of their training strategies, the model converges only in two devices. We suspect that this is because the data in these two devices have a better label balance than other devices. However, the linear and ring topologies cannot solve such imbalance issues due to the nature of their training. The model, however, converges in all devices over star and mesh topologies, i.e., they offer the best performance as in the case of IID distribution. Concurrent training seems beneficial over sequential training in DFL deployments with non-IID data distribution, where participating devices learn from others. Table <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S7.T11" title="TABLE XI ‣ VII-B Impact of Non-IID data Distribution ‣ VII Evaluation Results ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">XI</span></a> presents the performance of the logistic regression model with level 2 non-IID data. The performance trend in linear and ring topologies is similar to that of the SVM model, i.e., the model converges on two devices with a better label distribution compared to the other devices. Similarly, star and mesh topologies offer the best performance due to their concurrent communications. The performance trend in Level 1 distribution is similar to that of Level 2 due to the same reasons. However, Level 3 results exhibit an interesting but expected performance trend, where the trend just reversed between linear and ring vs. star and mesh. The first two devices in linear and ring topologies complement each other with their labelled data, which is carried over to the remaining devices. However, star and mesh experience an oscillation in this distribution and suffer from performance degradation.</p> </div> <figure class="ltx_table" id="S7.T10"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S7.T10.2.1.1" style="font-size:90%;">TABLE X</span>: </span><span class="ltx_text" id="S7.T10.3.2" style="font-size:90%;">SVM convergence rate with non-IID data.</span></figcaption> <table class="ltx_tabular ltx_centering ltx_align_middle" id="S7.T10.4"> <tr class="ltx_tr" id="S7.T10.4.1"> <td class="ltx_td ltx_align_left ltx_border_tt" id="S7.T10.4.1.1"><span class="ltx_text ltx_font_bold" id="S7.T10.4.1.1.1">Topology</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T10.4.1.2"><span class="ltx_text ltx_font_bold" id="S7.T10.4.1.2.1">Client1</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T10.4.1.3"><span class="ltx_text ltx_font_bold" id="S7.T10.4.1.3.1">Client2</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T10.4.1.4"><span class="ltx_text ltx_font_bold" id="S7.T10.4.1.4.1">Client3</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T10.4.1.5"><span class="ltx_text ltx_font_bold" id="S7.T10.4.1.5.1">Client4</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T10.4.1.6"><span class="ltx_text ltx_font_bold" id="S7.T10.4.1.6.1">Client5</span></td> </tr> <tr class="ltx_tr" id="S7.T10.4.2"> <td class="ltx_td ltx_align_left ltx_border_t" id="S7.T10.4.2.1">C_linear</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T10.4.2.2">NC</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T10.4.2.3">NC</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T10.4.2.4">260</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T10.4.2.5">301</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T10.4.2.6">NC</td> </tr> <tr class="ltx_tr" id="S7.T10.4.3"> <td class="ltx_td ltx_align_left" id="S7.T10.4.3.1">C_ring</td> <td class="ltx_td ltx_align_center" id="S7.T10.4.3.2">NC</td> <td class="ltx_td ltx_align_center" id="S7.T10.4.3.3">NC</td> <td class="ltx_td ltx_align_center" id="S7.T10.4.3.4">400</td> <td class="ltx_td ltx_align_center" id="S7.T10.4.3.5">450</td> <td class="ltx_td ltx_align_center" id="S7.T10.4.3.6">NC</td> </tr> <tr class="ltx_tr" id="S7.T10.4.4"> <td class="ltx_td ltx_align_left" id="S7.T10.4.4.1">A_linear</td> <td class="ltx_td ltx_align_center" id="S7.T10.4.4.2">NC</td> <td class="ltx_td ltx_align_center" id="S7.T10.4.4.3">NC</td> <td class="ltx_td ltx_align_center" id="S7.T10.4.4.4">299</td> <td class="ltx_td ltx_align_center" id="S7.T10.4.4.5">302</td> <td class="ltx_td ltx_align_center" id="S7.T10.4.4.6">NC</td> </tr> <tr class="ltx_tr" id="S7.T10.4.5"> <td class="ltx_td ltx_align_left" id="S7.T10.4.5.1">A_ring</td> <td class="ltx_td ltx_align_center" id="S7.T10.4.5.2">NC</td> <td class="ltx_td ltx_align_center" id="S7.T10.4.5.3">NC</td> <td class="ltx_td ltx_align_center" id="S7.T10.4.5.4">398</td> <td class="ltx_td ltx_align_center" id="S7.T10.4.5.5">452</td> <td class="ltx_td ltx_align_center" id="S7.T10.4.5.6">NC</td> </tr> <tr class="ltx_tr" id="S7.T10.4.6"> <td class="ltx_td ltx_align_left" id="S7.T10.4.6.1">Star</td> <td class="ltx_td ltx_align_center" id="S7.T10.4.6.2">402</td> <td class="ltx_td ltx_align_center" id="S7.T10.4.6.3">400</td> <td class="ltx_td ltx_align_center" id="S7.T10.4.6.4">401</td> <td class="ltx_td ltx_align_center" id="S7.T10.4.6.5">400</td> <td class="ltx_td ltx_align_center" id="S7.T10.4.6.6">402</td> </tr> <tr class="ltx_tr" id="S7.T10.4.7"> <td class="ltx_td ltx_align_left ltx_border_bb" id="S7.T10.4.7.1">Mesh</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T10.4.7.2">403</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T10.4.7.3">400</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T10.4.7.4">401</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T10.4.7.5">400</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T10.4.7.6">401</td> </tr> </table> </figure> <figure class="ltx_table" id="S7.T11"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S7.T11.2.1.1" style="font-size:90%;">TABLE XI</span>: </span><span class="ltx_text" id="S7.T11.3.2" style="font-size:90%;">Logic Regression convergence rate with non-IID data.</span></figcaption> <table class="ltx_tabular ltx_centering ltx_align_middle" id="S7.T11.4"> <tr class="ltx_tr" id="S7.T11.4.1"> <td class="ltx_td ltx_align_left ltx_border_tt" id="S7.T11.4.1.1"><span class="ltx_text ltx_font_bold" id="S7.T11.4.1.1.1">Topology</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T11.4.1.2"><span class="ltx_text ltx_font_bold" id="S7.T11.4.1.2.1">Client1</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T11.4.1.3"><span class="ltx_text ltx_font_bold" id="S7.T11.4.1.3.1">Client2</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T11.4.1.4"><span class="ltx_text ltx_font_bold" id="S7.T11.4.1.4.1">Client3</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T11.4.1.5"><span class="ltx_text ltx_font_bold" id="S7.T11.4.1.5.1">Client4</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T11.4.1.6"><span class="ltx_text ltx_font_bold" id="S7.T11.4.1.6.1">Client5</span></td> </tr> <tr class="ltx_tr" id="S7.T11.4.2"> <td class="ltx_td ltx_align_left ltx_border_t" id="S7.T11.4.2.1">C_linear</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T11.4.2.2">NC</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T11.4.2.3">NC</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T11.4.2.4">599</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T11.4.2.5">799</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T11.4.2.6">NC</td> </tr> <tr class="ltx_tr" id="S7.T11.4.3"> <td class="ltx_td ltx_align_left" id="S7.T11.4.3.1">C_ring</td> <td class="ltx_td ltx_align_center" id="S7.T11.4.3.2">NC</td> <td class="ltx_td ltx_align_center" id="S7.T11.4.3.3">NC</td> <td class="ltx_td ltx_align_center" id="S7.T11.4.3.4">799</td> <td class="ltx_td ltx_align_center" id="S7.T11.4.3.5">845</td> <td class="ltx_td ltx_align_center" id="S7.T11.4.3.6">NC</td> </tr> <tr class="ltx_tr" id="S7.T11.4.4"> <td class="ltx_td ltx_align_left" id="S7.T11.4.4.1">A_linear</td> <td class="ltx_td ltx_align_center" id="S7.T11.4.4.2">NC</td> <td class="ltx_td ltx_align_center" id="S7.T11.4.4.3">NC</td> <td class="ltx_td ltx_align_center" id="S7.T11.4.4.4">599</td> <td class="ltx_td ltx_align_center" id="S7.T11.4.4.5">799</td> <td class="ltx_td ltx_align_center" id="S7.T11.4.4.6">NC</td> </tr> <tr class="ltx_tr" id="S7.T11.4.5"> <td class="ltx_td ltx_align_left" id="S7.T11.4.5.1">A_ring</td> <td class="ltx_td ltx_align_center" id="S7.T11.4.5.2">NC</td> <td class="ltx_td ltx_align_center" id="S7.T11.4.5.3">NC</td> <td class="ltx_td ltx_align_center" id="S7.T11.4.5.4">799</td> <td class="ltx_td ltx_align_center" id="S7.T11.4.5.5">845</td> <td class="ltx_td ltx_align_center" id="S7.T11.4.5.6">NC</td> </tr> <tr class="ltx_tr" id="S7.T11.4.6"> <td class="ltx_td ltx_align_left" id="S7.T11.4.6.1">Star</td> <td class="ltx_td ltx_align_center" id="S7.T11.4.6.2">851</td> <td class="ltx_td ltx_align_center" id="S7.T11.4.6.3">850</td> <td class="ltx_td ltx_align_center" id="S7.T11.4.6.4">852</td> <td class="ltx_td ltx_align_center" id="S7.T11.4.6.5">851</td> <td class="ltx_td ltx_align_center" id="S7.T11.4.6.6">849</td> </tr> <tr class="ltx_tr" id="S7.T11.4.7"> <td class="ltx_td ltx_align_left ltx_border_bb" id="S7.T11.4.7.1">Mesh</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T11.4.7.2">830</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T11.4.7.3">831</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T11.4.7.4">832</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T11.4.7.5">831</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T11.4.7.6">831</td> </tr> </table> </figure> <div class="ltx_para" id="S7.SS2.p3"> <p class="ltx_p" id="S7.SS2.p3.1"><span class="ltx_text ltx_font_bold" id="S7.SS2.p3.1.1">Deep neural network models:</span> Table <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S7.T12" title="TABLE XII ‣ VII-B Impact of Non-IID data Distribution ‣ VII Evaluation Results ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">XII</span></a> and Table <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S7.T13" title="TABLE XIII ‣ VII-B Impact of Non-IID data Distribution ‣ VII Evaluation Results ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">XIII</span></a> present the convergence rate of ResNet and DistilBERT over Level 2 non-IID data distribution, respectively. For Resnet, we observe that at Non-IID level 1, all DFL models can converge. This indicates that at level 1, various topologies perform well. In both cases, star and mesh topologies offer the best performance, as we have seen in the case of traditional models. Similarly, the performance of linear and ring suffers. Specifically, devices with better-balanced labels converge. Level 1 also offers a similar performance trend, while level 3 distribution is the worst in terms of performance across all topologies. We found that in DFLs with non-star or non-mesh topologies, convergence is often complex when the data distribution on the devices is highly uneven. In other words, star and mesh are the most suitable deployment strategies to host deep learning models in DFL deployments.</p> </div> <figure class="ltx_table" id="S7.T12"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S7.T12.2.1.1" style="font-size:90%;">TABLE XII</span>: </span><span class="ltx_text" id="S7.T12.3.2" style="font-size:90%;"> Convergence rate of ResNet on non-IID data. </span></figcaption> <table class="ltx_tabular ltx_centering ltx_align_middle" id="S7.T12.4"> <tr class="ltx_tr" id="S7.T12.4.1"> <td class="ltx_td ltx_align_left ltx_border_tt" id="S7.T12.4.1.1"><span class="ltx_text ltx_font_bold" id="S7.T12.4.1.1.1">Topology</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T12.4.1.2"><span class="ltx_text ltx_font_bold" id="S7.T12.4.1.2.1">Client1</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T12.4.1.3"><span class="ltx_text ltx_font_bold" id="S7.T12.4.1.3.1">Client2</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T12.4.1.4"><span class="ltx_text ltx_font_bold" id="S7.T12.4.1.4.1">Client3</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T12.4.1.5"><span class="ltx_text ltx_font_bold" id="S7.T12.4.1.5.1">Client4</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T12.4.1.6"><span class="ltx_text ltx_font_bold" id="S7.T12.4.1.6.1">Client5</span></td> </tr> <tr class="ltx_tr" id="S7.T12.4.2"> <td class="ltx_td ltx_align_left ltx_border_t" id="S7.T12.4.2.1">C_linear</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T12.4.2.2">NC</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T12.4.2.3">39</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T12.4.2.4">59</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T12.4.2.5">NC</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T12.4.2.6">NC</td> </tr> <tr class="ltx_tr" id="S7.T12.4.3"> <td class="ltx_td ltx_align_left" id="S7.T12.4.3.1">C_ring</td> <td class="ltx_td ltx_align_center" id="S7.T12.4.3.2">NC</td> <td class="ltx_td ltx_align_center" id="S7.T12.4.3.3">20</td> <td class="ltx_td ltx_align_center" id="S7.T12.4.3.4">30</td> <td class="ltx_td ltx_align_center" id="S7.T12.4.3.5">40</td> <td class="ltx_td ltx_align_center" id="S7.T12.4.3.6">NC</td> </tr> <tr class="ltx_tr" id="S7.T12.4.4"> <td class="ltx_td ltx_align_left" id="S7.T12.4.4.1">A_linear</td> <td class="ltx_td ltx_align_center" id="S7.T12.4.4.2">19</td> <td class="ltx_td ltx_align_center" id="S7.T12.4.4.3">39</td> <td class="ltx_td ltx_align_center" id="S7.T12.4.4.4">59</td> <td class="ltx_td ltx_align_center" id="S7.T12.4.4.5">79</td> <td class="ltx_td ltx_align_center" id="S7.T12.4.4.6">99</td> </tr> <tr class="ltx_tr" id="S7.T12.4.5"> <td class="ltx_td ltx_align_left" id="S7.T12.4.5.1">A_ring</td> <td class="ltx_td ltx_align_center" id="S7.T12.4.5.2">NC</td> <td class="ltx_td ltx_align_center" id="S7.T12.4.5.3">20</td> <td class="ltx_td ltx_align_center" id="S7.T12.4.5.4">30</td> <td class="ltx_td ltx_align_center" id="S7.T12.4.5.5">40</td> <td class="ltx_td ltx_align_center" id="S7.T12.4.5.6">NC</td> </tr> <tr class="ltx_tr" id="S7.T12.4.6"> <td class="ltx_td ltx_align_left" id="S7.T12.4.6.1">Star</td> <td class="ltx_td ltx_align_center" id="S7.T12.4.6.2">24</td> <td class="ltx_td ltx_align_center" id="S7.T12.4.6.3">24</td> <td class="ltx_td ltx_align_center" id="S7.T12.4.6.4">24</td> <td class="ltx_td ltx_align_center" id="S7.T12.4.6.5">24</td> <td class="ltx_td ltx_align_center" id="S7.T12.4.6.6">24</td> </tr> <tr class="ltx_tr" id="S7.T12.4.7"> <td class="ltx_td ltx_align_left ltx_border_bb" id="S7.T12.4.7.1">Mesh</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T12.4.7.2">24</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T12.4.7.3">24</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T12.4.7.4">24</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T12.4.7.5">24</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T12.4.7.6">24</td> </tr> </table> </figure> <figure class="ltx_table" id="S7.T13"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S7.T13.2.1.1" style="font-size:90%;">TABLE XIII</span>: </span><span class="ltx_text" id="S7.T13.3.2" style="font-size:90%;"> Convergence rate of DistilBERT on non-IID data. </span></figcaption> <table class="ltx_tabular ltx_centering ltx_align_middle" id="S7.T13.4"> <tr class="ltx_tr" id="S7.T13.4.1"> <td class="ltx_td ltx_align_left ltx_border_tt" id="S7.T13.4.1.1"><span class="ltx_text ltx_font_bold" id="S7.T13.4.1.1.1">Topology</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T13.4.1.2"><span class="ltx_text ltx_font_bold" id="S7.T13.4.1.2.1">Client1</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T13.4.1.3"><span class="ltx_text ltx_font_bold" id="S7.T13.4.1.3.1">Client2</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T13.4.1.4"><span class="ltx_text ltx_font_bold" id="S7.T13.4.1.4.1">Client3</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T13.4.1.5"><span class="ltx_text ltx_font_bold" id="S7.T13.4.1.5.1">Client4</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S7.T13.4.1.6"><span class="ltx_text ltx_font_bold" id="S7.T13.4.1.6.1">Client5</span></td> </tr> <tr class="ltx_tr" id="S7.T13.4.2"> <td class="ltx_td ltx_align_left ltx_border_t" id="S7.T13.4.2.1">C_linear</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T13.4.2.2">NC</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T13.4.2.3">8</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T13.4.2.4">12</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T13.4.2.5">16</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S7.T13.4.2.6">20</td> </tr> <tr class="ltx_tr" id="S7.T13.4.3"> <td class="ltx_td ltx_align_left" id="S7.T13.4.3.1">C_ring</td> <td class="ltx_td ltx_align_center" id="S7.T13.4.3.2">NC</td> <td class="ltx_td ltx_align_center" id="S7.T13.4.3.3">12</td> <td class="ltx_td ltx_align_center" id="S7.T13.4.3.4">14</td> <td class="ltx_td ltx_align_center" id="S7.T13.4.3.5">18</td> <td class="ltx_td ltx_align_center" id="S7.T13.4.3.6">NC</td> </tr> <tr class="ltx_tr" id="S7.T13.4.4"> <td class="ltx_td ltx_align_left" id="S7.T13.4.4.1">A_linear</td> <td class="ltx_td ltx_align_center" id="S7.T13.4.4.2">NC</td> <td class="ltx_td ltx_align_center" id="S7.T13.4.4.3">NC</td> <td class="ltx_td ltx_align_center" id="S7.T13.4.4.4">12</td> <td class="ltx_td ltx_align_center" id="S7.T13.4.4.5">16</td> <td class="ltx_td ltx_align_center" id="S7.T13.4.4.6">20</td> </tr> <tr class="ltx_tr" id="S7.T13.4.5"> <td class="ltx_td ltx_align_left" id="S7.T13.4.5.1">A_ring</td> <td class="ltx_td ltx_align_center" id="S7.T13.4.5.2">NC</td> <td class="ltx_td ltx_align_center" id="S7.T13.4.5.3">12</td> <td class="ltx_td ltx_align_center" id="S7.T13.4.5.4">14</td> <td class="ltx_td ltx_align_center" id="S7.T13.4.5.5">18</td> <td class="ltx_td ltx_align_center" id="S7.T13.4.5.6">NC</td> </tr> <tr class="ltx_tr" id="S7.T13.4.6"> <td class="ltx_td ltx_align_left" id="S7.T13.4.6.1">Star</td> <td class="ltx_td ltx_align_center" id="S7.T13.4.6.2">20</td> <td class="ltx_td ltx_align_center" id="S7.T13.4.6.3">20</td> <td class="ltx_td ltx_align_center" id="S7.T13.4.6.4">20</td> <td class="ltx_td ltx_align_center" id="S7.T13.4.6.5">20</td> <td class="ltx_td ltx_align_center" id="S7.T13.4.6.6">20</td> </tr> <tr class="ltx_tr" id="S7.T13.4.7"> <td class="ltx_td ltx_align_left ltx_border_bb" id="S7.T13.4.7.1">Mesh</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T13.4.7.2">20</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T13.4.7.3">20</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T13.4.7.4">20</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T13.4.7.5">20</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S7.T13.4.7.6">20</td> </tr> </table> </figure> <div class="ltx_para" id="S7.SS2.p4"> <p class="ltx_p" id="S7.SS2.p4.1"><span class="ltx_text ltx_font_bold" id="S7.SS2.p4.1.1">F1 Score:</span> The average F1 score of different DFL deployments for the Level 2 non-IID data distribution is shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.11828v1#S7.T14" title="TABLE XIV ‣ VII-B Impact of Non-IID data Distribution ‣ VII Evaluation Results ‣ Performance Analysis of Decentralized Federated Learning Deployments"><span class="ltx_text ltx_ref_tag">XIV</span></a>. The results indicate that the F1 score decreases when the level of non-IID data distribution increases. Also, their F1 score is proportional to the convergence rate once the models converge. The impact of non-IID data on different topologies for model performance is minimal, indicating that these deployments exhibit good stability and consistency on non-IID data. Specifically, the maximum F1 score difference for the SVM model is 0.029, with the best performance in the C_ring topology (0.825) and the lowest in the A_linear topology (0.796). The maximum difference for Logistic regression is 0.099, performing best in the Mesh topology (0.834) but worst in the C_linear and A_linear topologies (0.742 and 0.735, respectively). ResNet shows the minimal F1 score variation, with a maximum difference of 0.021, demonstrating stable performance. However, DistilBERT has a maximum difference of 0.109, with performance fluctuations, particularly in the star and mesh topologies. ResNet performs best, while SVM and logistic regression models show minimal fluctuation, with DistilBERT offering the worst.</p> </div> <figure class="ltx_table" id="S7.T14"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S7.T14.2.1.1" style="font-size:90%;">TABLE XIV</span>: </span><span class="ltx_text" id="S7.T14.3.2" style="font-size:90%;">The average F1 score of different DFL deployments for non-IID data.</span></figcaption> <p class="ltx_p ltx_align_center" id="S7.T14.4"><span class="ltx_text" id="S7.T14.4.1"> <span class="ltx_inline-block ltx_transformed_outer" id="S7.T14.4.1.1" style="width:252.4pt;height:123.1pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(0.0pt,0.0pt) scale(1,1) ;"> <span class="ltx_p" id="S7.T14.4.1.1.1"><span class="ltx_text" id="S7.T14.4.1.1.1.1"> <span class="ltx_tabular ltx_align_middle" id="S7.T14.4.1.1.1.1.1"> <span class="ltx_tr" id="S7.T14.4.1.1.1.1.1.1"> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_l ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.1.1"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.1.1.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.1.1.1.1" style="width:34.1pt;"><span class="ltx_text ltx_font_bold" id="S7.T14.4.1.1.1.1.1.1.1.1.1.1">Model</span></span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.1.2"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.1.2.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.1.2.1.1" style="width:25.6pt;"><span class="ltx_text ltx_font_bold" id="S7.T14.4.1.1.1.1.1.1.2.1.1.1">C_linear</span></span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.1.3"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.1.3.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.1.3.1.1" style="width:22.8pt;"><span class="ltx_text ltx_font_bold" id="S7.T14.4.1.1.1.1.1.1.3.1.1.1">C_ring</span></span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.1.4"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.1.4.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.1.4.1.1" style="width:25.6pt;"><span class="ltx_text ltx_font_bold" id="S7.T14.4.1.1.1.1.1.1.4.1.1.1">A_linear</span></span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.1.5"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.1.5.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.1.5.1.1" style="width:22.8pt;"><span class="ltx_text ltx_font_bold" id="S7.T14.4.1.1.1.1.1.1.5.1.1.1">A_ring</span></span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.1.6"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.1.6.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.1.6.1.1" style="width:17.1pt;"><span class="ltx_text ltx_font_bold" id="S7.T14.4.1.1.1.1.1.1.6.1.1.1">Star</span></span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.1.7"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.1.7.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.1.7.1.1" style="width:17.1pt;"><span class="ltx_text ltx_font_bold" id="S7.T14.4.1.1.1.1.1.1.7.1.1.1">Mesh</span></span> </span></span></span> <span class="ltx_tr" id="S7.T14.4.1.1.1.1.1.2"> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_l ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.2.1"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.2.1.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.2.1.1.1" style="width:34.1pt;">SVM</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.2.2"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.2.2.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.2.2.1.1" style="width:25.6pt;">0.813</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.2.3"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.2.3.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.2.3.1.1" style="width:22.8pt;">0.825</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.2.4"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.2.4.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.2.4.1.1" style="width:25.6pt;">0.796</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.2.5"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.2.5.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.2.5.1.1" style="width:22.8pt;">0.809</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.2.6"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.2.6.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.2.6.1.1" style="width:17.1pt;">0.806</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.2.7"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.2.7.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.2.7.1.1" style="width:17.1pt;">0.798</span> </span></span></span> <span class="ltx_tr" id="S7.T14.4.1.1.1.1.1.3"> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_l ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.3.1"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.3.1.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.3.1.1.1" style="width:34.1pt;">Logistic</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.3.2"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.3.2.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.3.2.1.1" style="width:25.6pt;">0.742</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.3.3"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.3.3.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.3.3.1.1" style="width:22.8pt;">0.791</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.3.4"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.3.4.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.3.4.1.1" style="width:25.6pt;">0.735</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.3.5"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.3.5.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.3.5.1.1" style="width:22.8pt;">0.783</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.3.6"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.3.6.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.3.6.1.1" style="width:17.1pt;">0.783</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.3.7"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.3.7.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.3.7.1.1" style="width:17.1pt;">0.834</span> </span></span></span> <span class="ltx_tr" id="S7.T14.4.1.1.1.1.1.4"> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_l ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.4.1"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.4.1.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.4.1.1.1" style="width:34.1pt;">ResNet</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.4.2"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.4.2.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.4.2.1.1" style="width:25.6pt;">0.967</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.4.3"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.4.3.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.4.3.1.1" style="width:22.8pt;">0.954</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.4.4"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.4.4.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.4.4.1.1" style="width:25.6pt;">0.975</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.4.5"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.4.5.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.4.5.1.1" style="width:22.8pt;">0.967</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.4.6"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.4.6.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.4.6.1.1" style="width:17.1pt;">0.974</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.4.7"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.4.7.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.4.7.1.1" style="width:17.1pt;">0.963</span> </span></span></span> <span class="ltx_tr" id="S7.T14.4.1.1.1.1.1.5"> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.5.1"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.5.1.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.5.1.1.1" style="width:34.1pt;">DistilBERT</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_b ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.5.2"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.5.2.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.5.2.1.1" style="width:25.6pt;">0.818</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_b ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.5.3"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.5.3.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.5.3.1.1" style="width:22.8pt;">0.744</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_b ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.5.4"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.5.4.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.5.4.1.1" style="width:25.6pt;">0.834</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_b ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.5.5"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.5.5.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.5.5.1.1" style="width:22.8pt;">0.743</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_b ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.5.6"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.5.6.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.5.6.1.1" style="width:17.1pt;">0.730</span> </span></span> <span class="ltx_td ltx_align_justify ltx_align_top ltx_border_b ltx_border_r ltx_border_t" id="S7.T14.4.1.1.1.1.1.5.7"> <span class="ltx_inline-block ltx_align_top" id="S7.T14.4.1.1.1.1.1.5.7.1"> <span class="ltx_p" id="S7.T14.4.1.1.1.1.1.5.7.1.1" style="width:17.1pt;">0.725</span> </span></span></span> </span></span></span> </span></span></span></p> </figure> <div class="ltx_para ltx_noindent" id="S7.SS2.p5"> <svg class="ltx_picture" height="232.48" id="S7.SS2.p5.pic1" overflow="visible" version="1.1" width="600"><g fill="#000000" stroke="#000000" stroke-width="0.4pt" transform="translate(0,232.48) matrix(1 0 0 -1 0 0)"><g fill="#757677" fill-opacity="1.0"><path d="M 0 3.94 L 0 228.55 C 0 230.72 1.76 232.48 3.94 232.48 L 596.06 232.48 C 598.24 232.48 600 230.72 600 228.55 L 600 3.94 C 600 1.76 598.24 0 596.06 0 L 3.94 0 C 1.76 0 0 1.76 0 3.94 Z" style="stroke:none"></path></g><g fill="#DEE0E3" fill-opacity="1.0"><path d="M 8.3 3.94 L 8.3 228.55 C 8.3 230.72 10.06 232.48 12.24 232.48 L 596.06 232.48 C 598.24 232.48 600 230.72 600 228.55 L 600 3.94 C 600 1.76 598.24 0 596.06 0 L 12.24 0 C 10.06 0 8.3 1.76 8.3 3.94 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 27.99 11.81)"><foreignobject color="#000000" height="208.86" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="552.33"> <span class="ltx_inline-block ltx_minipage ltx_align_bottom" id="S7.SS2.p5.pic1.1.1.1.1.1" style="width:399.2pt;"> <span class="ltx_p" id="S7.SS2.p5.pic1.1.1.1.1.1.1"><span class="ltx_text ltx_font_bold" id="S7.SS2.p5.pic1.1.1.1.1.1.1.1">Key Takeaways:</span> If data is evenly distributed on each device, traditional and deep learning algorithms better converge with high F1 scores. We also observe that the impact of network topology on device convergence is more dominant than the aggregation strategies. In the traditional model, factors like the initiating device and data distribution both have an impact on the convergence. In particular, if devices can chronologically learn from preceding neighbours, they exhibit better performance in the linear and ring. On the contrary, stars and rings perform well if devices can concurrently train their datasets and aid others. Both traditional and deep learning models converge with high performance with evenly distributed data on each device. Specifically, deep learning models perform poorly with a high degree of non-IID data. This suggests that devices with incomplete label datasets in real-world applications should be excluded, as they negatively impact the convergence of the global model. When devices have complete label sets, star and mesh topologies perform better with uneven label distributions.</span> </span></foreignobject></g></g></svg> </div> </section> </section> <section class="ltx_section" id="S8"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">VIII </span><span class="ltx_text ltx_font_smallcaps" id="S8.1.1">Conclusion</span> </h2> <div class="ltx_para" id="S8.p1"> <p class="ltx_p" id="S8.p1.1">This work investigated the impact of network topology, non-IID data, and training strategies on the convergence of DFL. Specifically, we conducted both mathematical analysis and experimental evaluations to study these factors comprehensively. Using convex optimization, we analyzed the convergence of six different DFL deployments. We found that when data distribution across devices is IID, the difference between the ideal and actual solutions is constant. As the degree of non-IID data increases, this difference becomes larger. Next, we conducted two experiments to analyze the convergence rates of traditional, deep learning, and large language models in six different DFL deployments. The evaluation results aligned with the mathematical analysis, i.e., all models converge on each DFL deployment with IID data. However, in the case of non-IID data, the convergence rate is inversely proportional to the degree of non-IID data. In the future, we plan to leverage software-defined networking to realize agile DFL deployments.</p> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_tag_bibitem">[1]</span> <span class="ltx_bibblock"> B. McMahan, E. Moore, D. Ramage, S. Hampson, and B. 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