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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback</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/2405.19686v1/"/></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/2405.19686v1#S1" title="In Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S2" title="In Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Related Works</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S2.SS0.SSS0.Px1" title="In 2 Related Works ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title">KG-enhanced LLM</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S2.SS0.SSS0.Px2" title="In 2 Related Works ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title">LLM Personalization and Knowledge Editing</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S3" title="In Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Preliminary</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S3.SS0.SSS0.Px1" title="In 3 Preliminary ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title">Knowledge Graph (KG)</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S3.SS0.SSS0.Px2" title="In 3 Preliminary ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title">KG-enhanced LLM</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S4" title="In Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Method</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/2405.19686v1#S4.SS1" title="In 4 Method ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>Knowledge Graph Tuning</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S4.SS1.SSS0.Px1" title="In 4.1 Knowledge Graph Tuning ‣ 4 Method ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title">Example</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S4.SS2" title="In 4 Method ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>Objective Formulation</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S4.SS2.SSS0.Px1" title="In 4.2 Objective Formulation ‣ 4 Method ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title">Knowledge Retrieval</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S4.SS2.SSS0.Px2" title="In 4.2 Objective Formulation ‣ 4 Method ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title">Knowledge-enhanced Reasoning</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S4.SS3" title="In 4 Method ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3 </span>Optimization Algorithm</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S5" title="In Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Experiments</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/2405.19686v1#S5.SS1" title="In 5 Experiments ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1 </span>Experimental Setup</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S5.SS1.SSS0.Px1" title="In 5.1 Experimental Setup ‣ 5 Experiments ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title">Datasets</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S5.SS1.SSS0.Px2" title="In 5.1 Experimental Setup ‣ 5 Experiments ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title">Baselines and Configurations</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S5.SS1.SSS0.Px3" title="In 5.1 Experimental Setup ‣ 5 Experiments ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title">Metrics</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S5.SS2" title="In 5 Experiments ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.2 </span>Personalization Performance</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S5.SS3" title="In 5 Experiments ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.3 </span>Efficiency Comparison</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S5.SS4" title="In 5 Experiments ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.4 </span>Ablation Study</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/2405.19686v1#S5.SS4.SSS1" title="In 5.4 Ablation Study ‣ 5 Experiments ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.4.1 </span>Does User Need to Provide Feedback of Relations?</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S5.SS4.SSS2" title="In 5.4 Ablation Study ‣ 5 Experiments ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.4.2 </span>Effect of the Size of Query Set</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S6" title="In Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Conclusion and Limitation</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#A1" title="In Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A </span>Derivation of <math alttext="\mathcal{L}_{\text{retrieve}}" class="ltx_Math" display="inline"><semantics><msub><mi class="ltx_font_mathcaligraphic">ℒ</mi><mtext>retrieve</mtext></msub><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>ℒ</ci><ci><mtext mathsize="70%">retrieve</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex">\mathcal{L}_{\text{retrieve}}</annotation><annotation encoding="application/x-llamapun">caligraphic_L start_POSTSUBSCRIPT retrieve end_POSTSUBSCRIPT</annotation></semantics></math></span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#A2" title="In Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B </span>Details on the <span class="ltx_text ltx_font_italic">CounterFactExtension</span> Dataset</span></a> <ol class="ltx_toclist ltx_toclist_appendix"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#A2.SS1" title="In Appendix B Details on the CounterFactExtension Dataset ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B.1 </span>Dataset Samples</span></a></li> </ol> </li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <div class="ltx_para" id="p1"> <span class="ltx_ERROR undefined" id="p1.1">\newmdenv</span> <p class="ltx_p" id="p1.2">[ innerlinewidth=0.5pt, roundcorner=5pt, innerleftmargin=10pt, innerrightmargin=10pt, innertopmargin=5pt, innerbottommargin=5pt, outerlinewidth=0.5pt, linecolor=black, backgroundcolor=white ]custombox</p> </div> <h1 class="ltx_title ltx_title_document">Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Jingwei Sun, Zhixu Du<span class="ltx_note ltx_role_footnotemark" id="footnotex1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_note_type">footnotemark: </span><span class="ltx_tag ltx_tag_note">1</span></span></span></span>, Yiran Chen <br class="ltx_break"/>Department of Electrical and Computer Engineering <br class="ltx_break"/>Duke University <br class="ltx_break"/>Durham, NC 27705 <br class="ltx_break"/><span class="ltx_text ltx_font_typewriter" id="id1.1.id1">{jingwei.sun, zhixu.du, yiran.chen}@duke.edu</span> <br class="ltx_break"/> </span><span class="ltx_author_notes">equal contribution</span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id2.id1">Large language models (LLMs) have demonstrated remarkable proficiency in a range of natural language processing tasks. Once deployed, LLMs encounter users with personalized factual knowledge, and such personalized knowledge is consistently reflected through users’ interactions with the LLMs. To enhance user experience, real-time model personalization is essential, allowing LLMs to adapt user-specific knowledge based on user feedback during human-LLM interactions. Existing methods mostly require back-propagation to finetune the model parameters, which incurs high computational and memory costs. In addition, these methods suffer from low interpretability, which will cause unforeseen impacts on model performance during long-term use, where the user’s personalized knowledge is accumulated extensively. To address these challenges, we propose Knowledge Graph Tuning (KGT), a novel approach that leverages knowledge graphs (KGs) to personalize LLMs. KGT extracts personalized factual knowledge triples from users’ queries and feedback and optimizes KGs without modifying the LLM parameters. Our method improves computational and memory efficiency by avoiding back-propagation and ensures interpretability by making the KG adjustments comprehensible to humans. Experiments with state-of-the-art LLMs, including GPT-2, Llama2, and Llama3, show that KGT significantly improves personalization performance while reducing latency and GPU memory costs. Ultimately, KGT offers a promising solution of effective, efficient, and interpretable real-time LLM personalization during user interactions with the LLMs.</p> </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">Large language models (LLM) have shown increasing power on various NLP tasks <cite class="ltx_cite ltx_citemacro_citep">(Devlin et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib7" title="">2018</a>; Raffel et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib24" title="">2020</a>; Brown et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib3" title="">2020</a>; Fedus et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib10" title="">2022</a>; Zhang et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib37" title="">2021</a>; Zeng et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib35" title="">2021</a>)</cite>. The development of LLMs typically encompasses several key phases <cite class="ltx_cite ltx_citemacro_citep">(Hyperight, <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib12" title="">2024</a>)</cite>: pre-training on vast corpora, alignment to ensure adherence to ethical guidelines, and domain-specific fine-tuning. While these steps are crucial, there is often an oversight in recognizing the need for further personalization during the deployment phase. Once deployed, LLMs encounter users with personalized factual knowledge <cite class="ltx_cite ltx_citemacro_citep">(Petroni et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib22" title="">2019</a>; Jiang et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib14" title="">2020</a>; Roberts et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib25" title="">2020</a>)</cite>. These individual knowledge bases are consistently reflected through users’ feedback on various queries and the model’s responses during interactions, as shown in the interaction example in <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S1.F1" title="In 1 Introduction ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">1</span></a>. To enhance user experience, real-time model personalization is essential, allowing LLMs to adapt and incorporate user-specific knowledge based on persistent feedback throughout human-LLM interactions.</p> </div> <figure class="ltx_figure" id="S1.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="191" id="S1.F1.g1" src="x1.png" width="512"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>Pipeline of the development of an LLM. In the deployment phase, the model is personalized based on human feedback during the human-LLM interactions. The LLM in the figure is personalized to adapt to the knowledge that the user’s dog is vegetarian from the interaction. Then, in the later interactions, the LLM agent will recommend vegetarian dog food for the user given the same query.</figcaption> </figure> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">Several technologies are applicable to personalize the LLM, including Parameter Efficient Finetuning (PEFT) <cite class="ltx_cite ltx_citemacro_citep">(Ding et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib8" title="">2023</a>)</cite>, Knowledge Editing (KE) <cite class="ltx_cite ltx_citemacro_citep">(Geva et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib11" title="">2020</a>; Dai et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib5" title="">2022</a>; Meng et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib19" title="">2022</a>)</cite>, and in-context learning <cite class="ltx_cite ltx_citemacro_citep">(Brown et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib3" title="">2020</a>)</cite>. Nevertheless, these methodologies are often plagued by low efficiency, poor interpretability, or a combination of both drawbacks. PEFT and KE require back-propagation to optimize the model parameters. Back-propagation incurs unacceptable GPU memory and computational costs for the daily use of LLMs, especially for on-device applications where the onboard resources are limited. The high computational overhead also makes it difficult to realize real-time personalization. In addition to the low efficiency, these parameter-based methods also lack interpretability. Modifying the model parameters to satisfy the user’s personalized need for the current query may lead to corruption in model parameters, adversely affecting the responses to other queries. This adjustment fails to meet the long-term needs of users who rely on the accumulation of extensive personalized knowledge during interactions with large language models (LLMs). In-context learning has higher interpretability and does not need back-propagation, but its computational overhead, memory cost, and response latency increase drastically with the length of the reference context. Therefore, it is neither efficient nor scalable for long-term use by users.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">Knowledge graphs (KGs) <cite class="ltx_cite ltx_citemacro_citep">(Speer et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib29" title="">2017</a>; Sap et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib27" title="">2019</a>)</cite>, which store factual knowledge in a structured format, are increasingly integrated into LLMs to enhance the inference with external knowledge <cite class="ltx_cite ltx_citemacro_citep">(Chang et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib4" title="">2021</a>; Xu et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib33" title="">2021</a>; Yao et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib34" title="">2022</a>; Song et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib28" title="">2021</a>)</cite> and are becoming a standard component of LLM systems. In this paper, we propose a new paradigm of model personalization by tuning the KGs based on user feedback. Our proposed method, Knowledge Graph Tuning (KGT), extracts the personalized knowledge triples from the query and the user’s feedback. Instead of optimizing the LLM parameters, KGT optimizes the KGs based on the extracted personalized factual knowledge. We formulate the training objective based on the <span class="ltx_text ltx_font_italic" id="S1.p3.1.1">evidence lower bound (ELBO)</span> and derive an optimization goal of achieving a <span class="ltx_text ltx_font_italic" id="S1.p3.1.2">high personalized knowledge retrieval probability</span> and a <span class="ltx_text ltx_font_italic" id="S1.p3.1.3">high knowledge-enhanced reasoning probability</span>. We propose a heuristic optimization algorithm to finetune the KG by adding and removing knowledge triples without touching the model parameters, significantly improving computational and memory efficiency. Additionally, the added and removed knowledge triples are comprehensible to humans, ensuring the method’s interpretability.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.2">We conduct experiments on multiple datasets using SOTA pre-trained LLMs. The results demonstrate that KGT significantly improves personalization performance compared with baselines while reducing the latency and GPU memory cost by up to <math alttext="84\%" class="ltx_Math" display="inline" id="S1.p4.1.m1.1"><semantics id="S1.p4.1.m1.1a"><mrow id="S1.p4.1.m1.1.1" xref="S1.p4.1.m1.1.1.cmml"><mn id="S1.p4.1.m1.1.1.2" xref="S1.p4.1.m1.1.1.2.cmml">84</mn><mo id="S1.p4.1.m1.1.1.1" xref="S1.p4.1.m1.1.1.1.cmml">%</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.1.m1.1b"><apply id="S1.p4.1.m1.1.1.cmml" xref="S1.p4.1.m1.1.1"><csymbol cd="latexml" id="S1.p4.1.m1.1.1.1.cmml" xref="S1.p4.1.m1.1.1.1">percent</csymbol><cn id="S1.p4.1.m1.1.1.2.cmml" type="integer" xref="S1.p4.1.m1.1.1.2">84</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.1.m1.1c">84\%</annotation><annotation encoding="application/x-llamapun" id="S1.p4.1.m1.1d">84 %</annotation></semantics></math> and <math alttext="77\%" class="ltx_Math" display="inline" id="S1.p4.2.m2.1"><semantics id="S1.p4.2.m2.1a"><mrow id="S1.p4.2.m2.1.1" xref="S1.p4.2.m2.1.1.cmml"><mn id="S1.p4.2.m2.1.1.2" xref="S1.p4.2.m2.1.1.2.cmml">77</mn><mo id="S1.p4.2.m2.1.1.1" xref="S1.p4.2.m2.1.1.1.cmml">%</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.2.m2.1b"><apply id="S1.p4.2.m2.1.1.cmml" xref="S1.p4.2.m2.1.1"><csymbol cd="latexml" id="S1.p4.2.m2.1.1.1.cmml" xref="S1.p4.2.m2.1.1.1">percent</csymbol><cn id="S1.p4.2.m2.1.1.2.cmml" type="integer" xref="S1.p4.2.m2.1.1.2">77</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.2.m2.1c">77\%</annotation><annotation encoding="application/x-llamapun" id="S1.p4.2.m2.1d">77 %</annotation></semantics></math>, respectively. By proposing KGT, we offer an effective, efficient, and interpretable solution for realizing real-time LLM personalization during human-LLM interactions. We summarize our contributions as follows:</p> </div> <div class="ltx_para" id="S1.p5"> <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">We present the necessity of model personalization during the human-LLM interaction and the challenges of adopting existing technologies: low efficiency and poor interpretability.</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 propose a method, KGT, that realizes real-time LLM personalization based on user feedback by optimizing the knowledge graph rather than the model parameters.</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">We conduct experiments on multiple datasets with SOTA pre-trained LLMs. Compared with baselines, KGT achieves significantly better personalization performance while reducing computational and memory costs. KGT demonstrates considerable scalability as the volume of queries increases, a critical attribute for fulfilling the long-term needs of users who require accumulating extensive personalized knowledge during their interactions with LLMs.</p> </div> </li> </ul> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Related Works</h2> <section class="ltx_paragraph" id="S2.SS0.SSS0.Px1"> <h5 class="ltx_title ltx_title_paragraph">KG-enhanced LLM</h5> <div class="ltx_para" id="S2.SS0.SSS0.Px1.p1"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.p1.1">Large language models (LLMs) are acclaimed for their proficiency in assimilating knowledge from expansive corpora and achieving groundbreaking results in numerous natural language processing (NLP) tasks. Nonetheless, these models are frequently critiqued for the hallucination issues <cite class="ltx_cite ltx_citemacro_citep">(Lee et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib16" title="">2018</a>; Bang et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib2" title="">2023</a>)</cite> and their lack of interpretability. In response to these shortcomings, the integration of knowledge graphs (KGs) with LLMs has been proposed <cite class="ltx_cite ltx_citemacro_citep">(Chang et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib4" title="">2021</a>; Xu et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib33" title="">2021</a>; Yao et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib34" title="">2022</a>; Song et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib28" title="">2021</a>)</cite>. KGs, by storing extensive knowledge in a structured and explicit format, enhance the knowledge capacity of LLMs. Some approaches <cite class="ltx_cite ltx_citemacro_citep">(Zhang et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib36" title="">2019</a>; Rosset et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib26" title="">2020</a>)</cite> advocate for incorporating KGs during the <span class="ltx_text ltx_font_bold" id="S2.SS0.SSS0.Px1.p1.1.1">pre-training phase</span> of LLMs, enabling direct knowledge acquisition from these graphs. Others <cite class="ltx_cite ltx_citemacro_citep">(Chang et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib4" title="">2021</a>; Xu et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib33" title="">2021</a>)</cite> suggest the use of KGs during the <span class="ltx_text ltx_font_bold" id="S2.SS0.SSS0.Px1.p1.1.2">inference phase</span> to improve access to domain-specific knowledge, thereby substantially enhancing the models’ performance. Furthermore, to advance the interpretability of LLMs, researchers are utilizing KGs to elucidate both the factual knowledge <cite class="ltx_cite ltx_citemacro_citep">(Petroni et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib22" title="">2019</a>)</cite> and the reasoning processes <cite class="ltx_cite ltx_citemacro_citep">(Lin et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib18" title="">2019</a>)</cite> of these models. In this paper, we focus on the setting in which KG enhances the inference phase of an LLM.</p> </div> </section> <section class="ltx_paragraph" id="S2.SS0.SSS0.Px2"> <h5 class="ltx_title ltx_title_paragraph">LLM Personalization and Knowledge Editing</h5> <div class="ltx_para" id="S2.SS0.SSS0.Px2.p1"> <p class="ltx_p" id="S2.SS0.SSS0.Px2.p1.1">Recently, Knowledge Editing (KE) approaches have been proposed to personalize LLMs by localizing and modifying the factual knowledge within transformers. <cite class="ltx_cite ltx_citemacro_citet">Geva et al. (<a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib11" title="">2020</a>)</cite> suggests that the MLP layers within a transformer for masked language models serve as memory units for entities and their associated information. Extending this idea, KN <cite class="ltx_cite ltx_citemacro_citep">(Dai et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib5" title="">2022</a>)</cite> developed a technique for updating facts in BERT by manipulating specific rows of the MLP matrix with the embedding of the target object. They pinpoint crucial neurons for storing knowledge through gradient-based methods. <cite class="ltx_cite ltx_citemacro_citet">De Cao et al. (<a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib6" title="">2021</a>)</cite> implement a hyper-network that predicts necessary weight adjustments during test time to modify facts. Their experiments involve BERT and BART <cite class="ltx_cite ltx_citemacro_citep">(Lewis et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib17" title="">2019</a>)</cite>, particularly in models tailored for question-answering tasks. <cite class="ltx_cite ltx_citemacro_citet">Mitchell et al. (<a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib20" title="">2022</a>)</cite> introduce a hyper-network approach that adjusts gradient decomposition terms to efficiently update knowledge, showcasing scalability across large models like T5 <cite class="ltx_cite ltx_citemacro_citep">(Raffel et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib24" title="">2020</a>)</cite> and GPT-J <cite class="ltx_cite ltx_citemacro_citep">(Wang and Komatsuzaki, <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib32" title="">2022</a>)</cite>. The ROME method <cite class="ltx_cite ltx_citemacro_citep">(Meng et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib19" title="">2022</a>)</cite> employed causal tracing to pinpoint knowledge-relevant layers and then edit its FFN module, achieving superior outcomes. All these techniques employ gradient-based optimization methods, which are inefficient and unsuitable for achieving real-time personalization under constrained computational resources. Moreover, modifying model parameters lacks interpretability and may lead to unforeseen impacts on model performance. In contrast, our method solely necessitates model inference, which is significantly more efficient. By adjusting the knowledge graph (KG), our approach ensures that the personalization process is interpretable and the modifications in model performance are comprehensible.</p> </div> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Preliminary</h2> <section class="ltx_paragraph" id="S3.SS0.SSS0.Px1"> <h5 class="ltx_title ltx_title_paragraph">Knowledge Graph (KG)</h5> <div class="ltx_para" id="S3.SS0.SSS0.Px1.p1"> <p class="ltx_p" id="S3.SS0.SSS0.Px1.p1.5">A knowledge graph <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px1.p1.1.m1.1"><semantics id="S3.SS0.SSS0.Px1.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS0.SSS0.Px1.p1.1.m1.1.1" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p1.1.m1.1b"><ci id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.cmml" 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xref="S3.SS0.SSS0.Px1.p1.2.m2.5.5.2.2.2.2.2.3">ℛ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p1.2.m2.5c">\mathcal{G}=\{(e,r,e^{\prime})|e,e^{\prime}\in\mathcal{E},r\in\mathcal{R}\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px1.p1.2.m2.5d">caligraphic_G = { ( italic_e , italic_r , italic_e start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) | italic_e , italic_e start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_E , italic_r ∈ caligraphic_R }</annotation></semantics></math>, where <math alttext="\mathcal{E}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px1.p1.3.m3.1"><semantics id="S3.SS0.SSS0.Px1.p1.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS0.SSS0.Px1.p1.3.m3.1.1" xref="S3.SS0.SSS0.Px1.p1.3.m3.1.1.cmml">ℰ</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p1.3.m3.1b"><ci id="S3.SS0.SSS0.Px1.p1.3.m3.1.1.cmml" 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<math alttext="(e,r,e^{\prime})" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px1.p1.5.m5.3"><semantics id="S3.SS0.SSS0.Px1.p1.5.m5.3a"><mrow id="S3.SS0.SSS0.Px1.p1.5.m5.3.3.1" xref="S3.SS0.SSS0.Px1.p1.5.m5.3.3.2.cmml"><mo id="S3.SS0.SSS0.Px1.p1.5.m5.3.3.1.2" stretchy="false" xref="S3.SS0.SSS0.Px1.p1.5.m5.3.3.2.cmml">(</mo><mi id="S3.SS0.SSS0.Px1.p1.5.m5.1.1" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.cmml">e</mi><mo id="S3.SS0.SSS0.Px1.p1.5.m5.3.3.1.3" xref="S3.SS0.SSS0.Px1.p1.5.m5.3.3.2.cmml">,</mo><mi id="S3.SS0.SSS0.Px1.p1.5.m5.2.2" xref="S3.SS0.SSS0.Px1.p1.5.m5.2.2.cmml">r</mi><mo id="S3.SS0.SSS0.Px1.p1.5.m5.3.3.1.4" xref="S3.SS0.SSS0.Px1.p1.5.m5.3.3.2.cmml">,</mo><msup id="S3.SS0.SSS0.Px1.p1.5.m5.3.3.1.1" xref="S3.SS0.SSS0.Px1.p1.5.m5.3.3.1.1.cmml"><mi id="S3.SS0.SSS0.Px1.p1.5.m5.3.3.1.1.2" xref="S3.SS0.SSS0.Px1.p1.5.m5.3.3.1.1.2.cmml">e</mi><mo id="S3.SS0.SSS0.Px1.p1.5.m5.3.3.1.1.3" xref="S3.SS0.SSS0.Px1.p1.5.m5.3.3.1.1.3.cmml">′</mo></msup><mo id="S3.SS0.SSS0.Px1.p1.5.m5.3.3.1.5" stretchy="false" xref="S3.SS0.SSS0.Px1.p1.5.m5.3.3.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p1.5.m5.3b"><vector id="S3.SS0.SSS0.Px1.p1.5.m5.3.3.2.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.3.3.1"><ci id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1">𝑒</ci><ci id="S3.SS0.SSS0.Px1.p1.5.m5.2.2.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.2.2">𝑟</ci><apply id="S3.SS0.SSS0.Px1.p1.5.m5.3.3.1.1.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.3.3.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p1.5.m5.3.3.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.3.3.1.1">superscript</csymbol><ci id="S3.SS0.SSS0.Px1.p1.5.m5.3.3.1.1.2.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.3.3.1.1.2">𝑒</ci><ci id="S3.SS0.SSS0.Px1.p1.5.m5.3.3.1.1.3.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.3.3.1.1.3">′</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p1.5.m5.3c">(e,r,e^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px1.p1.5.m5.3d">( italic_e , italic_r , italic_e start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> is referred to as a knowledge triple.</p> </div> </section> <section class="ltx_paragraph" id="S3.SS0.SSS0.Px2"> <h5 class="ltx_title ltx_title_paragraph">KG-enhanced LLM</h5> <div class="ltx_para" id="S3.SS0.SSS0.Px2.p1"> <p class="ltx_p" id="S3.SS0.SSS0.Px2.p1.8">A KG-enhanced LLM <math alttext="f_{\theta,\mathcal{G}}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p1.1.m1.2"><semantics id="S3.SS0.SSS0.Px2.p1.1.m1.2a"><msub id="S3.SS0.SSS0.Px2.p1.1.m1.2.3" xref="S3.SS0.SSS0.Px2.p1.1.m1.2.3.cmml"><mi id="S3.SS0.SSS0.Px2.p1.1.m1.2.3.2" xref="S3.SS0.SSS0.Px2.p1.1.m1.2.3.2.cmml">f</mi><mrow id="S3.SS0.SSS0.Px2.p1.1.m1.2.2.2.4" xref="S3.SS0.SSS0.Px2.p1.1.m1.2.2.2.3.cmml"><mi id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.cmml">θ</mi><mo id="S3.SS0.SSS0.Px2.p1.1.m1.2.2.2.4.1" xref="S3.SS0.SSS0.Px2.p1.1.m1.2.2.2.3.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS0.SSS0.Px2.p1.1.m1.2.2.2.2" xref="S3.SS0.SSS0.Px2.p1.1.m1.2.2.2.2.cmml">𝒢</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p1.1.m1.2b"><apply id="S3.SS0.SSS0.Px2.p1.1.m1.2.3.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.2.3"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p1.1.m1.2.3.1.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.2.3">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p1.1.m1.2.3.2.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.2.3.2">𝑓</ci><list id="S3.SS0.SSS0.Px2.p1.1.m1.2.2.2.3.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.2.2.2.4"><ci id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1">𝜃</ci><ci id="S3.SS0.SSS0.Px2.p1.1.m1.2.2.2.2.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.2.2.2.2">𝒢</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p1.1.m1.2c">f_{\theta,\mathcal{G}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p1.1.m1.2d">italic_f start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT</annotation></semantics></math> is parameterized by both the LLM parameters <math alttext="\theta" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p1.2.m2.1"><semantics id="S3.SS0.SSS0.Px2.p1.2.m2.1a"><mi id="S3.SS0.SSS0.Px2.p1.2.m2.1.1" xref="S3.SS0.SSS0.Px2.p1.2.m2.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p1.2.m2.1b"><ci id="S3.SS0.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.2.m2.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p1.2.m2.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p1.2.m2.1d">italic_θ</annotation></semantics></math> and a KG <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p1.3.m3.1"><semantics id="S3.SS0.SSS0.Px2.p1.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS0.SSS0.Px2.p1.3.m3.1.1" xref="S3.SS0.SSS0.Px2.p1.3.m3.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p1.3.m3.1b"><ci id="S3.SS0.SSS0.Px2.p1.3.m3.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.3.m3.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p1.3.m3.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p1.3.m3.1d">caligraphic_G</annotation></semantics></math>. Given a natural language query <math alttext="q" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p1.4.m4.1"><semantics id="S3.SS0.SSS0.Px2.p1.4.m4.1a"><mi id="S3.SS0.SSS0.Px2.p1.4.m4.1.1" xref="S3.SS0.SSS0.Px2.p1.4.m4.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p1.4.m4.1b"><ci id="S3.SS0.SSS0.Px2.p1.4.m4.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.4.m4.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p1.4.m4.1c">q</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p1.4.m4.1d">italic_q</annotation></semantics></math>, the KG-enhanced LLM generates an answer <math alttext="f_{\theta,\mathcal{G}}(q)" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p1.5.m5.3"><semantics id="S3.SS0.SSS0.Px2.p1.5.m5.3a"><mrow id="S3.SS0.SSS0.Px2.p1.5.m5.3.4" xref="S3.SS0.SSS0.Px2.p1.5.m5.3.4.cmml"><msub id="S3.SS0.SSS0.Px2.p1.5.m5.3.4.2" xref="S3.SS0.SSS0.Px2.p1.5.m5.3.4.2.cmml"><mi id="S3.SS0.SSS0.Px2.p1.5.m5.3.4.2.2" xref="S3.SS0.SSS0.Px2.p1.5.m5.3.4.2.2.cmml">f</mi><mrow id="S3.SS0.SSS0.Px2.p1.5.m5.2.2.2.4" xref="S3.SS0.SSS0.Px2.p1.5.m5.2.2.2.3.cmml"><mi id="S3.SS0.SSS0.Px2.p1.5.m5.1.1.1.1" xref="S3.SS0.SSS0.Px2.p1.5.m5.1.1.1.1.cmml">θ</mi><mo id="S3.SS0.SSS0.Px2.p1.5.m5.2.2.2.4.1" xref="S3.SS0.SSS0.Px2.p1.5.m5.2.2.2.3.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS0.SSS0.Px2.p1.5.m5.2.2.2.2" xref="S3.SS0.SSS0.Px2.p1.5.m5.2.2.2.2.cmml">𝒢</mi></mrow></msub><mo id="S3.SS0.SSS0.Px2.p1.5.m5.3.4.1" xref="S3.SS0.SSS0.Px2.p1.5.m5.3.4.1.cmml">⁢</mo><mrow id="S3.SS0.SSS0.Px2.p1.5.m5.3.4.3.2" xref="S3.SS0.SSS0.Px2.p1.5.m5.3.4.cmml"><mo id="S3.SS0.SSS0.Px2.p1.5.m5.3.4.3.2.1" stretchy="false" xref="S3.SS0.SSS0.Px2.p1.5.m5.3.4.cmml">(</mo><mi id="S3.SS0.SSS0.Px2.p1.5.m5.3.3" xref="S3.SS0.SSS0.Px2.p1.5.m5.3.3.cmml">q</mi><mo id="S3.SS0.SSS0.Px2.p1.5.m5.3.4.3.2.2" stretchy="false" xref="S3.SS0.SSS0.Px2.p1.5.m5.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p1.5.m5.3b"><apply id="S3.SS0.SSS0.Px2.p1.5.m5.3.4.cmml" xref="S3.SS0.SSS0.Px2.p1.5.m5.3.4"><times id="S3.SS0.SSS0.Px2.p1.5.m5.3.4.1.cmml" xref="S3.SS0.SSS0.Px2.p1.5.m5.3.4.1"></times><apply id="S3.SS0.SSS0.Px2.p1.5.m5.3.4.2.cmml" xref="S3.SS0.SSS0.Px2.p1.5.m5.3.4.2"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p1.5.m5.3.4.2.1.cmml" xref="S3.SS0.SSS0.Px2.p1.5.m5.3.4.2">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p1.5.m5.3.4.2.2.cmml" xref="S3.SS0.SSS0.Px2.p1.5.m5.3.4.2.2">𝑓</ci><list id="S3.SS0.SSS0.Px2.p1.5.m5.2.2.2.3.cmml" xref="S3.SS0.SSS0.Px2.p1.5.m5.2.2.2.4"><ci id="S3.SS0.SSS0.Px2.p1.5.m5.1.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.5.m5.1.1.1.1">𝜃</ci><ci id="S3.SS0.SSS0.Px2.p1.5.m5.2.2.2.2.cmml" xref="S3.SS0.SSS0.Px2.p1.5.m5.2.2.2.2">𝒢</ci></list></apply><ci id="S3.SS0.SSS0.Px2.p1.5.m5.3.3.cmml" xref="S3.SS0.SSS0.Px2.p1.5.m5.3.3">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p1.5.m5.3c">f_{\theta,\mathcal{G}}(q)</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p1.5.m5.3d">italic_f start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_q )</annotation></semantics></math> based on both the model parameters and the retrieved knowledge triples from <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p1.6.m6.1"><semantics id="S3.SS0.SSS0.Px2.p1.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS0.SSS0.Px2.p1.6.m6.1.1" xref="S3.SS0.SSS0.Px2.p1.6.m6.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p1.6.m6.1b"><ci id="S3.SS0.SSS0.Px2.p1.6.m6.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.6.m6.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p1.6.m6.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p1.6.m6.1d">caligraphic_G</annotation></semantics></math>. We follow previous works <cite class="ltx_cite ltx_citemacro_citep">(Sun et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib30" title="">2019</a>; Jiang et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib13" title="">2022</a>)</cite> and assume that the entities <math alttext="e_{q}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p1.7.m7.1"><semantics id="S3.SS0.SSS0.Px2.p1.7.m7.1a"><msub id="S3.SS0.SSS0.Px2.p1.7.m7.1.1" xref="S3.SS0.SSS0.Px2.p1.7.m7.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p1.7.m7.1.1.2" xref="S3.SS0.SSS0.Px2.p1.7.m7.1.1.2.cmml">e</mi><mi id="S3.SS0.SSS0.Px2.p1.7.m7.1.1.3" xref="S3.SS0.SSS0.Px2.p1.7.m7.1.1.3.cmml">q</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p1.7.m7.1b"><apply id="S3.SS0.SSS0.Px2.p1.7.m7.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p1.7.m7.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.7.m7.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p1.7.m7.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p1.7.m7.1.1.2">𝑒</ci><ci id="S3.SS0.SSS0.Px2.p1.7.m7.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p1.7.m7.1.1.3">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p1.7.m7.1c">e_{q}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p1.7.m7.1d">italic_e start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT</annotation></semantics></math> in the query and <math alttext="e_{a}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p1.8.m8.1"><semantics id="S3.SS0.SSS0.Px2.p1.8.m8.1a"><msub id="S3.SS0.SSS0.Px2.p1.8.m8.1.1" xref="S3.SS0.SSS0.Px2.p1.8.m8.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p1.8.m8.1.1.2" xref="S3.SS0.SSS0.Px2.p1.8.m8.1.1.2.cmml">e</mi><mi id="S3.SS0.SSS0.Px2.p1.8.m8.1.1.3" xref="S3.SS0.SSS0.Px2.p1.8.m8.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p1.8.m8.1b"><apply id="S3.SS0.SSS0.Px2.p1.8.m8.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p1.8.m8.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.8.m8.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p1.8.m8.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p1.8.m8.1.1.2">𝑒</ci><ci id="S3.SS0.SSS0.Px2.p1.8.m8.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p1.8.m8.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p1.8.m8.1c">e_{a}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p1.8.m8.1d">italic_e start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> in the answer are labeled. In this paper, we focus on the setting where the LLM retrieves one-depth triples.</p> </div> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>Method</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">This section introduces our method, KGT (Knowledge Graph Tuning), which enables real-time LLM personalization based on user feedback. Instead of finetuning the model parameters, we edit the user’s <span class="ltx_text ltx_font_bold" id="S4.p1.1.1">personalized knowledge graph</span>, which provides the user’s personalized factual knowledge to enhance the customization capability of the LLM. KGT does not need to conduct back-propagation of the LLM, and only the inference is required, which significantly reduces the latency and computational cost. The edited knowledge triples are comprehensible, ensuring the method’s interpretability. The overview of KGT is shown in <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S4.F2" title="In 4 Method ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">2</span></a>.</p> </div> <figure class="ltx_figure" id="S4.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="210" id="S4.F2.g1" src="x2.png" width="539"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>The overview of KGT. The LLM extracts the posterior distribution of the personalized knowledge triples <math alttext="Q(z|q,a)" class="ltx_Math" display="inline" id="S4.F2.2.m1.3"><semantics id="S4.F2.2.m1.3b"><mrow id="S4.F2.2.m1.3.3" xref="S4.F2.2.m1.3.3.cmml"><mi id="S4.F2.2.m1.3.3.3" xref="S4.F2.2.m1.3.3.3.cmml">Q</mi><mo id="S4.F2.2.m1.3.3.2" xref="S4.F2.2.m1.3.3.2.cmml">⁢</mo><mrow id="S4.F2.2.m1.3.3.1.1" xref="S4.F2.2.m1.3.3.1.1.1.cmml"><mo id="S4.F2.2.m1.3.3.1.1.2" stretchy="false" xref="S4.F2.2.m1.3.3.1.1.1.cmml">(</mo><mrow id="S4.F2.2.m1.3.3.1.1.1" xref="S4.F2.2.m1.3.3.1.1.1.cmml"><mi id="S4.F2.2.m1.3.3.1.1.1.2" xref="S4.F2.2.m1.3.3.1.1.1.2.cmml">z</mi><mo fence="false" id="S4.F2.2.m1.3.3.1.1.1.1" xref="S4.F2.2.m1.3.3.1.1.1.1.cmml">|</mo><mrow id="S4.F2.2.m1.3.3.1.1.1.3.2" xref="S4.F2.2.m1.3.3.1.1.1.3.1.cmml"><mi id="S4.F2.2.m1.1.1" xref="S4.F2.2.m1.1.1.cmml">q</mi><mo id="S4.F2.2.m1.3.3.1.1.1.3.2.1" xref="S4.F2.2.m1.3.3.1.1.1.3.1.cmml">,</mo><mi id="S4.F2.2.m1.2.2" xref="S4.F2.2.m1.2.2.cmml">a</mi></mrow></mrow><mo id="S4.F2.2.m1.3.3.1.1.3" stretchy="false" xref="S4.F2.2.m1.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F2.2.m1.3c"><apply id="S4.F2.2.m1.3.3.cmml" xref="S4.F2.2.m1.3.3"><times id="S4.F2.2.m1.3.3.2.cmml" xref="S4.F2.2.m1.3.3.2"></times><ci id="S4.F2.2.m1.3.3.3.cmml" xref="S4.F2.2.m1.3.3.3">𝑄</ci><apply id="S4.F2.2.m1.3.3.1.1.1.cmml" xref="S4.F2.2.m1.3.3.1.1"><csymbol cd="latexml" id="S4.F2.2.m1.3.3.1.1.1.1.cmml" xref="S4.F2.2.m1.3.3.1.1.1.1">conditional</csymbol><ci id="S4.F2.2.m1.3.3.1.1.1.2.cmml" xref="S4.F2.2.m1.3.3.1.1.1.2">𝑧</ci><list id="S4.F2.2.m1.3.3.1.1.1.3.1.cmml" xref="S4.F2.2.m1.3.3.1.1.1.3.2"><ci id="S4.F2.2.m1.1.1.cmml" xref="S4.F2.2.m1.1.1">𝑞</ci><ci id="S4.F2.2.m1.2.2.cmml" xref="S4.F2.2.m1.2.2">𝑎</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F2.2.m1.3d">Q(z|q,a)</annotation><annotation encoding="application/x-llamapun" id="S4.F2.2.m1.3e">italic_Q ( italic_z | italic_q , italic_a )</annotation></semantics></math> from the human-LLM interaction. The personalized triples are utilized to optimize the KG to achieve two goals: The model can (1) retrieve the personalized triples with high probability and (2) generate the user’s feedback with the retrieved triples in high confidence.</figcaption> </figure> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1 </span>Knowledge Graph Tuning</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1">Recently, many techniques have been developed to efficiently finetune or personalize the pre-trained LLM parameters based on the user’s data <cite class="ltx_cite ltx_citemacro_citep">(Ding et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib8" title="">2023</a>; Geva et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib11" title="">2020</a>; Brown et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib3" title="">2020</a>)</cite>. Instead of finetuning the model parameters, we modify the user’s KG to inject personalized knowledge into the LLM.</p> </div> <div class="ltx_para" id="S4.SS1.p2"> <span class="ltx_ERROR undefined" id="S4.SS1.p2.1">{custombox}</span> </div> <section class="ltx_paragraph" id="S4.SS1.SSS0.Px1"> <h5 class="ltx_title ltx_title_paragraph">Example</h5> <div class="ltx_para" id="S4.SS1.SSS0.Px1.p1"> <p class="ltx_p" id="S4.SS1.SSS0.Px1.p1.3">A user wants to let the LLM-based assistant order some food for her vegetarian dog. 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xref="S4.SS1.SSS0.Px1.p1.1.m1.3.3.3.3.5">𝑡</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p1.1.m1.3c">\left(Dog,Enjoy,Meat\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.1.m1.3d">( italic_D italic_o italic_g , italic_E italic_n italic_j italic_o italic_y , italic_M italic_e italic_a italic_t )</annotation></semantics></math> and add a triple <math alttext="\left(Dog,Enjoy,Vegetable\right)" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p1.2.m2.3"><semantics id="S4.SS1.SSS0.Px1.p1.2.m2.3a"><mrow id="S4.SS1.SSS0.Px1.p1.2.m2.3.3.3" xref="S4.SS1.SSS0.Px1.p1.2.m2.3.3.4.cmml"><mo id="S4.SS1.SSS0.Px1.p1.2.m2.3.3.3.4" xref="S4.SS1.SSS0.Px1.p1.2.m2.3.3.4.cmml">(</mo><mrow id="S4.SS1.SSS0.Px1.p1.2.m2.1.1.1.1" xref="S4.SS1.SSS0.Px1.p1.2.m2.1.1.1.1.cmml"><mi id="S4.SS1.SSS0.Px1.p1.2.m2.1.1.1.1.2" xref="S4.SS1.SSS0.Px1.p1.2.m2.1.1.1.1.2.cmml">D</mi><mo id="S4.SS1.SSS0.Px1.p1.2.m2.1.1.1.1.1" 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xref="S4.SS1.SSS0.Px1.p1.2.m2.3.3.3.3.3">𝑒</ci><ci id="S4.SS1.SSS0.Px1.p1.2.m2.3.3.3.3.4.cmml" xref="S4.SS1.SSS0.Px1.p1.2.m2.3.3.3.3.4">𝑔</ci><ci id="S4.SS1.SSS0.Px1.p1.2.m2.3.3.3.3.5.cmml" xref="S4.SS1.SSS0.Px1.p1.2.m2.3.3.3.3.5">𝑒</ci><ci id="S4.SS1.SSS0.Px1.p1.2.m2.3.3.3.3.6.cmml" xref="S4.SS1.SSS0.Px1.p1.2.m2.3.3.3.3.6">𝑡</ci><ci id="S4.SS1.SSS0.Px1.p1.2.m2.3.3.3.3.7.cmml" xref="S4.SS1.SSS0.Px1.p1.2.m2.3.3.3.3.7">𝑎</ci><ci id="S4.SS1.SSS0.Px1.p1.2.m2.3.3.3.3.8.cmml" xref="S4.SS1.SSS0.Px1.p1.2.m2.3.3.3.3.8">𝑏</ci><ci id="S4.SS1.SSS0.Px1.p1.2.m2.3.3.3.3.9.cmml" xref="S4.SS1.SSS0.Px1.p1.2.m2.3.3.3.3.9">𝑙</ci><ci id="S4.SS1.SSS0.Px1.p1.2.m2.3.3.3.3.10.cmml" xref="S4.SS1.SSS0.Px1.p1.2.m2.3.3.3.3.10">𝑒</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p1.2.m2.3c">\left(Dog,Enjoy,Vegetable\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.2.m2.3d">( italic_D italic_o italic_g , italic_E italic_n italic_j italic_o italic_y , italic_V italic_e italic_g italic_e italic_t italic_a italic_b italic_l italic_e )</annotation></semantics></math> in his/her personal knowledge graph. 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xref="S4.SS1.SSS0.Px1.p1.3.m3.3.3.3.3.6">𝑡</ci><ci id="S4.SS1.SSS0.Px1.p1.3.m3.3.3.3.3.7.cmml" xref="S4.SS1.SSS0.Px1.p1.3.m3.3.3.3.3.7">𝑎</ci><ci id="S4.SS1.SSS0.Px1.p1.3.m3.3.3.3.3.8.cmml" xref="S4.SS1.SSS0.Px1.p1.3.m3.3.3.3.3.8">𝑏</ci><ci id="S4.SS1.SSS0.Px1.p1.3.m3.3.3.3.3.9.cmml" xref="S4.SS1.SSS0.Px1.p1.3.m3.3.3.3.3.9">𝑙</ci><ci id="S4.SS1.SSS0.Px1.p1.3.m3.3.3.3.3.10.cmml" xref="S4.SS1.SSS0.Px1.p1.3.m3.3.3.3.3.10">𝑒</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p1.3.m3.3c">\left(Dog,Enjoy,Vegetable\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p1.3.m3.3d">( italic_D italic_o italic_g , italic_E italic_n italic_j italic_o italic_y , italic_V italic_e italic_g italic_e italic_t italic_a italic_b italic_l italic_e )</annotation></semantics></math> based on the user’s query, then recommend vegetarian dog food for the user based on the query and the retrieved personalized knowledge triple.</p> </div> <div class="ltx_para" id="S4.SS1.SSS0.Px1.p2"> <p class="ltx_p" id="S4.SS1.SSS0.Px1.p2.1">From the example and previous works <cite class="ltx_cite ltx_citemacro_citep">(Xu et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib33" title="">2021</a>; Chang et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib4" title="">2021</a>; Dai et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib5" title="">2022</a>)</cite>, we conclude that there are two steps in KG-enhanced LLM reasoning: 1. knowledge retrieval and 2. knowledge-enhanced reasoning. Following this insight, we formulate the KG-enhanced LLM reasoning probability by marginalizing the knowledge triple distribution. Then, our KGT is an optimization problem that aims to maximize the probability of reasoning the answer from a KG-enhanced LLM:</p> <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="P_{\theta,\mathcal{G}}(a|q)=\sum_{z\in\mathcal{G}}P_{\theta,\mathcal{G}}(a|q,z% )P_{\theta,\mathcal{G}}(z|q)," class="ltx_Math" display="block" id="S4.E1.m1.9"><semantics id="S4.E1.m1.9a"><mrow id="S4.E1.m1.9.9.1" xref="S4.E1.m1.9.9.1.1.cmml"><mrow id="S4.E1.m1.9.9.1.1" xref="S4.E1.m1.9.9.1.1.cmml"><mrow id="S4.E1.m1.9.9.1.1.1" xref="S4.E1.m1.9.9.1.1.1.cmml"><msub id="S4.E1.m1.9.9.1.1.1.3" xref="S4.E1.m1.9.9.1.1.1.3.cmml"><mi id="S4.E1.m1.9.9.1.1.1.3.2" mathsize="90%" xref="S4.E1.m1.9.9.1.1.1.3.2.cmml">P</mi><mrow id="S4.E1.m1.2.2.2.4" xref="S4.E1.m1.2.2.2.3.cmml"><mi id="S4.E1.m1.1.1.1.1" mathsize="90%" 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italic_a | italic_q , italic_z ) italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_z | italic_q ) ,</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> </div> <div class="ltx_para" id="S4.SS1.SSS0.Px1.p3"> <p class="ltx_p" id="S4.SS1.SSS0.Px1.p3.9">where <math alttext="P_{\theta,\mathcal{G}}(z|q)" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p3.1.m1.3"><semantics id="S4.SS1.SSS0.Px1.p3.1.m1.3a"><mrow id="S4.SS1.SSS0.Px1.p3.1.m1.3.3" xref="S4.SS1.SSS0.Px1.p3.1.m1.3.3.cmml"><msub id="S4.SS1.SSS0.Px1.p3.1.m1.3.3.3" xref="S4.SS1.SSS0.Px1.p3.1.m1.3.3.3.cmml"><mi id="S4.SS1.SSS0.Px1.p3.1.m1.3.3.3.2" xref="S4.SS1.SSS0.Px1.p3.1.m1.3.3.3.2.cmml">P</mi><mrow id="S4.SS1.SSS0.Px1.p3.1.m1.2.2.2.4" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.2.2.3.cmml"><mi id="S4.SS1.SSS0.Px1.p3.1.m1.1.1.1.1" xref="S4.SS1.SSS0.Px1.p3.1.m1.1.1.1.1.cmml">θ</mi><mo id="S4.SS1.SSS0.Px1.p3.1.m1.2.2.2.4.1" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.2.2.3.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.SSS0.Px1.p3.1.m1.2.2.2.2" xref="S4.SS1.SSS0.Px1.p3.1.m1.2.2.2.2.cmml">𝒢</mi></mrow></msub><mo id="S4.SS1.SSS0.Px1.p3.1.m1.3.3.2" xref="S4.SS1.SSS0.Px1.p3.1.m1.3.3.2.cmml">⁢</mo><mrow id="S4.SS1.SSS0.Px1.p3.1.m1.3.3.1.1" xref="S4.SS1.SSS0.Px1.p3.1.m1.3.3.1.1.1.cmml"><mo id="S4.SS1.SSS0.Px1.p3.1.m1.3.3.1.1.2" stretchy="false" xref="S4.SS1.SSS0.Px1.p3.1.m1.3.3.1.1.1.cmml">(</mo><mrow id="S4.SS1.SSS0.Px1.p3.1.m1.3.3.1.1.1" xref="S4.SS1.SSS0.Px1.p3.1.m1.3.3.1.1.1.cmml"><mi id="S4.SS1.SSS0.Px1.p3.1.m1.3.3.1.1.1.2" xref="S4.SS1.SSS0.Px1.p3.1.m1.3.3.1.1.1.2.cmml">z</mi><mo fence="false" id="S4.SS1.SSS0.Px1.p3.1.m1.3.3.1.1.1.1" xref="S4.SS1.SSS0.Px1.p3.1.m1.3.3.1.1.1.1.cmml">|</mo><mi id="S4.SS1.SSS0.Px1.p3.1.m1.3.3.1.1.1.3" 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xref="S4.SS1.SSS0.Px1.p3.1.m1.2.2.2.2">𝒢</ci></list></apply><apply id="S4.SS1.SSS0.Px1.p3.1.m1.3.3.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.p3.1.m1.3.3.1.1"><csymbol cd="latexml" id="S4.SS1.SSS0.Px1.p3.1.m1.3.3.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px1.p3.1.m1.3.3.1.1.1.1">conditional</csymbol><ci id="S4.SS1.SSS0.Px1.p3.1.m1.3.3.1.1.1.2.cmml" xref="S4.SS1.SSS0.Px1.p3.1.m1.3.3.1.1.1.2">𝑧</ci><ci id="S4.SS1.SSS0.Px1.p3.1.m1.3.3.1.1.1.3.cmml" xref="S4.SS1.SSS0.Px1.p3.1.m1.3.3.1.1.1.3">𝑞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p3.1.m1.3c">P_{\theta,\mathcal{G}}(z|q)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p3.1.m1.3d">italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_z | italic_q )</annotation></semantics></math> is the probability of retrieving the knowledge triple <math alttext="z" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p3.2.m2.1"><semantics id="S4.SS1.SSS0.Px1.p3.2.m2.1a"><mi id="S4.SS1.SSS0.Px1.p3.2.m2.1.1" xref="S4.SS1.SSS0.Px1.p3.2.m2.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p3.2.m2.1b"><ci id="S4.SS1.SSS0.Px1.p3.2.m2.1.1.cmml" xref="S4.SS1.SSS0.Px1.p3.2.m2.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p3.2.m2.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p3.2.m2.1d">italic_z</annotation></semantics></math> given the user’s query <math alttext="q" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p3.3.m3.1"><semantics id="S4.SS1.SSS0.Px1.p3.3.m3.1a"><mi id="S4.SS1.SSS0.Px1.p3.3.m3.1.1" xref="S4.SS1.SSS0.Px1.p3.3.m3.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p3.3.m3.1b"><ci id="S4.SS1.SSS0.Px1.p3.3.m3.1.1.cmml" xref="S4.SS1.SSS0.Px1.p3.3.m3.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p3.3.m3.1c">q</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p3.3.m3.1d">italic_q</annotation></semantics></math>, which represents the <span class="ltx_text ltx_font_bold" id="S4.SS1.SSS0.Px1.p3.9.1">knowledge retrieval</span> step. 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xref="S4.SS1.SSS0.Px1.p3.5.m5.4.4">𝑧</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p3.5.m5.5c">P_{\theta,\mathcal{G}}(a|q,z)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p3.5.m5.5d">italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_a | italic_q , italic_z )</annotation></semantics></math> is the probability of generating the answer <math alttext="a" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p3.6.m6.1"><semantics id="S4.SS1.SSS0.Px1.p3.6.m6.1a"><mi id="S4.SS1.SSS0.Px1.p3.6.m6.1.1" xref="S4.SS1.SSS0.Px1.p3.6.m6.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p3.6.m6.1b"><ci id="S4.SS1.SSS0.Px1.p3.6.m6.1.1.cmml" xref="S4.SS1.SSS0.Px1.p3.6.m6.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p3.6.m6.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p3.6.m6.1d">italic_a</annotation></semantics></math> given the user’s query and the retrieved triple <math alttext="z" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p3.7.m7.1"><semantics id="S4.SS1.SSS0.Px1.p3.7.m7.1a"><mi id="S4.SS1.SSS0.Px1.p3.7.m7.1.1" xref="S4.SS1.SSS0.Px1.p3.7.m7.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p3.7.m7.1b"><ci id="S4.SS1.SSS0.Px1.p3.7.m7.1.1.cmml" xref="S4.SS1.SSS0.Px1.p3.7.m7.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p3.7.m7.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p3.7.m7.1d">italic_z</annotation></semantics></math>, which stands for the <span class="ltx_text ltx_font_bold" id="S4.SS1.SSS0.Px1.p3.9.2">knowledge-enhanced reasoning</span> step. Instead of optimizing the model parameters <math alttext="\theta" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p3.8.m8.1"><semantics id="S4.SS1.SSS0.Px1.p3.8.m8.1a"><mi id="S4.SS1.SSS0.Px1.p3.8.m8.1.1" xref="S4.SS1.SSS0.Px1.p3.8.m8.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p3.8.m8.1b"><ci id="S4.SS1.SSS0.Px1.p3.8.m8.1.1.cmml" xref="S4.SS1.SSS0.Px1.p3.8.m8.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p3.8.m8.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p3.8.m8.1d">italic_θ</annotation></semantics></math>, we finetune the knowledge graph <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px1.p3.9.m9.1"><semantics id="S4.SS1.SSS0.Px1.p3.9.m9.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.SSS0.Px1.p3.9.m9.1.1" xref="S4.SS1.SSS0.Px1.p3.9.m9.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px1.p3.9.m9.1b"><ci id="S4.SS1.SSS0.Px1.p3.9.m9.1.1.cmml" xref="S4.SS1.SSS0.Px1.p3.9.m9.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px1.p3.9.m9.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px1.p3.9.m9.1d">caligraphic_G</annotation></semantics></math> in KGT.</p> </div> </section> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.2 </span>Objective Formulation</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.1">The training objective in <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S4.E1" title="In Example ‣ 4.1 Knowledge Graph Tuning ‣ 4 Method ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_tag">Equation</span> <span class="ltx_text ltx_ref_tag">1</span></a> can be optimized by maximizing the evidence lower bound (ELBO) <cite class="ltx_cite ltx_citemacro_citep">(Jordan et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib15" title="">1999</a>)</cite>, which is formulated as</p> <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="\log P_{\theta,\mathcal{G}}(a|q)\geq\underbrace{\mathbb{E}_{z\sim Q(z)}\left[% \log P_{\theta,\mathcal{G}}(a|q,z)\right]}_{\text{knowledge-enhanced reasoning% }}\underbrace{-D_{KL}\left(Q(z)||P_{\theta,\mathcal{G}}\left(z|q\right)\right)% }_{\text{knowledge retrieval}}," class="ltx_math_unparsed" display="block" id="S4.E2.m1.12"><semantics id="S4.E2.m1.12a"><mrow id="S4.E2.m1.12.12.1"><mrow id="S4.E2.m1.12.12.1.1"><mrow id="S4.E2.m1.12.12.1.1.1"><mrow id="S4.E2.m1.12.12.1.1.1.3"><mi id="S4.E2.m1.12.12.1.1.1.3.1" 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mathsize="90%">,</mo></mrow><annotation encoding="application/x-tex" id="S4.E2.m1.12b">\log P_{\theta,\mathcal{G}}(a|q)\geq\underbrace{\mathbb{E}_{z\sim Q(z)}\left[% \log P_{\theta,\mathcal{G}}(a|q,z)\right]}_{\text{knowledge-enhanced reasoning% }}\underbrace{-D_{KL}\left(Q(z)||P_{\theta,\mathcal{G}}\left(z|q\right)\right)% }_{\text{knowledge retrieval}},</annotation><annotation encoding="application/x-llamapun" id="S4.E2.m1.12c">roman_log italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_a | italic_q ) ≥ under⏟ start_ARG blackboard_E start_POSTSUBSCRIPT italic_z ∼ italic_Q ( italic_z ) end_POSTSUBSCRIPT [ roman_log italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_a | italic_q , italic_z ) ] end_ARG start_POSTSUBSCRIPT knowledge-enhanced reasoning end_POSTSUBSCRIPT under⏟ start_ARG - italic_D start_POSTSUBSCRIPT italic_K italic_L end_POSTSUBSCRIPT ( italic_Q ( italic_z ) | | italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_z | italic_q ) ) end_ARG start_POSTSUBSCRIPT knowledge retrieval 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> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.4">where <math alttext="Q(z)\triangleq Q(z|q,a)" class="ltx_Math" display="inline" id="S4.SS2.p2.1.m1.4"><semantics id="S4.SS2.p2.1.m1.4a"><mrow id="S4.SS2.p2.1.m1.4.4" xref="S4.SS2.p2.1.m1.4.4.cmml"><mrow id="S4.SS2.p2.1.m1.4.4.3" xref="S4.SS2.p2.1.m1.4.4.3.cmml"><mi id="S4.SS2.p2.1.m1.4.4.3.2" xref="S4.SS2.p2.1.m1.4.4.3.2.cmml">Q</mi><mo id="S4.SS2.p2.1.m1.4.4.3.1" xref="S4.SS2.p2.1.m1.4.4.3.1.cmml">⁢</mo><mrow id="S4.SS2.p2.1.m1.4.4.3.3.2" xref="S4.SS2.p2.1.m1.4.4.3.cmml"><mo id="S4.SS2.p2.1.m1.4.4.3.3.2.1" stretchy="false" xref="S4.SS2.p2.1.m1.4.4.3.cmml">(</mo><mi id="S4.SS2.p2.1.m1.1.1" xref="S4.SS2.p2.1.m1.1.1.cmml">z</mi><mo id="S4.SS2.p2.1.m1.4.4.3.3.2.2" stretchy="false" xref="S4.SS2.p2.1.m1.4.4.3.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p2.1.m1.4.4.2" xref="S4.SS2.p2.1.m1.4.4.2.cmml">≜</mo><mrow id="S4.SS2.p2.1.m1.4.4.1" xref="S4.SS2.p2.1.m1.4.4.1.cmml"><mi id="S4.SS2.p2.1.m1.4.4.1.3" xref="S4.SS2.p2.1.m1.4.4.1.3.cmml">Q</mi><mo id="S4.SS2.p2.1.m1.4.4.1.2" xref="S4.SS2.p2.1.m1.4.4.1.2.cmml">⁢</mo><mrow id="S4.SS2.p2.1.m1.4.4.1.1.1" xref="S4.SS2.p2.1.m1.4.4.1.1.1.1.cmml"><mo id="S4.SS2.p2.1.m1.4.4.1.1.1.2" stretchy="false" xref="S4.SS2.p2.1.m1.4.4.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.p2.1.m1.4.4.1.1.1.1" xref="S4.SS2.p2.1.m1.4.4.1.1.1.1.cmml"><mi id="S4.SS2.p2.1.m1.4.4.1.1.1.1.2" xref="S4.SS2.p2.1.m1.4.4.1.1.1.1.2.cmml">z</mi><mo fence="false" id="S4.SS2.p2.1.m1.4.4.1.1.1.1.1" xref="S4.SS2.p2.1.m1.4.4.1.1.1.1.1.cmml">|</mo><mrow id="S4.SS2.p2.1.m1.4.4.1.1.1.1.3.2" xref="S4.SS2.p2.1.m1.4.4.1.1.1.1.3.1.cmml"><mi id="S4.SS2.p2.1.m1.2.2" xref="S4.SS2.p2.1.m1.2.2.cmml">q</mi><mo id="S4.SS2.p2.1.m1.4.4.1.1.1.1.3.2.1" xref="S4.SS2.p2.1.m1.4.4.1.1.1.1.3.1.cmml">,</mo><mi id="S4.SS2.p2.1.m1.3.3" xref="S4.SS2.p2.1.m1.3.3.cmml">a</mi></mrow></mrow><mo id="S4.SS2.p2.1.m1.4.4.1.1.1.3" stretchy="false" xref="S4.SS2.p2.1.m1.4.4.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.1.m1.4b"><apply id="S4.SS2.p2.1.m1.4.4.cmml" xref="S4.SS2.p2.1.m1.4.4"><ci id="S4.SS2.p2.1.m1.4.4.2.cmml" xref="S4.SS2.p2.1.m1.4.4.2">≜</ci><apply id="S4.SS2.p2.1.m1.4.4.3.cmml" xref="S4.SS2.p2.1.m1.4.4.3"><times id="S4.SS2.p2.1.m1.4.4.3.1.cmml" xref="S4.SS2.p2.1.m1.4.4.3.1"></times><ci id="S4.SS2.p2.1.m1.4.4.3.2.cmml" xref="S4.SS2.p2.1.m1.4.4.3.2">𝑄</ci><ci id="S4.SS2.p2.1.m1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1">𝑧</ci></apply><apply id="S4.SS2.p2.1.m1.4.4.1.cmml" xref="S4.SS2.p2.1.m1.4.4.1"><times id="S4.SS2.p2.1.m1.4.4.1.2.cmml" xref="S4.SS2.p2.1.m1.4.4.1.2"></times><ci id="S4.SS2.p2.1.m1.4.4.1.3.cmml" xref="S4.SS2.p2.1.m1.4.4.1.3">𝑄</ci><apply id="S4.SS2.p2.1.m1.4.4.1.1.1.1.cmml" xref="S4.SS2.p2.1.m1.4.4.1.1.1"><csymbol cd="latexml" id="S4.SS2.p2.1.m1.4.4.1.1.1.1.1.cmml" xref="S4.SS2.p2.1.m1.4.4.1.1.1.1.1">conditional</csymbol><ci id="S4.SS2.p2.1.m1.4.4.1.1.1.1.2.cmml" xref="S4.SS2.p2.1.m1.4.4.1.1.1.1.2">𝑧</ci><list id="S4.SS2.p2.1.m1.4.4.1.1.1.1.3.1.cmml" xref="S4.SS2.p2.1.m1.4.4.1.1.1.1.3.2"><ci id="S4.SS2.p2.1.m1.2.2.cmml" xref="S4.SS2.p2.1.m1.2.2">𝑞</ci><ci id="S4.SS2.p2.1.m1.3.3.cmml" xref="S4.SS2.p2.1.m1.3.3">𝑎</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.4c">Q(z)\triangleq Q(z|q,a)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.4d">italic_Q ( italic_z ) ≜ italic_Q ( italic_z | italic_q , italic_a )</annotation></semantics></math> is the posterior distribution of <math alttext="z" class="ltx_Math" display="inline" id="S4.SS2.p2.2.m2.1"><semantics id="S4.SS2.p2.2.m2.1a"><mi id="S4.SS2.p2.2.m2.1.1" xref="S4.SS2.p2.2.m2.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.2.m2.1b"><ci id="S4.SS2.p2.2.m2.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.2.m2.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.2.m2.1d">italic_z</annotation></semantics></math> given the user’s query and the user’s feedback <math alttext="a" class="ltx_Math" display="inline" id="S4.SS2.p2.3.m3.1"><semantics id="S4.SS2.p2.3.m3.1a"><mi id="S4.SS2.p2.3.m3.1.1" xref="S4.SS2.p2.3.m3.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.3.m3.1b"><ci id="S4.SS2.p2.3.m3.1.1.cmml" xref="S4.SS2.p2.3.m3.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.3.m3.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.3.m3.1d">italic_a</annotation></semantics></math>. This posterior distribution gives a larger probability to the triples containing personalized factual knowledge based on the user’s feedback (e.g., the example <math alttext="\left(Dog,Enjoy,Vegetable\right)" class="ltx_Math" display="inline" id="S4.SS2.p2.4.m4.3"><semantics id="S4.SS2.p2.4.m4.3a"><mrow id="S4.SS2.p2.4.m4.3.3.3" xref="S4.SS2.p2.4.m4.3.3.4.cmml"><mo id="S4.SS2.p2.4.m4.3.3.3.4" xref="S4.SS2.p2.4.m4.3.3.4.cmml">(</mo><mrow id="S4.SS2.p2.4.m4.1.1.1.1" xref="S4.SS2.p2.4.m4.1.1.1.1.cmml"><mi id="S4.SS2.p2.4.m4.1.1.1.1.2" xref="S4.SS2.p2.4.m4.1.1.1.1.2.cmml">D</mi><mo id="S4.SS2.p2.4.m4.1.1.1.1.1" xref="S4.SS2.p2.4.m4.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS2.p2.4.m4.1.1.1.1.3" xref="S4.SS2.p2.4.m4.1.1.1.1.3.cmml">o</mi><mo id="S4.SS2.p2.4.m4.1.1.1.1.1a" xref="S4.SS2.p2.4.m4.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS2.p2.4.m4.1.1.1.1.4" xref="S4.SS2.p2.4.m4.1.1.1.1.4.cmml">g</mi></mrow><mo id="S4.SS2.p2.4.m4.3.3.3.5" xref="S4.SS2.p2.4.m4.3.3.4.cmml">,</mo><mrow 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ltx_ref_tag">4.1</span></a>), which are referred to the <span class="ltx_text ltx_font_bold" id="S4.SS2.p2.4.1">personalized triples</span>. The later term minimizes the KL divergence between the posterior distribution and the prior distribution of the knowledge triple retrieved by the LLM. The former term maximizes the expectation that LLM generates correct answers based on the knowledge triples sampled from the posterior distribution.</p> </div> <section class="ltx_paragraph" id="S4.SS2.SSS0.Px1"> <h5 class="ltx_title ltx_title_paragraph">Knowledge Retrieval</h5> <div class="ltx_para" id="S4.SS2.SSS0.Px1.p1"> <p class="ltx_p" id="S4.SS2.SSS0.Px1.p1.10">The goal of the knowledge retrieval term is to finetune the KG such that the LLM can retrieve personalized triples based on the user’s feedback. 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id="S4.SS2.SSS0.Px1.p1.7.m7.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p1.7.m7.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p1.7.m7.1c">e_{a}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p1.7.m7.1d">italic_e start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> in this query’s context. We construct the personalized triple as <math alttext="\{e_{q},r_{k},e_{a}\}_{k\in[K]}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p1.8.m8.4"><semantics id="S4.SS2.SSS0.Px1.p1.8.m8.4a"><msub id="S4.SS2.SSS0.Px1.p1.8.m8.4.4" xref="S4.SS2.SSS0.Px1.p1.8.m8.4.4.cmml"><mrow id="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.3" xref="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.4.cmml"><mo id="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.3.4" stretchy="false" xref="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.4.cmml">{</mo><msub id="S4.SS2.SSS0.Px1.p1.8.m8.2.2.1.1.1" xref="S4.SS2.SSS0.Px1.p1.8.m8.2.2.1.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p1.8.m8.2.2.1.1.1.2" xref="S4.SS2.SSS0.Px1.p1.8.m8.2.2.1.1.1.2.cmml">e</mi><mi id="S4.SS2.SSS0.Px1.p1.8.m8.2.2.1.1.1.3" xref="S4.SS2.SSS0.Px1.p1.8.m8.2.2.1.1.1.3.cmml">q</mi></msub><mo id="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.3.5" xref="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.4.cmml">,</mo><msub id="S4.SS2.SSS0.Px1.p1.8.m8.3.3.2.2.2" xref="S4.SS2.SSS0.Px1.p1.8.m8.3.3.2.2.2.cmml"><mi id="S4.SS2.SSS0.Px1.p1.8.m8.3.3.2.2.2.2" xref="S4.SS2.SSS0.Px1.p1.8.m8.3.3.2.2.2.2.cmml">r</mi><mi id="S4.SS2.SSS0.Px1.p1.8.m8.3.3.2.2.2.3" xref="S4.SS2.SSS0.Px1.p1.8.m8.3.3.2.2.2.3.cmml">k</mi></msub><mo id="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.3.6" xref="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.4.cmml">,</mo><msub id="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.3.3" xref="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.3.3.cmml"><mi id="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.3.3.2" xref="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.3.3.2.cmml">e</mi><mi id="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.3.3.3" xref="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.3.3.3.cmml">a</mi></msub><mo id="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.3.7" stretchy="false" xref="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.4.cmml">}</mo></mrow><mrow id="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1" xref="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.3" xref="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.3.cmml">k</mi><mo id="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.2" xref="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.2.cmml">∈</mo><mrow id="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.4.2" xref="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.4.1.cmml"><mo id="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.4.2.1" stretchy="false" xref="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.4.1.1.cmml">[</mo><mi id="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.1" xref="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.1.cmml">K</mi><mo id="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.4.2.2" stretchy="false" xref="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p1.8.m8.4b"><apply id="S4.SS2.SSS0.Px1.p1.8.m8.4.4.cmml" xref="S4.SS2.SSS0.Px1.p1.8.m8.4.4"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p1.8.m8.4.4.4.cmml" xref="S4.SS2.SSS0.Px1.p1.8.m8.4.4">subscript</csymbol><set id="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.4.cmml" xref="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.3"><apply id="S4.SS2.SSS0.Px1.p1.8.m8.2.2.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.8.m8.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p1.8.m8.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.8.m8.2.2.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p1.8.m8.2.2.1.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p1.8.m8.2.2.1.1.1.2">𝑒</ci><ci id="S4.SS2.SSS0.Px1.p1.8.m8.2.2.1.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p1.8.m8.2.2.1.1.1.3">𝑞</ci></apply><apply id="S4.SS2.SSS0.Px1.p1.8.m8.3.3.2.2.2.cmml" xref="S4.SS2.SSS0.Px1.p1.8.m8.3.3.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p1.8.m8.3.3.2.2.2.1.cmml" xref="S4.SS2.SSS0.Px1.p1.8.m8.3.3.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p1.8.m8.3.3.2.2.2.2.cmml" xref="S4.SS2.SSS0.Px1.p1.8.m8.3.3.2.2.2.2">𝑟</ci><ci id="S4.SS2.SSS0.Px1.p1.8.m8.3.3.2.2.2.3.cmml" xref="S4.SS2.SSS0.Px1.p1.8.m8.3.3.2.2.2.3">𝑘</ci></apply><apply id="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.3.3.cmml" xref="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.3.3"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.3.3.1.cmml" xref="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.3.3">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.3.3.2.cmml" xref="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.3.3.2">𝑒</ci><ci id="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.3.3.3.cmml" xref="S4.SS2.SSS0.Px1.p1.8.m8.4.4.3.3.3.3">𝑎</ci></apply></set><apply id="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1"><in id="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.2"></in><ci id="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.3">𝑘</ci><apply id="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.4.1.cmml" xref="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.4.2"><csymbol cd="latexml" id="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.4.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.4.2.1">delimited-[]</csymbol><ci id="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.8.m8.1.1.1.1">𝐾</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p1.8.m8.4c">\{e_{q},r_{k},e_{a}\}_{k\in[K]}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p1.8.m8.4d">{ italic_e start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_e start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_k ∈ [ italic_K ] end_POSTSUBSCRIPT</annotation></semantics></math> and denote this triple set as <math alttext="\mathcal{H}(q,a,K)" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p1.9.m9.3"><semantics id="S4.SS2.SSS0.Px1.p1.9.m9.3a"><mrow id="S4.SS2.SSS0.Px1.p1.9.m9.3.4" xref="S4.SS2.SSS0.Px1.p1.9.m9.3.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.SSS0.Px1.p1.9.m9.3.4.2" xref="S4.SS2.SSS0.Px1.p1.9.m9.3.4.2.cmml">ℋ</mi><mo id="S4.SS2.SSS0.Px1.p1.9.m9.3.4.1" xref="S4.SS2.SSS0.Px1.p1.9.m9.3.4.1.cmml">⁢</mo><mrow id="S4.SS2.SSS0.Px1.p1.9.m9.3.4.3.2" xref="S4.SS2.SSS0.Px1.p1.9.m9.3.4.3.1.cmml"><mo id="S4.SS2.SSS0.Px1.p1.9.m9.3.4.3.2.1" stretchy="false" xref="S4.SS2.SSS0.Px1.p1.9.m9.3.4.3.1.cmml">(</mo><mi id="S4.SS2.SSS0.Px1.p1.9.m9.1.1" xref="S4.SS2.SSS0.Px1.p1.9.m9.1.1.cmml">q</mi><mo id="S4.SS2.SSS0.Px1.p1.9.m9.3.4.3.2.2" xref="S4.SS2.SSS0.Px1.p1.9.m9.3.4.3.1.cmml">,</mo><mi id="S4.SS2.SSS0.Px1.p1.9.m9.2.2" xref="S4.SS2.SSS0.Px1.p1.9.m9.2.2.cmml">a</mi><mo id="S4.SS2.SSS0.Px1.p1.9.m9.3.4.3.2.3" xref="S4.SS2.SSS0.Px1.p1.9.m9.3.4.3.1.cmml">,</mo><mi id="S4.SS2.SSS0.Px1.p1.9.m9.3.3" xref="S4.SS2.SSS0.Px1.p1.9.m9.3.3.cmml">K</mi><mo id="S4.SS2.SSS0.Px1.p1.9.m9.3.4.3.2.4" stretchy="false" xref="S4.SS2.SSS0.Px1.p1.9.m9.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p1.9.m9.3b"><apply id="S4.SS2.SSS0.Px1.p1.9.m9.3.4.cmml" xref="S4.SS2.SSS0.Px1.p1.9.m9.3.4"><times id="S4.SS2.SSS0.Px1.p1.9.m9.3.4.1.cmml" xref="S4.SS2.SSS0.Px1.p1.9.m9.3.4.1"></times><ci id="S4.SS2.SSS0.Px1.p1.9.m9.3.4.2.cmml" xref="S4.SS2.SSS0.Px1.p1.9.m9.3.4.2">ℋ</ci><vector id="S4.SS2.SSS0.Px1.p1.9.m9.3.4.3.1.cmml" xref="S4.SS2.SSS0.Px1.p1.9.m9.3.4.3.2"><ci id="S4.SS2.SSS0.Px1.p1.9.m9.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.9.m9.1.1">𝑞</ci><ci id="S4.SS2.SSS0.Px1.p1.9.m9.2.2.cmml" xref="S4.SS2.SSS0.Px1.p1.9.m9.2.2">𝑎</ci><ci id="S4.SS2.SSS0.Px1.p1.9.m9.3.3.cmml" xref="S4.SS2.SSS0.Px1.p1.9.m9.3.3">𝐾</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p1.9.m9.3c">\mathcal{H}(q,a,K)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p1.9.m9.3d">caligraphic_H ( italic_q , italic_a , italic_K )</annotation></semantics></math>. Then, the posterior distribution <math alttext="Q(z)" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p1.10.m10.1"><semantics id="S4.SS2.SSS0.Px1.p1.10.m10.1a"><mrow id="S4.SS2.SSS0.Px1.p1.10.m10.1.2" xref="S4.SS2.SSS0.Px1.p1.10.m10.1.2.cmml"><mi id="S4.SS2.SSS0.Px1.p1.10.m10.1.2.2" xref="S4.SS2.SSS0.Px1.p1.10.m10.1.2.2.cmml">Q</mi><mo id="S4.SS2.SSS0.Px1.p1.10.m10.1.2.1" xref="S4.SS2.SSS0.Px1.p1.10.m10.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS0.Px1.p1.10.m10.1.2.3.2" xref="S4.SS2.SSS0.Px1.p1.10.m10.1.2.cmml"><mo id="S4.SS2.SSS0.Px1.p1.10.m10.1.2.3.2.1" stretchy="false" xref="S4.SS2.SSS0.Px1.p1.10.m10.1.2.cmml">(</mo><mi id="S4.SS2.SSS0.Px1.p1.10.m10.1.1" xref="S4.SS2.SSS0.Px1.p1.10.m10.1.1.cmml">z</mi><mo id="S4.SS2.SSS0.Px1.p1.10.m10.1.2.3.2.2" stretchy="false" xref="S4.SS2.SSS0.Px1.p1.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p1.10.m10.1b"><apply id="S4.SS2.SSS0.Px1.p1.10.m10.1.2.cmml" xref="S4.SS2.SSS0.Px1.p1.10.m10.1.2"><times id="S4.SS2.SSS0.Px1.p1.10.m10.1.2.1.cmml" xref="S4.SS2.SSS0.Px1.p1.10.m10.1.2.1"></times><ci id="S4.SS2.SSS0.Px1.p1.10.m10.1.2.2.cmml" xref="S4.SS2.SSS0.Px1.p1.10.m10.1.2.2">𝑄</ci><ci id="S4.SS2.SSS0.Px1.p1.10.m10.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.10.m10.1.1">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p1.10.m10.1c">Q(z)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p1.10.m10.1d">italic_Q ( italic_z )</annotation></semantics></math> can be approximated, formally,</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="Q(z|q,a)\simeq\begin{cases}\frac{1}{K}&\text{if }z\in\mathcal{H}(q,a,K),\\ 0&\text{else},\end{cases}" class="ltx_Math" display="block" id="S4.E3.m1.7"><semantics 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xref="S4.E3.m1.2.2.2.2.2.1.3">𝐾</ci></vector></apply></apply><cn id="S4.E3.m1.3.3.3.3.1.1.cmml" type="integer" xref="S4.E3.m1.3.3.3.3.1.1">0</cn><ci id="S4.E3.m1.4.4.4.4.2.1.1a.cmml" xref="S4.E3.m1.4.4.4.4.2.1.3"><mtext id="S4.E3.m1.4.4.4.4.2.1.1.cmml" mathsize="90%" xref="S4.E3.m1.4.4.4.4.2.1.1">else</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E3.m1.7c">Q(z|q,a)\simeq\begin{cases}\frac{1}{K}&\text{if }z\in\mathcal{H}(q,a,K),\\ 0&\text{else},\end{cases}</annotation><annotation encoding="application/x-llamapun" id="S4.E3.m1.7d">italic_Q ( italic_z | italic_q , italic_a ) ≃ { start_ROW start_CELL divide start_ARG 1 end_ARG start_ARG italic_K end_ARG end_CELL start_CELL if italic_z ∈ caligraphic_H ( italic_q , italic_a , italic_K ) , end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL else , 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">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS0.Px1.p1.11">where we assume a uniform distribution over the subset of personalized triples <math alttext="\mathcal{H}(q,a,K)" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p1.11.m1.3"><semantics id="S4.SS2.SSS0.Px1.p1.11.m1.3a"><mrow id="S4.SS2.SSS0.Px1.p1.11.m1.3.4" xref="S4.SS2.SSS0.Px1.p1.11.m1.3.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.SSS0.Px1.p1.11.m1.3.4.2" xref="S4.SS2.SSS0.Px1.p1.11.m1.3.4.2.cmml">ℋ</mi><mo id="S4.SS2.SSS0.Px1.p1.11.m1.3.4.1" xref="S4.SS2.SSS0.Px1.p1.11.m1.3.4.1.cmml">⁢</mo><mrow id="S4.SS2.SSS0.Px1.p1.11.m1.3.4.3.2" xref="S4.SS2.SSS0.Px1.p1.11.m1.3.4.3.1.cmml"><mo id="S4.SS2.SSS0.Px1.p1.11.m1.3.4.3.2.1" stretchy="false" xref="S4.SS2.SSS0.Px1.p1.11.m1.3.4.3.1.cmml">(</mo><mi id="S4.SS2.SSS0.Px1.p1.11.m1.1.1" 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id="S4.SS2.SSS0.Px1.p1.11.m1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.11.m1.1.1">𝑞</ci><ci id="S4.SS2.SSS0.Px1.p1.11.m1.2.2.cmml" xref="S4.SS2.SSS0.Px1.p1.11.m1.2.2">𝑎</ci><ci id="S4.SS2.SSS0.Px1.p1.11.m1.3.3.cmml" xref="S4.SS2.SSS0.Px1.p1.11.m1.3.3">𝐾</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p1.11.m1.3c">\mathcal{H}(q,a,K)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p1.11.m1.3d">caligraphic_H ( italic_q , italic_a , italic_K )</annotation></semantics></math>. Therefore, the knowledge retrieval loss term can be calculated as</p> <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="\mathcal{L}_{\text{retrieve}}=D_{KL}\left(Q(z)||P_{\theta,\mathcal{G}}\left(z|% q\right)\right)=-\frac{1}{K}\sum_{z\in\mathcal{H}(q,a,K)}\log P_{\theta,% \mathcal{G}}\left(z|q\right)." class="ltx_math_unparsed" display="block" id="S4.E4.m1.8"><semantics id="S4.E4.m1.8a"><mrow id="S4.E4.m1.8b"><msub id="S4.E4.m1.8.9"><mi class="ltx_font_mathcaligraphic" id="S4.E4.m1.8.9.2" mathsize="90%">ℒ</mi><mtext id="S4.E4.m1.8.9.3" mathsize="90%">retrieve</mtext></msub><mo id="S4.E4.m1.8.10" mathsize="90%">=</mo><msub id="S4.E4.m1.8.11"><mi id="S4.E4.m1.8.11.2" mathsize="90%">D</mi><mrow id="S4.E4.m1.8.11.3"><mi id="S4.E4.m1.8.11.3.2" mathsize="90%">K</mi><mo id="S4.E4.m1.8.11.3.1">⁢</mo><mi id="S4.E4.m1.8.11.3.3" mathsize="90%">L</mi></mrow></msub><mrow id="S4.E4.m1.8.12"><mo id="S4.E4.m1.8.12.1">(</mo><mi id="S4.E4.m1.8.12.2" mathsize="90%">Q</mi><mrow id="S4.E4.m1.8.12.3"><mo id="S4.E4.m1.8.12.3.1" maxsize="90%" minsize="90%">(</mo><mi id="S4.E4.m1.8.8" mathsize="90%">z</mi><mo id="S4.E4.m1.8.12.3.2" maxsize="90%" minsize="90%">)</mo></mrow><mo fence="false" id="S4.E4.m1.8.12.4" maxsize="90%" minsize="90%" rspace="0.167em">|</mo><mo fence="false" id="S4.E4.m1.8.12.5" maxsize="90%" minsize="90%" rspace="0.167em">|</mo><msub id="S4.E4.m1.8.12.6"><mi id="S4.E4.m1.8.12.6.2" mathsize="90%">P</mi><mrow id="S4.E4.m1.2.2.2.4"><mi id="S4.E4.m1.1.1.1.1" mathsize="90%">θ</mi><mo id="S4.E4.m1.2.2.2.4.1" mathsize="90%">,</mo><mi class="ltx_font_mathcaligraphic" id="S4.E4.m1.2.2.2.2" mathsize="90%">𝒢</mi></mrow></msub><mrow id="S4.E4.m1.8.12.7"><mo id="S4.E4.m1.8.12.7.1">(</mo><mi id="S4.E4.m1.8.12.7.2" mathsize="90%">z</mi><mo fence="false" id="S4.E4.m1.8.12.7.3" maxsize="90%" minsize="90%" rspace="0.167em">|</mo><mi id="S4.E4.m1.8.12.7.4" mathsize="90%">q</mi><mo id="S4.E4.m1.8.12.7.5">)</mo></mrow><mo id="S4.E4.m1.8.12.8">)</mo></mrow><mo id="S4.E4.m1.8.13" mathsize="90%" rspace="0em">=</mo><mo id="S4.E4.m1.8.14" lspace="0em" mathsize="90%">−</mo><mfrac id="S4.E4.m1.8.15"><mn id="S4.E4.m1.8.15.2" mathsize="90%">1</mn><mi id="S4.E4.m1.8.15.3" mathsize="90%">K</mi></mfrac><munder id="S4.E4.m1.8.16"><mo id="S4.E4.m1.8.16.2" maxsize="90%" minsize="90%" movablelimits="false" stretchy="true">∑</mo><mrow id="S4.E4.m1.5.5.3"><mi id="S4.E4.m1.5.5.3.5" mathsize="90%">z</mi><mo id="S4.E4.m1.5.5.3.4" mathsize="90%">∈</mo><mrow id="S4.E4.m1.5.5.3.6"><mi class="ltx_font_mathcaligraphic" id="S4.E4.m1.5.5.3.6.2" mathsize="90%">ℋ</mi><mo id="S4.E4.m1.5.5.3.6.1">⁢</mo><mrow id="S4.E4.m1.5.5.3.6.3.2"><mo id="S4.E4.m1.5.5.3.6.3.2.1" maxsize="90%" minsize="90%">(</mo><mi id="S4.E4.m1.3.3.1.1" mathsize="90%">q</mi><mo id="S4.E4.m1.5.5.3.6.3.2.2" mathsize="90%">,</mo><mi id="S4.E4.m1.4.4.2.2" mathsize="90%">a</mi><mo id="S4.E4.m1.5.5.3.6.3.2.3" mathsize="90%">,</mo><mi id="S4.E4.m1.5.5.3.3" mathsize="90%">K</mi><mo id="S4.E4.m1.5.5.3.6.3.2.4" maxsize="90%" minsize="90%">)</mo></mrow></mrow></mrow></munder><mi id="S4.E4.m1.8.17" mathsize="90%">log</mi><msub id="S4.E4.m1.8.18"><mi id="S4.E4.m1.8.18.2" mathsize="90%">P</mi><mrow id="S4.E4.m1.7.7.2.4"><mi id="S4.E4.m1.6.6.1.1" mathsize="90%">θ</mi><mo id="S4.E4.m1.7.7.2.4.1" mathsize="90%">,</mo><mi class="ltx_font_mathcaligraphic" id="S4.E4.m1.7.7.2.2" mathsize="90%">𝒢</mi></mrow></msub><mrow id="S4.E4.m1.8.19"><mo id="S4.E4.m1.8.19.1">(</mo><mi id="S4.E4.m1.8.19.2" mathsize="90%">z</mi><mo fence="false" id="S4.E4.m1.8.19.3" maxsize="90%" minsize="90%" rspace="0.167em">|</mo><mi id="S4.E4.m1.8.19.4" mathsize="90%">q</mi><mo id="S4.E4.m1.8.19.5">)</mo></mrow><mo id="S4.E4.m1.8.20" lspace="0em" mathsize="90%">.</mo></mrow><annotation encoding="application/x-tex" id="S4.E4.m1.8c">\mathcal{L}_{\text{retrieve}}=D_{KL}\left(Q(z)||P_{\theta,\mathcal{G}}\left(z|% q\right)\right)=-\frac{1}{K}\sum_{z\in\mathcal{H}(q,a,K)}\log P_{\theta,% \mathcal{G}}\left(z|q\right).</annotation><annotation encoding="application/x-llamapun" id="S4.E4.m1.8d">caligraphic_L start_POSTSUBSCRIPT retrieve end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_K italic_L end_POSTSUBSCRIPT ( italic_Q ( italic_z ) | | italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_z | italic_q ) ) = - divide start_ARG 1 end_ARG start_ARG italic_K end_ARG ∑ start_POSTSUBSCRIPT italic_z ∈ caligraphic_H ( italic_q , italic_a , italic_K ) end_POSTSUBSCRIPT roman_log italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_z | italic_q ) .</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> <p class="ltx_p" id="S4.SS2.SSS0.Px1.p1.13">The detailed derivation can be found in <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#A1" title='Appendix A Derivation of ℒ_"retrieve" ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback'><span class="ltx_text ltx_ref_tag">Appendix</span> <span class="ltx_text ltx_ref_tag">A</span></a>. By reducing <math alttext="\mathcal{L}_{\text{retrieve}}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p1.12.m1.1"><semantics id="S4.SS2.SSS0.Px1.p1.12.m1.1a"><msub id="S4.SS2.SSS0.Px1.p1.12.m1.1.1" xref="S4.SS2.SSS0.Px1.p1.12.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.SSS0.Px1.p1.12.m1.1.1.2" xref="S4.SS2.SSS0.Px1.p1.12.m1.1.1.2.cmml">ℒ</mi><mtext id="S4.SS2.SSS0.Px1.p1.12.m1.1.1.3" xref="S4.SS2.SSS0.Px1.p1.12.m1.1.1.3a.cmml">retrieve</mtext></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p1.12.m1.1b"><apply id="S4.SS2.SSS0.Px1.p1.12.m1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.12.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p1.12.m1.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.12.m1.1.1">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p1.12.m1.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p1.12.m1.1.1.2">ℒ</ci><ci id="S4.SS2.SSS0.Px1.p1.12.m1.1.1.3a.cmml" xref="S4.SS2.SSS0.Px1.p1.12.m1.1.1.3"><mtext id="S4.SS2.SSS0.Px1.p1.12.m1.1.1.3.cmml" mathsize="70%" xref="S4.SS2.SSS0.Px1.p1.12.m1.1.1.3">retrieve</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p1.12.m1.1c">\mathcal{L}_{\text{retrieve}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p1.12.m1.1d">caligraphic_L start_POSTSUBSCRIPT retrieve end_POSTSUBSCRIPT</annotation></semantics></math>, we finetune the user’s KG <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p1.13.m2.1"><semantics id="S4.SS2.SSS0.Px1.p1.13.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.SSS0.Px1.p1.13.m2.1.1" xref="S4.SS2.SSS0.Px1.p1.13.m2.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p1.13.m2.1b"><ci id="S4.SS2.SSS0.Px1.p1.13.m2.1.1.cmml" xref="S4.SS2.SSS0.Px1.p1.13.m2.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p1.13.m2.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p1.13.m2.1d">caligraphic_G</annotation></semantics></math> such that the LLM can retrieve the personalized triples with higher confidence.</p> </div> <div class="ltx_para" id="S4.SS2.SSS0.Px1.p2"> <p class="ltx_p" id="S4.SS2.SSS0.Px1.p2.7">To collect <math alttext="\mathcal{H}(q,a,K)" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p2.1.m1.3"><semantics id="S4.SS2.SSS0.Px1.p2.1.m1.3a"><mrow id="S4.SS2.SSS0.Px1.p2.1.m1.3.4" xref="S4.SS2.SSS0.Px1.p2.1.m1.3.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.SSS0.Px1.p2.1.m1.3.4.2" xref="S4.SS2.SSS0.Px1.p2.1.m1.3.4.2.cmml">ℋ</mi><mo id="S4.SS2.SSS0.Px1.p2.1.m1.3.4.1" xref="S4.SS2.SSS0.Px1.p2.1.m1.3.4.1.cmml">⁢</mo><mrow id="S4.SS2.SSS0.Px1.p2.1.m1.3.4.3.2" xref="S4.SS2.SSS0.Px1.p2.1.m1.3.4.3.1.cmml"><mo id="S4.SS2.SSS0.Px1.p2.1.m1.3.4.3.2.1" stretchy="false" xref="S4.SS2.SSS0.Px1.p2.1.m1.3.4.3.1.cmml">(</mo><mi id="S4.SS2.SSS0.Px1.p2.1.m1.1.1" xref="S4.SS2.SSS0.Px1.p2.1.m1.1.1.cmml">q</mi><mo id="S4.SS2.SSS0.Px1.p2.1.m1.3.4.3.2.2" xref="S4.SS2.SSS0.Px1.p2.1.m1.3.4.3.1.cmml">,</mo><mi id="S4.SS2.SSS0.Px1.p2.1.m1.2.2" xref="S4.SS2.SSS0.Px1.p2.1.m1.2.2.cmml">a</mi><mo id="S4.SS2.SSS0.Px1.p2.1.m1.3.4.3.2.3" xref="S4.SS2.SSS0.Px1.p2.1.m1.3.4.3.1.cmml">,</mo><mi id="S4.SS2.SSS0.Px1.p2.1.m1.3.3" xref="S4.SS2.SSS0.Px1.p2.1.m1.3.3.cmml">K</mi><mo id="S4.SS2.SSS0.Px1.p2.1.m1.3.4.3.2.4" stretchy="false" xref="S4.SS2.SSS0.Px1.p2.1.m1.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p2.1.m1.3b"><apply id="S4.SS2.SSS0.Px1.p2.1.m1.3.4.cmml" xref="S4.SS2.SSS0.Px1.p2.1.m1.3.4"><times id="S4.SS2.SSS0.Px1.p2.1.m1.3.4.1.cmml" xref="S4.SS2.SSS0.Px1.p2.1.m1.3.4.1"></times><ci id="S4.SS2.SSS0.Px1.p2.1.m1.3.4.2.cmml" xref="S4.SS2.SSS0.Px1.p2.1.m1.3.4.2">ℋ</ci><vector id="S4.SS2.SSS0.Px1.p2.1.m1.3.4.3.1.cmml" xref="S4.SS2.SSS0.Px1.p2.1.m1.3.4.3.2"><ci id="S4.SS2.SSS0.Px1.p2.1.m1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p2.1.m1.1.1">𝑞</ci><ci id="S4.SS2.SSS0.Px1.p2.1.m1.2.2.cmml" xref="S4.SS2.SSS0.Px1.p2.1.m1.2.2">𝑎</ci><ci id="S4.SS2.SSS0.Px1.p2.1.m1.3.3.cmml" xref="S4.SS2.SSS0.Px1.p2.1.m1.3.3">𝐾</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p2.1.m1.3c">\mathcal{H}(q,a,K)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p2.1.m1.3d">caligraphic_H ( italic_q , italic_a , italic_K )</annotation></semantics></math>, users can either provide feedback about the relations between <math alttext="e_{q}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p2.2.m2.1"><semantics id="S4.SS2.SSS0.Px1.p2.2.m2.1a"><msub id="S4.SS2.SSS0.Px1.p2.2.m2.1.1" xref="S4.SS2.SSS0.Px1.p2.2.m2.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p2.2.m2.1.1.2" xref="S4.SS2.SSS0.Px1.p2.2.m2.1.1.2.cmml">e</mi><mi id="S4.SS2.SSS0.Px1.p2.2.m2.1.1.3" xref="S4.SS2.SSS0.Px1.p2.2.m2.1.1.3.cmml">q</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p2.2.m2.1b"><apply id="S4.SS2.SSS0.Px1.p2.2.m2.1.1.cmml" xref="S4.SS2.SSS0.Px1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p2.2.m2.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p2.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p2.2.m2.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p2.2.m2.1.1.2">𝑒</ci><ci id="S4.SS2.SSS0.Px1.p2.2.m2.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p2.2.m2.1.1.3">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p2.2.m2.1c">e_{q}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p2.2.m2.1d">italic_e start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="e_{a}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p2.3.m3.1"><semantics id="S4.SS2.SSS0.Px1.p2.3.m3.1a"><msub id="S4.SS2.SSS0.Px1.p2.3.m3.1.1" xref="S4.SS2.SSS0.Px1.p2.3.m3.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p2.3.m3.1.1.2" xref="S4.SS2.SSS0.Px1.p2.3.m3.1.1.2.cmml">e</mi><mi id="S4.SS2.SSS0.Px1.p2.3.m3.1.1.3" xref="S4.SS2.SSS0.Px1.p2.3.m3.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p2.3.m3.1b"><apply id="S4.SS2.SSS0.Px1.p2.3.m3.1.1.cmml" xref="S4.SS2.SSS0.Px1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p2.3.m3.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p2.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p2.3.m3.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p2.3.m3.1.1.2">𝑒</ci><ci id="S4.SS2.SSS0.Px1.p2.3.m3.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p2.3.m3.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p2.3.m3.1c">e_{a}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p2.3.m3.1d">italic_e start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> in this query’s context or extract the relations using the model <math alttext="f_{\theta,\mathcal{G}}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p2.4.m4.2"><semantics id="S4.SS2.SSS0.Px1.p2.4.m4.2a"><msub id="S4.SS2.SSS0.Px1.p2.4.m4.2.3" xref="S4.SS2.SSS0.Px1.p2.4.m4.2.3.cmml"><mi id="S4.SS2.SSS0.Px1.p2.4.m4.2.3.2" xref="S4.SS2.SSS0.Px1.p2.4.m4.2.3.2.cmml">f</mi><mrow id="S4.SS2.SSS0.Px1.p2.4.m4.2.2.2.4" xref="S4.SS2.SSS0.Px1.p2.4.m4.2.2.2.3.cmml"><mi id="S4.SS2.SSS0.Px1.p2.4.m4.1.1.1.1" xref="S4.SS2.SSS0.Px1.p2.4.m4.1.1.1.1.cmml">θ</mi><mo id="S4.SS2.SSS0.Px1.p2.4.m4.2.2.2.4.1" xref="S4.SS2.SSS0.Px1.p2.4.m4.2.2.2.3.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.SSS0.Px1.p2.4.m4.2.2.2.2" xref="S4.SS2.SSS0.Px1.p2.4.m4.2.2.2.2.cmml">𝒢</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p2.4.m4.2b"><apply id="S4.SS2.SSS0.Px1.p2.4.m4.2.3.cmml" xref="S4.SS2.SSS0.Px1.p2.4.m4.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p2.4.m4.2.3.1.cmml" xref="S4.SS2.SSS0.Px1.p2.4.m4.2.3">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p2.4.m4.2.3.2.cmml" xref="S4.SS2.SSS0.Px1.p2.4.m4.2.3.2">𝑓</ci><list id="S4.SS2.SSS0.Px1.p2.4.m4.2.2.2.3.cmml" xref="S4.SS2.SSS0.Px1.p2.4.m4.2.2.2.4"><ci id="S4.SS2.SSS0.Px1.p2.4.m4.1.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p2.4.m4.1.1.1.1">𝜃</ci><ci id="S4.SS2.SSS0.Px1.p2.4.m4.2.2.2.2.cmml" xref="S4.SS2.SSS0.Px1.p2.4.m4.2.2.2.2">𝒢</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p2.4.m4.2c">f_{\theta,\mathcal{G}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p2.4.m4.2d">italic_f start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT</annotation></semantics></math>. We design an instruction template to utilize the LLM to extract the relations <math alttext="\{r_{k}\}_{k\in[K]}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p2.5.m5.2"><semantics id="S4.SS2.SSS0.Px1.p2.5.m5.2a"><msub id="S4.SS2.SSS0.Px1.p2.5.m5.2.2" xref="S4.SS2.SSS0.Px1.p2.5.m5.2.2.cmml"><mrow id="S4.SS2.SSS0.Px1.p2.5.m5.2.2.1.1" xref="S4.SS2.SSS0.Px1.p2.5.m5.2.2.1.2.cmml"><mo id="S4.SS2.SSS0.Px1.p2.5.m5.2.2.1.1.2" stretchy="false" xref="S4.SS2.SSS0.Px1.p2.5.m5.2.2.1.2.cmml">{</mo><msub id="S4.SS2.SSS0.Px1.p2.5.m5.2.2.1.1.1" xref="S4.SS2.SSS0.Px1.p2.5.m5.2.2.1.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p2.5.m5.2.2.1.1.1.2" xref="S4.SS2.SSS0.Px1.p2.5.m5.2.2.1.1.1.2.cmml">r</mi><mi id="S4.SS2.SSS0.Px1.p2.5.m5.2.2.1.1.1.3" xref="S4.SS2.SSS0.Px1.p2.5.m5.2.2.1.1.1.3.cmml">k</mi></msub><mo id="S4.SS2.SSS0.Px1.p2.5.m5.2.2.1.1.3" stretchy="false" xref="S4.SS2.SSS0.Px1.p2.5.m5.2.2.1.2.cmml">}</mo></mrow><mrow id="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1" xref="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.3" xref="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.3.cmml">k</mi><mo id="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.2" xref="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.2.cmml">∈</mo><mrow id="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.4.2" xref="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.4.1.cmml"><mo id="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.4.2.1" stretchy="false" xref="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.4.1.1.cmml">[</mo><mi id="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.1" xref="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.1.cmml">K</mi><mo id="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.4.2.2" stretchy="false" xref="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p2.5.m5.2b"><apply id="S4.SS2.SSS0.Px1.p2.5.m5.2.2.cmml" xref="S4.SS2.SSS0.Px1.p2.5.m5.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p2.5.m5.2.2.2.cmml" xref="S4.SS2.SSS0.Px1.p2.5.m5.2.2">subscript</csymbol><set id="S4.SS2.SSS0.Px1.p2.5.m5.2.2.1.2.cmml" xref="S4.SS2.SSS0.Px1.p2.5.m5.2.2.1.1"><apply id="S4.SS2.SSS0.Px1.p2.5.m5.2.2.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p2.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p2.5.m5.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p2.5.m5.2.2.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p2.5.m5.2.2.1.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p2.5.m5.2.2.1.1.1.2">𝑟</ci><ci id="S4.SS2.SSS0.Px1.p2.5.m5.2.2.1.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p2.5.m5.2.2.1.1.1.3">𝑘</ci></apply></set><apply id="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1"><in id="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.2"></in><ci id="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.3">𝑘</ci><apply id="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.4.1.cmml" xref="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.4.2"><csymbol cd="latexml" id="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.4.1.1.cmml" xref="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.4.2.1">delimited-[]</csymbol><ci id="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p2.5.m5.1.1.1.1">𝐾</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p2.5.m5.2c">\{r_{k}\}_{k\in[K]}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p2.5.m5.2d">{ italic_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_k ∈ [ italic_K ] end_POSTSUBSCRIPT</annotation></semantics></math> between <math alttext="e_{q}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p2.6.m6.1"><semantics id="S4.SS2.SSS0.Px1.p2.6.m6.1a"><msub id="S4.SS2.SSS0.Px1.p2.6.m6.1.1" xref="S4.SS2.SSS0.Px1.p2.6.m6.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p2.6.m6.1.1.2" xref="S4.SS2.SSS0.Px1.p2.6.m6.1.1.2.cmml">e</mi><mi id="S4.SS2.SSS0.Px1.p2.6.m6.1.1.3" xref="S4.SS2.SSS0.Px1.p2.6.m6.1.1.3.cmml">q</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p2.6.m6.1b"><apply id="S4.SS2.SSS0.Px1.p2.6.m6.1.1.cmml" xref="S4.SS2.SSS0.Px1.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p2.6.m6.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p2.6.m6.1.1">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p2.6.m6.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p2.6.m6.1.1.2">𝑒</ci><ci id="S4.SS2.SSS0.Px1.p2.6.m6.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p2.6.m6.1.1.3">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p2.6.m6.1c">e_{q}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p2.6.m6.1d">italic_e start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="e_{a}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p2.7.m7.1"><semantics id="S4.SS2.SSS0.Px1.p2.7.m7.1a"><msub id="S4.SS2.SSS0.Px1.p2.7.m7.1.1" xref="S4.SS2.SSS0.Px1.p2.7.m7.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p2.7.m7.1.1.2" xref="S4.SS2.SSS0.Px1.p2.7.m7.1.1.2.cmml">e</mi><mi id="S4.SS2.SSS0.Px1.p2.7.m7.1.1.3" xref="S4.SS2.SSS0.Px1.p2.7.m7.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p2.7.m7.1b"><apply id="S4.SS2.SSS0.Px1.p2.7.m7.1.1.cmml" xref="S4.SS2.SSS0.Px1.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p2.7.m7.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p2.7.m7.1.1">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p2.7.m7.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p2.7.m7.1.1.2">𝑒</ci><ci id="S4.SS2.SSS0.Px1.p2.7.m7.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p2.7.m7.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p2.7.m7.1c">e_{a}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p2.7.m7.1d">italic_e start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math>:</p> </div> <div class="ltx_para" id="S4.SS2.SSS0.Px1.p3"> <span class="ltx_ERROR undefined" id="S4.SS2.SSS0.Px1.p3.5">{custombox}</span> <p class="ltx_p" id="S4.SS2.SSS0.Px1.p3.4">Based on the provided query and answer, identify <span class="ltx_text" id="S4.SS2.SSS0.Px1.p3.4.1" style="color:#0000FF;">K</span> types of relationships between the subject <subject> and the object <object>, considering the context of the query and answer. <query>: <math alttext="q" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p3.1.m1.1"><semantics id="S4.SS2.SSS0.Px1.p3.1.m1.1a"><mi id="S4.SS2.SSS0.Px1.p3.1.m1.1.1" mathcolor="#0000FF" xref="S4.SS2.SSS0.Px1.p3.1.m1.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p3.1.m1.1b"><ci id="S4.SS2.SSS0.Px1.p3.1.m1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p3.1.m1.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p3.1.m1.1c">q</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p3.1.m1.1d">italic_q</annotation></semantics></math> <answer>: <math alttext="a" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p3.2.m2.1"><semantics id="S4.SS2.SSS0.Px1.p3.2.m2.1a"><mi id="S4.SS2.SSS0.Px1.p3.2.m2.1.1" mathcolor="#0000FF" xref="S4.SS2.SSS0.Px1.p3.2.m2.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p3.2.m2.1b"><ci id="S4.SS2.SSS0.Px1.p3.2.m2.1.1.cmml" xref="S4.SS2.SSS0.Px1.p3.2.m2.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p3.2.m2.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p3.2.m2.1d">italic_a</annotation></semantics></math> <subject>: <math alttext="e_{q}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p3.3.m3.1"><semantics id="S4.SS2.SSS0.Px1.p3.3.m3.1a"><msub id="S4.SS2.SSS0.Px1.p3.3.m3.1.1" xref="S4.SS2.SSS0.Px1.p3.3.m3.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p3.3.m3.1.1.2" mathcolor="#0000FF" xref="S4.SS2.SSS0.Px1.p3.3.m3.1.1.2.cmml">e</mi><mi id="S4.SS2.SSS0.Px1.p3.3.m3.1.1.3" mathcolor="#0000FF" xref="S4.SS2.SSS0.Px1.p3.3.m3.1.1.3.cmml">q</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p3.3.m3.1b"><apply id="S4.SS2.SSS0.Px1.p3.3.m3.1.1.cmml" xref="S4.SS2.SSS0.Px1.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p3.3.m3.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p3.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p3.3.m3.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p3.3.m3.1.1.2">𝑒</ci><ci id="S4.SS2.SSS0.Px1.p3.3.m3.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p3.3.m3.1.1.3">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p3.3.m3.1c">e_{q}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p3.3.m3.1d">italic_e start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT</annotation></semantics></math> <object>: <math alttext="e_{a}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p3.4.m4.1"><semantics id="S4.SS2.SSS0.Px1.p3.4.m4.1a"><msub id="S4.SS2.SSS0.Px1.p3.4.m4.1.1" xref="S4.SS2.SSS0.Px1.p3.4.m4.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p3.4.m4.1.1.2" mathcolor="#0000FF" xref="S4.SS2.SSS0.Px1.p3.4.m4.1.1.2.cmml">e</mi><mi id="S4.SS2.SSS0.Px1.p3.4.m4.1.1.3" mathcolor="#0000FF" xref="S4.SS2.SSS0.Px1.p3.4.m4.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p3.4.m4.1b"><apply id="S4.SS2.SSS0.Px1.p3.4.m4.1.1.cmml" xref="S4.SS2.SSS0.Px1.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p3.4.m4.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p3.4.m4.1.1">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p3.4.m4.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p3.4.m4.1.1.2">𝑒</ci><ci id="S4.SS2.SSS0.Px1.p3.4.m4.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p3.4.m4.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p3.4.m4.1c">e_{a}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p3.4.m4.1d">italic_e start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> <relation>: [MASK]<span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>The LLMs in this paper are causal models that do not need a [MASK] token for generation. We include a [MASK] token within the templates in the paper to delineate the starting point for the model’s generation, thereby aiding reader comprehension.</span></span></span></p> </div> <div class="ltx_para" id="S4.SS2.SSS0.Px1.p4"> <p class="ltx_p" id="S4.SS2.SSS0.Px1.p4.1">The instruction is fed into the LLM to generate <math alttext="K" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p4.1.m1.1"><semantics id="S4.SS2.SSS0.Px1.p4.1.m1.1a"><mi id="S4.SS2.SSS0.Px1.p4.1.m1.1.1" xref="S4.SS2.SSS0.Px1.p4.1.m1.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p4.1.m1.1b"><ci id="S4.SS2.SSS0.Px1.p4.1.m1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p4.1.m1.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p4.1.m1.1c">K</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p4.1.m1.1d">italic_K</annotation></semantics></math> relations formatted as</p> </div> <div class="ltx_para" id="S4.SS2.SSS0.Px1.p5"> <span class="ltx_ERROR undefined" id="S4.SS2.SSS0.Px1.p5.4">{custombox}</span> <p class="ltx_p" id="S4.SS2.SSS0.Px1.p5.3"><math alttext="r_{1}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p5.1.m1.1"><semantics id="S4.SS2.SSS0.Px1.p5.1.m1.1a"><msub id="S4.SS2.SSS0.Px1.p5.1.m1.1.1" xref="S4.SS2.SSS0.Px1.p5.1.m1.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p5.1.m1.1.1.2" xref="S4.SS2.SSS0.Px1.p5.1.m1.1.1.2.cmml">r</mi><mn id="S4.SS2.SSS0.Px1.p5.1.m1.1.1.3" xref="S4.SS2.SSS0.Px1.p5.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p5.1.m1.1b"><apply id="S4.SS2.SSS0.Px1.p5.1.m1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p5.1.m1.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p5.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p5.1.m1.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p5.1.m1.1.1.2">𝑟</ci><cn id="S4.SS2.SSS0.Px1.p5.1.m1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS0.Px1.p5.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p5.1.m1.1c">r_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p5.1.m1.1d">italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> <sep> <math alttext="r_{2}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p5.2.m2.1"><semantics id="S4.SS2.SSS0.Px1.p5.2.m2.1a"><msub id="S4.SS2.SSS0.Px1.p5.2.m2.1.1" xref="S4.SS2.SSS0.Px1.p5.2.m2.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p5.2.m2.1.1.2" xref="S4.SS2.SSS0.Px1.p5.2.m2.1.1.2.cmml">r</mi><mn id="S4.SS2.SSS0.Px1.p5.2.m2.1.1.3" xref="S4.SS2.SSS0.Px1.p5.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p5.2.m2.1b"><apply id="S4.SS2.SSS0.Px1.p5.2.m2.1.1.cmml" xref="S4.SS2.SSS0.Px1.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p5.2.m2.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p5.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p5.2.m2.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p5.2.m2.1.1.2">𝑟</ci><cn id="S4.SS2.SSS0.Px1.p5.2.m2.1.1.3.cmml" type="integer" xref="S4.SS2.SSS0.Px1.p5.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p5.2.m2.1c">r_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p5.2.m2.1d">italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> <sep> … <sep> <math alttext="r_{K}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p5.3.m3.1"><semantics id="S4.SS2.SSS0.Px1.p5.3.m3.1a"><msub id="S4.SS2.SSS0.Px1.p5.3.m3.1.1" xref="S4.SS2.SSS0.Px1.p5.3.m3.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p5.3.m3.1.1.2" xref="S4.SS2.SSS0.Px1.p5.3.m3.1.1.2.cmml">r</mi><mi id="S4.SS2.SSS0.Px1.p5.3.m3.1.1.3" xref="S4.SS2.SSS0.Px1.p5.3.m3.1.1.3.cmml">K</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p5.3.m3.1b"><apply id="S4.SS2.SSS0.Px1.p5.3.m3.1.1.cmml" xref="S4.SS2.SSS0.Px1.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p5.3.m3.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p5.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p5.3.m3.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p5.3.m3.1.1.2">𝑟</ci><ci id="S4.SS2.SSS0.Px1.p5.3.m3.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p5.3.m3.1.1.3">𝐾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p5.3.m3.1c">r_{K}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p5.3.m3.1d">italic_r start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.SSS0.Px1.p6"> <p class="ltx_p" id="S4.SS2.SSS0.Px1.p6.2">Then, we formulate <math alttext="\mathcal{H}(q,a,K)" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p6.1.m1.3"><semantics id="S4.SS2.SSS0.Px1.p6.1.m1.3a"><mrow id="S4.SS2.SSS0.Px1.p6.1.m1.3.4" xref="S4.SS2.SSS0.Px1.p6.1.m1.3.4.cmml"><mi class="ltx_font_mathcaligraphic" 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id="S4.SS2.SSS0.Px1.p6.1.m1.3b"><apply id="S4.SS2.SSS0.Px1.p6.1.m1.3.4.cmml" xref="S4.SS2.SSS0.Px1.p6.1.m1.3.4"><times id="S4.SS2.SSS0.Px1.p6.1.m1.3.4.1.cmml" xref="S4.SS2.SSS0.Px1.p6.1.m1.3.4.1"></times><ci id="S4.SS2.SSS0.Px1.p6.1.m1.3.4.2.cmml" xref="S4.SS2.SSS0.Px1.p6.1.m1.3.4.2">ℋ</ci><vector id="S4.SS2.SSS0.Px1.p6.1.m1.3.4.3.1.cmml" xref="S4.SS2.SSS0.Px1.p6.1.m1.3.4.3.2"><ci id="S4.SS2.SSS0.Px1.p6.1.m1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p6.1.m1.1.1">𝑞</ci><ci id="S4.SS2.SSS0.Px1.p6.1.m1.2.2.cmml" xref="S4.SS2.SSS0.Px1.p6.1.m1.2.2">𝑎</ci><ci id="S4.SS2.SSS0.Px1.p6.1.m1.3.3.cmml" xref="S4.SS2.SSS0.Px1.p6.1.m1.3.3">𝐾</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p6.1.m1.3c">\mathcal{H}(q,a,K)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p6.1.m1.3d">caligraphic_H ( italic_q , italic_a , italic_K )</annotation></semantics></math> as the set <math alttext="\{e_{q},r_{k},e_{a}\}_{k\in[K]}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p6.2.m2.4"><semantics id="S4.SS2.SSS0.Px1.p6.2.m2.4a"><msub id="S4.SS2.SSS0.Px1.p6.2.m2.4.4" xref="S4.SS2.SSS0.Px1.p6.2.m2.4.4.cmml"><mrow id="S4.SS2.SSS0.Px1.p6.2.m2.4.4.3.3" xref="S4.SS2.SSS0.Px1.p6.2.m2.4.4.3.4.cmml"><mo id="S4.SS2.SSS0.Px1.p6.2.m2.4.4.3.3.4" stretchy="false" xref="S4.SS2.SSS0.Px1.p6.2.m2.4.4.3.4.cmml">{</mo><msub id="S4.SS2.SSS0.Px1.p6.2.m2.2.2.1.1.1" xref="S4.SS2.SSS0.Px1.p6.2.m2.2.2.1.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p6.2.m2.2.2.1.1.1.2" xref="S4.SS2.SSS0.Px1.p6.2.m2.2.2.1.1.1.2.cmml">e</mi><mi id="S4.SS2.SSS0.Px1.p6.2.m2.2.2.1.1.1.3" xref="S4.SS2.SSS0.Px1.p6.2.m2.2.2.1.1.1.3.cmml">q</mi></msub><mo id="S4.SS2.SSS0.Px1.p6.2.m2.4.4.3.3.5" xref="S4.SS2.SSS0.Px1.p6.2.m2.4.4.3.4.cmml">,</mo><msub id="S4.SS2.SSS0.Px1.p6.2.m2.3.3.2.2.2" xref="S4.SS2.SSS0.Px1.p6.2.m2.3.3.2.2.2.cmml"><mi id="S4.SS2.SSS0.Px1.p6.2.m2.3.3.2.2.2.2" xref="S4.SS2.SSS0.Px1.p6.2.m2.3.3.2.2.2.2.cmml">r</mi><mi 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id="S4.SS2.SSS0.Px1.p6.2.m2.1.1.1.4.2.1" stretchy="false" xref="S4.SS2.SSS0.Px1.p6.2.m2.1.1.1.4.1.1.cmml">[</mo><mi id="S4.SS2.SSS0.Px1.p6.2.m2.1.1.1.1" xref="S4.SS2.SSS0.Px1.p6.2.m2.1.1.1.1.cmml">K</mi><mo id="S4.SS2.SSS0.Px1.p6.2.m2.1.1.1.4.2.2" stretchy="false" xref="S4.SS2.SSS0.Px1.p6.2.m2.1.1.1.4.1.1.cmml">]</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p6.2.m2.4b"><apply id="S4.SS2.SSS0.Px1.p6.2.m2.4.4.cmml" xref="S4.SS2.SSS0.Px1.p6.2.m2.4.4"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p6.2.m2.4.4.4.cmml" xref="S4.SS2.SSS0.Px1.p6.2.m2.4.4">subscript</csymbol><set id="S4.SS2.SSS0.Px1.p6.2.m2.4.4.3.4.cmml" xref="S4.SS2.SSS0.Px1.p6.2.m2.4.4.3.3"><apply id="S4.SS2.SSS0.Px1.p6.2.m2.2.2.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p6.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p6.2.m2.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p6.2.m2.2.2.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p6.2.m2.2.2.1.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p6.2.m2.2.2.1.1.1.2">𝑒</ci><ci id="S4.SS2.SSS0.Px1.p6.2.m2.2.2.1.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p6.2.m2.2.2.1.1.1.3">𝑞</ci></apply><apply id="S4.SS2.SSS0.Px1.p6.2.m2.3.3.2.2.2.cmml" xref="S4.SS2.SSS0.Px1.p6.2.m2.3.3.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p6.2.m2.3.3.2.2.2.1.cmml" xref="S4.SS2.SSS0.Px1.p6.2.m2.3.3.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p6.2.m2.3.3.2.2.2.2.cmml" xref="S4.SS2.SSS0.Px1.p6.2.m2.3.3.2.2.2.2">𝑟</ci><ci id="S4.SS2.SSS0.Px1.p6.2.m2.3.3.2.2.2.3.cmml" xref="S4.SS2.SSS0.Px1.p6.2.m2.3.3.2.2.2.3">𝑘</ci></apply><apply id="S4.SS2.SSS0.Px1.p6.2.m2.4.4.3.3.3.cmml" xref="S4.SS2.SSS0.Px1.p6.2.m2.4.4.3.3.3"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p6.2.m2.4.4.3.3.3.1.cmml" xref="S4.SS2.SSS0.Px1.p6.2.m2.4.4.3.3.3">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p6.2.m2.4.4.3.3.3.2.cmml" xref="S4.SS2.SSS0.Px1.p6.2.m2.4.4.3.3.3.2">𝑒</ci><ci id="S4.SS2.SSS0.Px1.p6.2.m2.4.4.3.3.3.3.cmml" xref="S4.SS2.SSS0.Px1.p6.2.m2.4.4.3.3.3.3">𝑎</ci></apply></set><apply id="S4.SS2.SSS0.Px1.p6.2.m2.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p6.2.m2.1.1.1"><in id="S4.SS2.SSS0.Px1.p6.2.m2.1.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p6.2.m2.1.1.1.2"></in><ci id="S4.SS2.SSS0.Px1.p6.2.m2.1.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p6.2.m2.1.1.1.3">𝑘</ci><apply id="S4.SS2.SSS0.Px1.p6.2.m2.1.1.1.4.1.cmml" xref="S4.SS2.SSS0.Px1.p6.2.m2.1.1.1.4.2"><csymbol cd="latexml" id="S4.SS2.SSS0.Px1.p6.2.m2.1.1.1.4.1.1.cmml" xref="S4.SS2.SSS0.Px1.p6.2.m2.1.1.1.4.2.1">delimited-[]</csymbol><ci id="S4.SS2.SSS0.Px1.p6.2.m2.1.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p6.2.m2.1.1.1.1">𝐾</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p6.2.m2.4c">\{e_{q},r_{k},e_{a}\}_{k\in[K]}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p6.2.m2.4d">{ italic_e start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_e start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_k ∈ [ italic_K ] end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.SSS0.Px1.p7"> <p class="ltx_p" id="S4.SS2.SSS0.Px1.p7.2">To calculate the retrieval probability <math alttext="P_{\theta,\mathcal{G}}\left(z|q\right)" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p7.1.m1.3"><semantics id="S4.SS2.SSS0.Px1.p7.1.m1.3a"><mrow id="S4.SS2.SSS0.Px1.p7.1.m1.3.3" xref="S4.SS2.SSS0.Px1.p7.1.m1.3.3.cmml"><msub id="S4.SS2.SSS0.Px1.p7.1.m1.3.3.3" xref="S4.SS2.SSS0.Px1.p7.1.m1.3.3.3.cmml"><mi id="S4.SS2.SSS0.Px1.p7.1.m1.3.3.3.2" xref="S4.SS2.SSS0.Px1.p7.1.m1.3.3.3.2.cmml">P</mi><mrow id="S4.SS2.SSS0.Px1.p7.1.m1.2.2.2.4" xref="S4.SS2.SSS0.Px1.p7.1.m1.2.2.2.3.cmml"><mi id="S4.SS2.SSS0.Px1.p7.1.m1.1.1.1.1" xref="S4.SS2.SSS0.Px1.p7.1.m1.1.1.1.1.cmml">θ</mi><mo id="S4.SS2.SSS0.Px1.p7.1.m1.2.2.2.4.1" xref="S4.SS2.SSS0.Px1.p7.1.m1.2.2.2.3.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.SSS0.Px1.p7.1.m1.2.2.2.2" xref="S4.SS2.SSS0.Px1.p7.1.m1.2.2.2.2.cmml">𝒢</mi></mrow></msub><mo id="S4.SS2.SSS0.Px1.p7.1.m1.3.3.2" xref="S4.SS2.SSS0.Px1.p7.1.m1.3.3.2.cmml">⁢</mo><mrow id="S4.SS2.SSS0.Px1.p7.1.m1.3.3.1.1" xref="S4.SS2.SSS0.Px1.p7.1.m1.3.3.1.1.1.cmml"><mo id="S4.SS2.SSS0.Px1.p7.1.m1.3.3.1.1.2" xref="S4.SS2.SSS0.Px1.p7.1.m1.3.3.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS0.Px1.p7.1.m1.3.3.1.1.1" xref="S4.SS2.SSS0.Px1.p7.1.m1.3.3.1.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p7.1.m1.3.3.1.1.1.2" xref="S4.SS2.SSS0.Px1.p7.1.m1.3.3.1.1.1.2.cmml">z</mi><mo fence="false" id="S4.SS2.SSS0.Px1.p7.1.m1.3.3.1.1.1.1" xref="S4.SS2.SSS0.Px1.p7.1.m1.3.3.1.1.1.1.cmml">|</mo><mi id="S4.SS2.SSS0.Px1.p7.1.m1.3.3.1.1.1.3" xref="S4.SS2.SSS0.Px1.p7.1.m1.3.3.1.1.1.3.cmml">q</mi></mrow><mo id="S4.SS2.SSS0.Px1.p7.1.m1.3.3.1.1.3" xref="S4.SS2.SSS0.Px1.p7.1.m1.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p7.1.m1.3b"><apply id="S4.SS2.SSS0.Px1.p7.1.m1.3.3.cmml" xref="S4.SS2.SSS0.Px1.p7.1.m1.3.3"><times id="S4.SS2.SSS0.Px1.p7.1.m1.3.3.2.cmml" xref="S4.SS2.SSS0.Px1.p7.1.m1.3.3.2"></times><apply id="S4.SS2.SSS0.Px1.p7.1.m1.3.3.3.cmml" xref="S4.SS2.SSS0.Px1.p7.1.m1.3.3.3"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p7.1.m1.3.3.3.1.cmml" xref="S4.SS2.SSS0.Px1.p7.1.m1.3.3.3">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p7.1.m1.3.3.3.2.cmml" xref="S4.SS2.SSS0.Px1.p7.1.m1.3.3.3.2">𝑃</ci><list id="S4.SS2.SSS0.Px1.p7.1.m1.2.2.2.3.cmml" xref="S4.SS2.SSS0.Px1.p7.1.m1.2.2.2.4"><ci id="S4.SS2.SSS0.Px1.p7.1.m1.1.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p7.1.m1.1.1.1.1">𝜃</ci><ci id="S4.SS2.SSS0.Px1.p7.1.m1.2.2.2.2.cmml" xref="S4.SS2.SSS0.Px1.p7.1.m1.2.2.2.2">𝒢</ci></list></apply><apply id="S4.SS2.SSS0.Px1.p7.1.m1.3.3.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p7.1.m1.3.3.1.1"><csymbol cd="latexml" id="S4.SS2.SSS0.Px1.p7.1.m1.3.3.1.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p7.1.m1.3.3.1.1.1.1">conditional</csymbol><ci id="S4.SS2.SSS0.Px1.p7.1.m1.3.3.1.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p7.1.m1.3.3.1.1.1.2">𝑧</ci><ci id="S4.SS2.SSS0.Px1.p7.1.m1.3.3.1.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p7.1.m1.3.3.1.1.1.3">𝑞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p7.1.m1.3c">P_{\theta,\mathcal{G}}\left(z|q\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p7.1.m1.3d">italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_z | italic_q )</annotation></semantics></math>, we design an instruction template <math alttext="\mathcal{T}_{\text{retrieve}}(\cdot)" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p7.2.m2.1"><semantics id="S4.SS2.SSS0.Px1.p7.2.m2.1a"><mrow id="S4.SS2.SSS0.Px1.p7.2.m2.1.2" xref="S4.SS2.SSS0.Px1.p7.2.m2.1.2.cmml"><msub id="S4.SS2.SSS0.Px1.p7.2.m2.1.2.2" xref="S4.SS2.SSS0.Px1.p7.2.m2.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.SSS0.Px1.p7.2.m2.1.2.2.2" xref="S4.SS2.SSS0.Px1.p7.2.m2.1.2.2.2.cmml">𝒯</mi><mtext id="S4.SS2.SSS0.Px1.p7.2.m2.1.2.2.3" xref="S4.SS2.SSS0.Px1.p7.2.m2.1.2.2.3a.cmml">retrieve</mtext></msub><mo id="S4.SS2.SSS0.Px1.p7.2.m2.1.2.1" xref="S4.SS2.SSS0.Px1.p7.2.m2.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS0.Px1.p7.2.m2.1.2.3.2" xref="S4.SS2.SSS0.Px1.p7.2.m2.1.2.cmml"><mo id="S4.SS2.SSS0.Px1.p7.2.m2.1.2.3.2.1" stretchy="false" xref="S4.SS2.SSS0.Px1.p7.2.m2.1.2.cmml">(</mo><mo id="S4.SS2.SSS0.Px1.p7.2.m2.1.1" lspace="0em" rspace="0em" xref="S4.SS2.SSS0.Px1.p7.2.m2.1.1.cmml">⋅</mo><mo id="S4.SS2.SSS0.Px1.p7.2.m2.1.2.3.2.2" stretchy="false" xref="S4.SS2.SSS0.Px1.p7.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p7.2.m2.1b"><apply id="S4.SS2.SSS0.Px1.p7.2.m2.1.2.cmml" xref="S4.SS2.SSS0.Px1.p7.2.m2.1.2"><times id="S4.SS2.SSS0.Px1.p7.2.m2.1.2.1.cmml" xref="S4.SS2.SSS0.Px1.p7.2.m2.1.2.1"></times><apply id="S4.SS2.SSS0.Px1.p7.2.m2.1.2.2.cmml" xref="S4.SS2.SSS0.Px1.p7.2.m2.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p7.2.m2.1.2.2.1.cmml" xref="S4.SS2.SSS0.Px1.p7.2.m2.1.2.2">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p7.2.m2.1.2.2.2.cmml" xref="S4.SS2.SSS0.Px1.p7.2.m2.1.2.2.2">𝒯</ci><ci id="S4.SS2.SSS0.Px1.p7.2.m2.1.2.2.3a.cmml" xref="S4.SS2.SSS0.Px1.p7.2.m2.1.2.2.3"><mtext id="S4.SS2.SSS0.Px1.p7.2.m2.1.2.2.3.cmml" mathsize="70%" xref="S4.SS2.SSS0.Px1.p7.2.m2.1.2.2.3">retrieve</mtext></ci></apply><ci id="S4.SS2.SSS0.Px1.p7.2.m2.1.1.cmml" xref="S4.SS2.SSS0.Px1.p7.2.m2.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p7.2.m2.1c">\mathcal{T}_{\text{retrieve}}(\cdot)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p7.2.m2.1d">caligraphic_T start_POSTSUBSCRIPT retrieve end_POSTSUBSCRIPT ( ⋅ )</annotation></semantics></math> to instruct the LLM to predict what kind of the relationship is needed to answer this query:</p> </div> <div class="ltx_para" id="S4.SS2.SSS0.Px1.p8"> <span class="ltx_ERROR undefined" id="S4.SS2.SSS0.Px1.p8.3">{custombox}</span> <p class="ltx_p" id="S4.SS2.SSS0.Px1.p8.2">To answer the query: <math alttext="q" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p8.1.m1.1"><semantics id="S4.SS2.SSS0.Px1.p8.1.m1.1a"><mi id="S4.SS2.SSS0.Px1.p8.1.m1.1.1" mathcolor="#0000FF" xref="S4.SS2.SSS0.Px1.p8.1.m1.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p8.1.m1.1b"><ci id="S4.SS2.SSS0.Px1.p8.1.m1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p8.1.m1.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p8.1.m1.1c">q</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p8.1.m1.1d">italic_q</annotation></semantics></math>, I need information <math alttext="e_{q}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p8.2.m2.1"><semantics id="S4.SS2.SSS0.Px1.p8.2.m2.1a"><msub id="S4.SS2.SSS0.Px1.p8.2.m2.1.1" xref="S4.SS2.SSS0.Px1.p8.2.m2.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p8.2.m2.1.1.2" mathcolor="#0000FF" xref="S4.SS2.SSS0.Px1.p8.2.m2.1.1.2.cmml">e</mi><mi id="S4.SS2.SSS0.Px1.p8.2.m2.1.1.3" mathcolor="#0000FF" xref="S4.SS2.SSS0.Px1.p8.2.m2.1.1.3.cmml">q</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p8.2.m2.1b"><apply id="S4.SS2.SSS0.Px1.p8.2.m2.1.1.cmml" xref="S4.SS2.SSS0.Px1.p8.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p8.2.m2.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p8.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p8.2.m2.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p8.2.m2.1.1.2">𝑒</ci><ci id="S4.SS2.SSS0.Px1.p8.2.m2.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p8.2.m2.1.1.3">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p8.2.m2.1c">e_{q}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p8.2.m2.1d">italic_e start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT</annotation></semantics></math> [MASK]</p> </div> <div class="ltx_para" id="S4.SS2.SSS0.Px1.p9"> <p class="ltx_p" id="S4.SS2.SSS0.Px1.p9.3">We calculate the probability of a specific relation <math alttext="r" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p9.1.m1.1"><semantics id="S4.SS2.SSS0.Px1.p9.1.m1.1a"><mi id="S4.SS2.SSS0.Px1.p9.1.m1.1.1" xref="S4.SS2.SSS0.Px1.p9.1.m1.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p9.1.m1.1b"><ci id="S4.SS2.SSS0.Px1.p9.1.m1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p9.1.m1.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p9.1.m1.1c">r</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p9.1.m1.1d">italic_r</annotation></semantics></math> as <math alttext="P_{\theta}\left(r|\mathcal{T}_{\text{retrieve}}\left(q\right)\right)" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p9.2.m2.2"><semantics id="S4.SS2.SSS0.Px1.p9.2.m2.2a"><mrow 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xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.cmml"><msub id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.2" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.2.2" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.2.2.cmml">𝒯</mi><mtext id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.2.3" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.2.3a.cmml">retrieve</mtext></msub><mo id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.1" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.1.cmml">⁢</mo><mrow id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.3.2" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.cmml"><mo id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.3.2.1" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.cmml">(</mo><mi id="S4.SS2.SSS0.Px1.p9.2.m2.1.1" xref="S4.SS2.SSS0.Px1.p9.2.m2.1.1.cmml">q</mi><mo id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.3.2.2" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.3" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p9.2.m2.2b"><apply id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.cmml" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2"><times id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.2.cmml" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.2"></times><apply id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.3.cmml" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.3.1.cmml" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.3">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.3.2.cmml" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.3.2">𝑃</ci><ci id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.3.3.cmml" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.3.3">𝜃</ci></apply><apply id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1"><csymbol cd="latexml" id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.1">conditional</csymbol><ci id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.2">𝑟</ci><apply id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3"><times id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.1.cmml" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.1"></times><apply id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.2.cmml" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.2.1.cmml" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.2">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.2.2.cmml" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.2.2">𝒯</ci><ci id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.2.3a.cmml" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.2.3"><mtext id="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.2.3.cmml" mathsize="70%" xref="S4.SS2.SSS0.Px1.p9.2.m2.2.2.1.1.1.3.2.3">retrieve</mtext></ci></apply><ci id="S4.SS2.SSS0.Px1.p9.2.m2.1.1.cmml" xref="S4.SS2.SSS0.Px1.p9.2.m2.1.1">𝑞</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p9.2.m2.2c">P_{\theta}\left(r|\mathcal{T}_{\text{retrieve}}\left(q\right)\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p9.2.m2.2d">italic_P start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( italic_r | caligraphic_T start_POSTSUBSCRIPT retrieve end_POSTSUBSCRIPT ( italic_q ) )</annotation></semantics></math>. Then, the retrieval probability of a knowledge triple <math alttext="z=\left(e,r,e^{\prime}\right)" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p9.3.m3.3"><semantics id="S4.SS2.SSS0.Px1.p9.3.m3.3a"><mrow id="S4.SS2.SSS0.Px1.p9.3.m3.3.3" xref="S4.SS2.SSS0.Px1.p9.3.m3.3.3.cmml"><mi id="S4.SS2.SSS0.Px1.p9.3.m3.3.3.3" xref="S4.SS2.SSS0.Px1.p9.3.m3.3.3.3.cmml">z</mi><mo id="S4.SS2.SSS0.Px1.p9.3.m3.3.3.2" xref="S4.SS2.SSS0.Px1.p9.3.m3.3.3.2.cmml">=</mo><mrow id="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.1" xref="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.2.cmml"><mo id="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.1.2" xref="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.2.cmml">(</mo><mi id="S4.SS2.SSS0.Px1.p9.3.m3.1.1" xref="S4.SS2.SSS0.Px1.p9.3.m3.1.1.cmml">e</mi><mo id="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.1.3" xref="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.2.cmml">,</mo><mi id="S4.SS2.SSS0.Px1.p9.3.m3.2.2" xref="S4.SS2.SSS0.Px1.p9.3.m3.2.2.cmml">r</mi><mo id="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.1.4" xref="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.2.cmml">,</mo><msup id="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.1.1" xref="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.1.1.2" xref="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.1.1.2.cmml">e</mi><mo id="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.1.1.3" xref="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.1.1.3.cmml">′</mo></msup><mo id="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.1.5" xref="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p9.3.m3.3b"><apply id="S4.SS2.SSS0.Px1.p9.3.m3.3.3.cmml" xref="S4.SS2.SSS0.Px1.p9.3.m3.3.3"><eq id="S4.SS2.SSS0.Px1.p9.3.m3.3.3.2.cmml" xref="S4.SS2.SSS0.Px1.p9.3.m3.3.3.2"></eq><ci id="S4.SS2.SSS0.Px1.p9.3.m3.3.3.3.cmml" xref="S4.SS2.SSS0.Px1.p9.3.m3.3.3.3">𝑧</ci><vector id="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.2.cmml" xref="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.1"><ci id="S4.SS2.SSS0.Px1.p9.3.m3.1.1.cmml" xref="S4.SS2.SSS0.Px1.p9.3.m3.1.1">𝑒</ci><ci id="S4.SS2.SSS0.Px1.p9.3.m3.2.2.cmml" xref="S4.SS2.SSS0.Px1.p9.3.m3.2.2">𝑟</ci><apply id="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.1.1.2">𝑒</ci><ci id="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p9.3.m3.3.3.1.1.1.3">′</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p9.3.m3.3c">z=\left(e,r,e^{\prime}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p9.3.m3.3d">italic_z = ( italic_e , italic_r , italic_e start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> is derived as</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="P_{\theta,\mathcal{G}}\left(z|q\right)=P_{\theta,\mathcal{G}}\left(\left(e,r,e% ^{\prime}\right)|q\right)=\begin{cases}\frac{P_{\theta}\left(r|\mathcal{T}_{% \text{retrieve}}\left(q\right)\right)}{\sum_{z\in\mathcal{G}}P_{\theta,% \mathcal{G}}\left(z|q\right)}&\text{if }e=e_{q}\text{ and }z\in\mathcal{G},\\ 0&\text{else}.\end{cases}" class="ltx_Math" display="block" id="S4.E5.m1.12"><semantics id="S4.E5.m1.12a"><mrow id="S4.E5.m1.12.12" xref="S4.E5.m1.12.12.cmml"><mrow id="S4.E5.m1.11.11.1" xref="S4.E5.m1.11.11.1.cmml"><msub id="S4.E5.m1.11.11.1.3" xref="S4.E5.m1.11.11.1.3.cmml"><mi id="S4.E5.m1.11.11.1.3.2" mathsize="90%" xref="S4.E5.m1.11.11.1.3.2.cmml">P</mi><mrow id="S4.E5.m1.6.6.2.4" xref="S4.E5.m1.6.6.2.3.cmml"><mi id="S4.E5.m1.5.5.1.1" mathsize="90%" xref="S4.E5.m1.5.5.1.1.cmml">θ</mi><mo id="S4.E5.m1.6.6.2.4.1" mathsize="90%" xref="S4.E5.m1.6.6.2.3.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S4.E5.m1.6.6.2.2" mathsize="90%" xref="S4.E5.m1.6.6.2.2.cmml">𝒢</mi></mrow></msub><mo id="S4.E5.m1.11.11.1.2" xref="S4.E5.m1.11.11.1.2.cmml">⁢</mo><mrow id="S4.E5.m1.11.11.1.1.1" xref="S4.E5.m1.11.11.1.1.1.1.cmml"><mo id="S4.E5.m1.11.11.1.1.1.2" xref="S4.E5.m1.11.11.1.1.1.1.cmml">(</mo><mrow id="S4.E5.m1.11.11.1.1.1.1" xref="S4.E5.m1.11.11.1.1.1.1.cmml"><mi id="S4.E5.m1.11.11.1.1.1.1.2" mathsize="90%" xref="S4.E5.m1.11.11.1.1.1.1.2.cmml">z</mi><mo fence="false" id="S4.E5.m1.11.11.1.1.1.1.1" mathsize="90%" xref="S4.E5.m1.11.11.1.1.1.1.1.cmml">|</mo><mi id="S4.E5.m1.11.11.1.1.1.1.3" mathsize="90%" xref="S4.E5.m1.11.11.1.1.1.1.3.cmml">q</mi></mrow><mo id="S4.E5.m1.11.11.1.1.1.3" xref="S4.E5.m1.11.11.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E5.m1.12.12.4" mathsize="90%" xref="S4.E5.m1.12.12.4.cmml">=</mo><mrow id="S4.E5.m1.12.12.2" xref="S4.E5.m1.12.12.2.cmml"><msub id="S4.E5.m1.12.12.2.3" xref="S4.E5.m1.12.12.2.3.cmml"><mi id="S4.E5.m1.12.12.2.3.2" mathsize="90%" xref="S4.E5.m1.12.12.2.3.2.cmml">P</mi><mrow id="S4.E5.m1.8.8.2.4" xref="S4.E5.m1.8.8.2.3.cmml"><mi id="S4.E5.m1.7.7.1.1" mathsize="90%" xref="S4.E5.m1.7.7.1.1.cmml">θ</mi><mo id="S4.E5.m1.8.8.2.4.1" mathsize="90%" xref="S4.E5.m1.8.8.2.3.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S4.E5.m1.8.8.2.2" mathsize="90%" xref="S4.E5.m1.8.8.2.2.cmml">𝒢</mi></mrow></msub><mo id="S4.E5.m1.12.12.2.2" xref="S4.E5.m1.12.12.2.2.cmml">⁢</mo><mrow id="S4.E5.m1.12.12.2.1.1" xref="S4.E5.m1.12.12.2.1.1.1.cmml"><mo id="S4.E5.m1.12.12.2.1.1.2" xref="S4.E5.m1.12.12.2.1.1.1.cmml">(</mo><mrow id="S4.E5.m1.12.12.2.1.1.1" xref="S4.E5.m1.12.12.2.1.1.1.cmml"><mrow id="S4.E5.m1.12.12.2.1.1.1.1.1" xref="S4.E5.m1.12.12.2.1.1.1.1.2.cmml"><mo id="S4.E5.m1.12.12.2.1.1.1.1.1.2" xref="S4.E5.m1.12.12.2.1.1.1.1.2.cmml">(</mo><mi id="S4.E5.m1.9.9" mathsize="90%" xref="S4.E5.m1.9.9.cmml">e</mi><mo id="S4.E5.m1.12.12.2.1.1.1.1.1.3" mathsize="90%" xref="S4.E5.m1.12.12.2.1.1.1.1.2.cmml">,</mo><mi id="S4.E5.m1.10.10" mathsize="90%" xref="S4.E5.m1.10.10.cmml">r</mi><mo id="S4.E5.m1.12.12.2.1.1.1.1.1.4" mathsize="90%" xref="S4.E5.m1.12.12.2.1.1.1.1.2.cmml">,</mo><msup id="S4.E5.m1.12.12.2.1.1.1.1.1.1" xref="S4.E5.m1.12.12.2.1.1.1.1.1.1.cmml"><mi id="S4.E5.m1.12.12.2.1.1.1.1.1.1.2" mathsize="90%" xref="S4.E5.m1.12.12.2.1.1.1.1.1.1.2.cmml">e</mi><mo id="S4.E5.m1.12.12.2.1.1.1.1.1.1.3" mathsize="90%" xref="S4.E5.m1.12.12.2.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.E5.m1.12.12.2.1.1.1.1.1.5" xref="S4.E5.m1.12.12.2.1.1.1.1.2.cmml">)</mo></mrow><mo fence="false" id="S4.E5.m1.12.12.2.1.1.1.2" mathsize="90%" xref="S4.E5.m1.12.12.2.1.1.1.2.cmml">|</mo><mi id="S4.E5.m1.12.12.2.1.1.1.3" mathsize="90%" xref="S4.E5.m1.12.12.2.1.1.1.3.cmml">q</mi></mrow><mo id="S4.E5.m1.12.12.2.1.1.3" xref="S4.E5.m1.12.12.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E5.m1.12.12.5" mathsize="90%" xref="S4.E5.m1.12.12.5.cmml">=</mo><mrow id="S4.E5.m1.4.4" xref="S4.E5.m1.12.12.6.1.cmml"><mo id="S4.E5.m1.4.4.5" xref="S4.E5.m1.12.12.6.1.1.cmml">{</mo><mtable columnspacing="5pt" displaystyle="true" id="S4.E5.m1.4.4.4" rowspacing="0pt" xref="S4.E5.m1.12.12.6.1.cmml"><mtr id="S4.E5.m1.4.4.4a" xref="S4.E5.m1.12.12.6.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S4.E5.m1.4.4.4b" xref="S4.E5.m1.12.12.6.1.cmml"><mstyle displaystyle="false" id="S4.E5.m1.1.1.1.1.1.1" xref="S4.E5.m1.1.1.1.1.1.1.cmml"><mfrac id="S4.E5.m1.1.1.1.1.1.1a" xref="S4.E5.m1.1.1.1.1.1.1.cmml"><mrow id="S4.E5.m1.1.1.1.1.1.1.2.2" xref="S4.E5.m1.1.1.1.1.1.1.2.2.cmml"><msub id="S4.E5.m1.1.1.1.1.1.1.2.2.4" xref="S4.E5.m1.1.1.1.1.1.1.2.2.4.cmml"><mi id="S4.E5.m1.1.1.1.1.1.1.2.2.4.2" mathsize="90%" xref="S4.E5.m1.1.1.1.1.1.1.2.2.4.2.cmml">P</mi><mi id="S4.E5.m1.1.1.1.1.1.1.2.2.4.3" mathsize="90%" xref="S4.E5.m1.1.1.1.1.1.1.2.2.4.3.cmml">θ</mi></msub><mo id="S4.E5.m1.1.1.1.1.1.1.2.2.3" xref="S4.E5.m1.1.1.1.1.1.1.2.2.3.cmml">⁢</mo><mrow id="S4.E5.m1.1.1.1.1.1.1.2.2.2.1" xref="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.cmml"><mo id="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.2" xref="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.cmml">(</mo><mrow id="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1" xref="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.cmml"><mi id="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.2" mathsize="90%" xref="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.2.cmml">r</mi><mo fence="false" id="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.1" mathsize="90%" xref="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.1.cmml">|</mo><mrow id="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.3" xref="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.3.cmml"><msub id="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.3.2" xref="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.3.2.2" mathsize="90%" xref="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.3.2.2.cmml">𝒯</mi><mtext id="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.3.2.3" mathsize="90%" xref="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.3.2.3a.cmml">retrieve</mtext></msub><mo id="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.3.1" xref="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.3.1.cmml">⁢</mo><mrow id="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.3.3.2" xref="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.3.cmml"><mo id="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.3.3.2.1" xref="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.3.cmml">(</mo><mi id="S4.E5.m1.1.1.1.1.1.1.1.1.1" mathsize="90%" xref="S4.E5.m1.1.1.1.1.1.1.1.1.1.cmml">q</mi><mo id="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.3.3.2.2" xref="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.3" xref="S4.E5.m1.1.1.1.1.1.1.2.2.2.1.1.cmml">)</mo></mrow></mrow><mrow id="S4.E5.m1.1.1.1.1.1.1.5.5" xref="S4.E5.m1.1.1.1.1.1.1.5.5.cmml"><mstyle displaystyle="false" id="S4.E5.m1.1.1.1.1.1.1.5.5.4" xref="S4.E5.m1.1.1.1.1.1.1.5.5.4.cmml"><msub id="S4.E5.m1.1.1.1.1.1.1.5.5.4a" xref="S4.E5.m1.1.1.1.1.1.1.5.5.4.cmml"><mo id="S4.E5.m1.1.1.1.1.1.1.5.5.4.2" maxsize="63%" minsize="63%" stretchy="true" xref="S4.E5.m1.1.1.1.1.1.1.5.5.4.2.cmml">∑</mo><mrow id="S4.E5.m1.1.1.1.1.1.1.5.5.4.3" xref="S4.E5.m1.1.1.1.1.1.1.5.5.4.3.cmml"><mi id="S4.E5.m1.1.1.1.1.1.1.5.5.4.3.2" mathsize="90%" xref="S4.E5.m1.1.1.1.1.1.1.5.5.4.3.2.cmml">z</mi><mo id="S4.E5.m1.1.1.1.1.1.1.5.5.4.3.1" mathsize="90%" xref="S4.E5.m1.1.1.1.1.1.1.5.5.4.3.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S4.E5.m1.1.1.1.1.1.1.5.5.4.3.3" mathsize="90%" xref="S4.E5.m1.1.1.1.1.1.1.5.5.4.3.3.cmml">𝒢</mi></mrow></msub></mstyle><mrow id="S4.E5.m1.1.1.1.1.1.1.5.5.3" xref="S4.E5.m1.1.1.1.1.1.1.5.5.3.cmml"><msub id="S4.E5.m1.1.1.1.1.1.1.5.5.3.3" xref="S4.E5.m1.1.1.1.1.1.1.5.5.3.3.cmml"><mi id="S4.E5.m1.1.1.1.1.1.1.5.5.3.3.2" mathsize="90%" xref="S4.E5.m1.1.1.1.1.1.1.5.5.3.3.2.cmml">P</mi><mrow id="S4.E5.m1.1.1.1.1.1.1.4.4.2.2.4" xref="S4.E5.m1.1.1.1.1.1.1.4.4.2.2.3.cmml"><mi id="S4.E5.m1.1.1.1.1.1.1.3.3.1.1.1" mathsize="90%" xref="S4.E5.m1.1.1.1.1.1.1.3.3.1.1.1.cmml">θ</mi><mo id="S4.E5.m1.1.1.1.1.1.1.4.4.2.2.4.1" mathsize="90%" xref="S4.E5.m1.1.1.1.1.1.1.4.4.2.2.3.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S4.E5.m1.1.1.1.1.1.1.4.4.2.2.2" 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0&\text{else}.\end{cases}</annotation><annotation encoding="application/x-llamapun" id="S4.E5.m1.12d">italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_z | italic_q ) = italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( ( italic_e , italic_r , italic_e start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) | italic_q ) = { start_ROW start_CELL divide start_ARG italic_P start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( italic_r | caligraphic_T start_POSTSUBSCRIPT retrieve end_POSTSUBSCRIPT ( italic_q ) ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_z ∈ caligraphic_G end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_z | italic_q ) end_ARG end_CELL start_CELL if italic_e = italic_e start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT and italic_z ∈ caligraphic_G , end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL else . 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> <p class="ltx_p" id="S4.SS2.SSS0.Px1.p9.5">Thus, only the knowledge triples in <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p9.4.m1.1"><semantics id="S4.SS2.SSS0.Px1.p9.4.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.SSS0.Px1.p9.4.m1.1.1" xref="S4.SS2.SSS0.Px1.p9.4.m1.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p9.4.m1.1b"><ci id="S4.SS2.SSS0.Px1.p9.4.m1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p9.4.m1.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p9.4.m1.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p9.4.m1.1d">caligraphic_G</annotation></semantics></math> that start from <math alttext="e_{q}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px1.p9.5.m2.1"><semantics id="S4.SS2.SSS0.Px1.p9.5.m2.1a"><msub id="S4.SS2.SSS0.Px1.p9.5.m2.1.1" xref="S4.SS2.SSS0.Px1.p9.5.m2.1.1.cmml"><mi id="S4.SS2.SSS0.Px1.p9.5.m2.1.1.2" xref="S4.SS2.SSS0.Px1.p9.5.m2.1.1.2.cmml">e</mi><mi id="S4.SS2.SSS0.Px1.p9.5.m2.1.1.3" xref="S4.SS2.SSS0.Px1.p9.5.m2.1.1.3.cmml">q</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px1.p9.5.m2.1b"><apply id="S4.SS2.SSS0.Px1.p9.5.m2.1.1.cmml" xref="S4.SS2.SSS0.Px1.p9.5.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px1.p9.5.m2.1.1.1.cmml" xref="S4.SS2.SSS0.Px1.p9.5.m2.1.1">subscript</csymbol><ci id="S4.SS2.SSS0.Px1.p9.5.m2.1.1.2.cmml" xref="S4.SS2.SSS0.Px1.p9.5.m2.1.1.2">𝑒</ci><ci id="S4.SS2.SSS0.Px1.p9.5.m2.1.1.3.cmml" xref="S4.SS2.SSS0.Px1.p9.5.m2.1.1.3">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px1.p9.5.m2.1c">e_{q}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px1.p9.5.m2.1d">italic_e start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT</annotation></semantics></math> have a retrieval probability larger than 0.</p> </div> </section> <section class="ltx_paragraph" id="S4.SS2.SSS0.Px2"> <h5 class="ltx_title ltx_title_paragraph">Knowledge-enhanced Reasoning</h5> <div class="ltx_para" id="S4.SS2.SSS0.Px2.p1"> <p class="ltx_p" id="S4.SS2.SSS0.Px2.p1.1">The goal of the knowledge-enhanced reasoning term is to finetune the KG such that the retrieved personalized triples can most encourage the LLM to generate the correct answer. With the approximated posterior distribution <math alttext="Q(z)" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px2.p1.1.m1.1"><semantics id="S4.SS2.SSS0.Px2.p1.1.m1.1a"><mrow id="S4.SS2.SSS0.Px2.p1.1.m1.1.2" xref="S4.SS2.SSS0.Px2.p1.1.m1.1.2.cmml"><mi id="S4.SS2.SSS0.Px2.p1.1.m1.1.2.2" xref="S4.SS2.SSS0.Px2.p1.1.m1.1.2.2.cmml">Q</mi><mo id="S4.SS2.SSS0.Px2.p1.1.m1.1.2.1" xref="S4.SS2.SSS0.Px2.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS0.Px2.p1.1.m1.1.2.3.2" xref="S4.SS2.SSS0.Px2.p1.1.m1.1.2.cmml"><mo id="S4.SS2.SSS0.Px2.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S4.SS2.SSS0.Px2.p1.1.m1.1.2.cmml">(</mo><mi id="S4.SS2.SSS0.Px2.p1.1.m1.1.1" xref="S4.SS2.SSS0.Px2.p1.1.m1.1.1.cmml">z</mi><mo id="S4.SS2.SSS0.Px2.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S4.SS2.SSS0.Px2.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px2.p1.1.m1.1b"><apply id="S4.SS2.SSS0.Px2.p1.1.m1.1.2.cmml" xref="S4.SS2.SSS0.Px2.p1.1.m1.1.2"><times id="S4.SS2.SSS0.Px2.p1.1.m1.1.2.1.cmml" xref="S4.SS2.SSS0.Px2.p1.1.m1.1.2.1"></times><ci id="S4.SS2.SSS0.Px2.p1.1.m1.1.2.2.cmml" xref="S4.SS2.SSS0.Px2.p1.1.m1.1.2.2">𝑄</ci><ci id="S4.SS2.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S4.SS2.SSS0.Px2.p1.1.m1.1.1">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px2.p1.1.m1.1c">Q(z)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px2.p1.1.m1.1d">italic_Q ( italic_z )</annotation></semantics></math> in <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S4.E3" title="In Knowledge Retrieval ‣ 4.2 Objective Formulation ‣ 4 Method ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_tag">Equation</span> <span class="ltx_text ltx_ref_tag">3</span></a>, the reasoning loss term can be formulated as</p> <table class="ltx_equation ltx_eqn_table" id="S4.E6"> <tbody><tr class="ltx_equation ltx_eqn_row 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encoding="application/x-llamapun" id="S4.E6.m1.13d">caligraphic_L start_POSTSUBSCRIPT reasoning end_POSTSUBSCRIPT = - blackboard_E start_POSTSUBSCRIPT italic_z ∼ italic_Q ( italic_z ) end_POSTSUBSCRIPT [ roman_log italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_a | italic_q , italic_z ) ] = - divide start_ARG 1 end_ARG start_ARG italic_K end_ARG ∑ start_POSTSUBSCRIPT italic_z ∈ caligraphic_H ( italic_q , italic_a , italic_K ) end_POSTSUBSCRIPT roman_log italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_a | italic_q , italic_z ) .</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> <p class="ltx_p" id="S4.SS2.SSS0.Px2.p1.4">We design an instruction template <math alttext="\mathcal{T}_{\text{reasoning}}" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px2.p1.2.m1.1"><semantics id="S4.SS2.SSS0.Px2.p1.2.m1.1a"><msub id="S4.SS2.SSS0.Px2.p1.2.m1.1.1" xref="S4.SS2.SSS0.Px2.p1.2.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.SSS0.Px2.p1.2.m1.1.1.2" xref="S4.SS2.SSS0.Px2.p1.2.m1.1.1.2.cmml">𝒯</mi><mtext id="S4.SS2.SSS0.Px2.p1.2.m1.1.1.3" xref="S4.SS2.SSS0.Px2.p1.2.m1.1.1.3a.cmml">reasoning</mtext></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px2.p1.2.m1.1b"><apply id="S4.SS2.SSS0.Px2.p1.2.m1.1.1.cmml" xref="S4.SS2.SSS0.Px2.p1.2.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px2.p1.2.m1.1.1.1.cmml" xref="S4.SS2.SSS0.Px2.p1.2.m1.1.1">subscript</csymbol><ci id="S4.SS2.SSS0.Px2.p1.2.m1.1.1.2.cmml" xref="S4.SS2.SSS0.Px2.p1.2.m1.1.1.2">𝒯</ci><ci id="S4.SS2.SSS0.Px2.p1.2.m1.1.1.3a.cmml" xref="S4.SS2.SSS0.Px2.p1.2.m1.1.1.3"><mtext id="S4.SS2.SSS0.Px2.p1.2.m1.1.1.3.cmml" mathsize="70%" xref="S4.SS2.SSS0.Px2.p1.2.m1.1.1.3">reasoning</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px2.p1.2.m1.1c">\mathcal{T}_{\text{reasoning}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px2.p1.2.m1.1d">caligraphic_T start_POSTSUBSCRIPT reasoning end_POSTSUBSCRIPT</annotation></semantics></math> to instruct the model to predict the answer based on both the query <math alttext="q" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px2.p1.3.m2.1"><semantics id="S4.SS2.SSS0.Px2.p1.3.m2.1a"><mi id="S4.SS2.SSS0.Px2.p1.3.m2.1.1" xref="S4.SS2.SSS0.Px2.p1.3.m2.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px2.p1.3.m2.1b"><ci id="S4.SS2.SSS0.Px2.p1.3.m2.1.1.cmml" xref="S4.SS2.SSS0.Px2.p1.3.m2.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px2.p1.3.m2.1c">q</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px2.p1.3.m2.1d">italic_q</annotation></semantics></math> and the knowledge triple <math alttext="z" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px2.p1.4.m3.1"><semantics id="S4.SS2.SSS0.Px2.p1.4.m3.1a"><mi id="S4.SS2.SSS0.Px2.p1.4.m3.1.1" xref="S4.SS2.SSS0.Px2.p1.4.m3.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px2.p1.4.m3.1b"><ci id="S4.SS2.SSS0.Px2.p1.4.m3.1.1.cmml" xref="S4.SS2.SSS0.Px2.p1.4.m3.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px2.p1.4.m3.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px2.p1.4.m3.1d">italic_z</annotation></semantics></math>:</p> </div> <div class="ltx_para" id="S4.SS2.SSS0.Px2.p2"> <span class="ltx_ERROR undefined" id="S4.SS2.SSS0.Px2.p2.3">{custombox}</span> <p class="ltx_p" id="S4.SS2.SSS0.Px2.p2.2">Answer the query considering the user’s personalized facts. <question>: <math alttext="q" class="ltx_Math" display="inline" 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encoding="application/x-tex" id="S4.SS2.SSS0.Px2.p2.2.m2.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px2.p2.2.m2.1d">italic_z</annotation></semantics></math> <answer>: [MASK]</p> </div> <div class="ltx_para" id="S4.SS2.SSS0.Px2.p3"> <p class="ltx_p" id="S4.SS2.SSS0.Px2.p3.1">The knowledge-enhanced reasoning probability is derived as <math alttext="P_{\theta,\mathcal{G}}\left(a|q,z\right)=P_{\theta}\left(a|\mathcal{T}_{\text{% reasoning}}\left(q,z\right)\right)" class="ltx_Math" display="inline" id="S4.SS2.SSS0.Px2.p3.1.m1.8"><semantics id="S4.SS2.SSS0.Px2.p3.1.m1.8a"><mrow id="S4.SS2.SSS0.Px2.p3.1.m1.8.8" xref="S4.SS2.SSS0.Px2.p3.1.m1.8.8.cmml"><mrow id="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1" xref="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.cmml"><msub id="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.3" xref="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.3.cmml"><mi id="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.3.2" xref="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.3.2.cmml">P</mi><mrow id="S4.SS2.SSS0.Px2.p3.1.m1.2.2.2.4" xref="S4.SS2.SSS0.Px2.p3.1.m1.2.2.2.3.cmml"><mi id="S4.SS2.SSS0.Px2.p3.1.m1.1.1.1.1" xref="S4.SS2.SSS0.Px2.p3.1.m1.1.1.1.1.cmml">θ</mi><mo id="S4.SS2.SSS0.Px2.p3.1.m1.2.2.2.4.1" xref="S4.SS2.SSS0.Px2.p3.1.m1.2.2.2.3.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.SSS0.Px2.p3.1.m1.2.2.2.2" xref="S4.SS2.SSS0.Px2.p3.1.m1.2.2.2.2.cmml">𝒢</mi></mrow></msub><mo id="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.2" xref="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.1.1" xref="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.1.1.1.cmml"><mo id="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.1.1.2" xref="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.1.1.1" xref="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.1.1.1.cmml"><mi id="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.1.1.1.2" xref="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.1.1.1.2.cmml">a</mi><mo fence="false" id="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.1.1.1.1" xref="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.1.1.1.1.cmml">|</mo><mrow id="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.1.1.1.3.2" xref="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.1.1.1.3.1.cmml"><mi id="S4.SS2.SSS0.Px2.p3.1.m1.3.3" xref="S4.SS2.SSS0.Px2.p3.1.m1.3.3.cmml">q</mi><mo id="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.1.1.1.3.2.1" xref="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.1.1.1.3.1.cmml">,</mo><mi id="S4.SS2.SSS0.Px2.p3.1.m1.4.4" xref="S4.SS2.SSS0.Px2.p3.1.m1.4.4.cmml">z</mi></mrow></mrow><mo id="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.1.1.3" xref="S4.SS2.SSS0.Px2.p3.1.m1.7.7.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS0.Px2.p3.1.m1.8.8.3" xref="S4.SS2.SSS0.Px2.p3.1.m1.8.8.3.cmml">=</mo><mrow id="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2" xref="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.cmml"><msub id="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.3" xref="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.3.cmml"><mi id="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.3.2" xref="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.3.2.cmml">P</mi><mi id="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.3.3" xref="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.3.3.cmml">θ</mi></msub><mo id="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.2" 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xref="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.1.1.1.3.2.3a.cmml">reasoning</mtext></msub><mo id="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.1.1.1.3.1" xref="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.1.1.1.3.1.cmml">⁢</mo><mrow id="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.1.1.1.3.3.2" xref="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.1.1.1.3.3.1.cmml"><mo id="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.1.1.1.3.3.2.1" xref="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.1.1.1.3.3.1.cmml">(</mo><mi id="S4.SS2.SSS0.Px2.p3.1.m1.5.5" xref="S4.SS2.SSS0.Px2.p3.1.m1.5.5.cmml">q</mi><mo id="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.1.1.1.3.3.2.2" xref="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.1.1.1.3.3.1.cmml">,</mo><mi id="S4.SS2.SSS0.Px2.p3.1.m1.6.6" xref="S4.SS2.SSS0.Px2.p3.1.m1.6.6.cmml">z</mi><mo id="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.1.1.1.3.3.2.3" xref="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.1.1.1.3.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.1.1.3" xref="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" 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xref="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.1.1.1.3.2">subscript</csymbol><ci id="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.1.1.1.3.2.2.cmml" xref="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.1.1.1.3.2.2">𝒯</ci><ci id="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.1.1.1.3.2.3a.cmml" xref="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.1.1.1.3.2.3"><mtext id="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.1.1.1.3.2.3.cmml" mathsize="70%" xref="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.1.1.1.3.2.3">reasoning</mtext></ci></apply><interval closure="open" id="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.1.1.1.3.3.1.cmml" xref="S4.SS2.SSS0.Px2.p3.1.m1.8.8.2.1.1.1.3.3.2"><ci id="S4.SS2.SSS0.Px2.p3.1.m1.5.5.cmml" xref="S4.SS2.SSS0.Px2.p3.1.m1.5.5">𝑞</ci><ci id="S4.SS2.SSS0.Px2.p3.1.m1.6.6.cmml" xref="S4.SS2.SSS0.Px2.p3.1.m1.6.6">𝑧</ci></interval></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px2.p3.1.m1.8c">P_{\theta,\mathcal{G}}\left(a|q,z\right)=P_{\theta}\left(a|\mathcal{T}_{\text{% reasoning}}\left(q,z\right)\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS0.Px2.p3.1.m1.8d">italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_a | italic_q , italic_z ) = italic_P start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( italic_a | caligraphic_T start_POSTSUBSCRIPT reasoning end_POSTSUBSCRIPT ( italic_q , italic_z ) )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.SSS0.Px2.p4"> <p class="ltx_p" id="S4.SS2.SSS0.Px2.p4.1">The final objective of KGT is the combination of knowledge retrieval optimization and knowledge-enhanced reasoning optimization, which is formulated as</p> <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="\mathcal{L}=\mathcal{L_{\text{retrieve}}}+\mathcal{L_{\text{reasoning}}}=-% \frac{1}{K}\sum_{z\in\mathcal{H}(q,a,K)}\log\left[P_{\theta,\mathcal{G}}\left(% a|q,z\right)P_{\theta,\mathcal{G}}\left(z|q\right)\right]." class="ltx_Math" display="block" id="S4.E7.m1.11"><semantics id="S4.E7.m1.11a"><mrow id="S4.E7.m1.11.11.1" xref="S4.E7.m1.11.11.1.1.cmml"><mrow id="S4.E7.m1.11.11.1.1" xref="S4.E7.m1.11.11.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E7.m1.11.11.1.1.3" mathsize="90%" xref="S4.E7.m1.11.11.1.1.3.cmml">ℒ</mi><mo id="S4.E7.m1.11.11.1.1.4" mathsize="90%" xref="S4.E7.m1.11.11.1.1.4.cmml">=</mo><mrow id="S4.E7.m1.11.11.1.1.5" xref="S4.E7.m1.11.11.1.1.5.cmml"><msub id="S4.E7.m1.11.11.1.1.5.2" xref="S4.E7.m1.11.11.1.1.5.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E7.m1.11.11.1.1.5.2.2" mathsize="90%" xref="S4.E7.m1.11.11.1.1.5.2.2.cmml">ℒ</mi><mtext id="S4.E7.m1.11.11.1.1.5.2.3" mathsize="90%" xref="S4.E7.m1.11.11.1.1.5.2.3a.cmml">retrieve</mtext></msub><mo id="S4.E7.m1.11.11.1.1.5.1" mathsize="90%" 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xref="S4.E7.m1.11.11.1.1.1.1.1.1.1.1.1.2.1"><csymbol cd="latexml" id="S4.E7.m1.11.11.1.1.1.1.1.1.1.1.1.2.1.1.1.cmml" xref="S4.E7.m1.11.11.1.1.1.1.1.1.1.1.1.2.1.1.1">conditional</csymbol><ci id="S4.E7.m1.11.11.1.1.1.1.1.1.1.1.1.2.1.1.2.cmml" xref="S4.E7.m1.11.11.1.1.1.1.1.1.1.1.1.2.1.1.2">𝑧</ci><ci id="S4.E7.m1.11.11.1.1.1.1.1.1.1.1.1.2.1.1.3.cmml" xref="S4.E7.m1.11.11.1.1.1.1.1.1.1.1.1.2.1.1.3">𝑞</ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E7.m1.11c">\mathcal{L}=\mathcal{L_{\text{retrieve}}}+\mathcal{L_{\text{reasoning}}}=-% \frac{1}{K}\sum_{z\in\mathcal{H}(q,a,K)}\log\left[P_{\theta,\mathcal{G}}\left(% a|q,z\right)P_{\theta,\mathcal{G}}\left(z|q\right)\right].</annotation><annotation encoding="application/x-llamapun" id="S4.E7.m1.11d">caligraphic_L = caligraphic_L start_POSTSUBSCRIPT retrieve end_POSTSUBSCRIPT + caligraphic_L start_POSTSUBSCRIPT reasoning end_POSTSUBSCRIPT = - divide start_ARG 1 end_ARG start_ARG italic_K end_ARG ∑ start_POSTSUBSCRIPT italic_z ∈ caligraphic_H ( italic_q , italic_a , italic_K ) end_POSTSUBSCRIPT roman_log [ italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_a | italic_q , italic_z ) italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_z | italic_q ) ] .</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> </div> </section> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.3 </span>Optimization Algorithm</h3> <div class="ltx_para" id="S4.SS3.p1"> <p class="ltx_p" id="S4.SS3.p1.19">Unlike finetuning the model parameters with gradient descent, KGT is achieved by adding and removing the knowledge triples to and from the KG. For a triple <math alttext="z" class="ltx_Math" display="inline" id="S4.SS3.p1.1.m1.1"><semantics id="S4.SS3.p1.1.m1.1a"><mi id="S4.SS3.p1.1.m1.1.1" xref="S4.SS3.p1.1.m1.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.1.m1.1b"><ci id="S4.SS3.p1.1.m1.1.1.cmml" xref="S4.SS3.p1.1.m1.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.1.m1.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.1.m1.1d">italic_z</annotation></semantics></math> and a KG <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S4.SS3.p1.2.m2.1"><semantics id="S4.SS3.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p1.2.m2.1.1" xref="S4.SS3.p1.2.m2.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.2.m2.1b"><ci id="S4.SS3.p1.2.m2.1.1.cmml" xref="S4.SS3.p1.2.m2.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.2.m2.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.2.m2.1d">caligraphic_G</annotation></semantics></math>, we have two operations: 1. <math alttext="\mathcal{G}\oplus z\triangleq\mathcal{G}\cup\{z\}" class="ltx_Math" display="inline" id="S4.SS3.p1.3.m3.1"><semantics id="S4.SS3.p1.3.m3.1a"><mrow id="S4.SS3.p1.3.m3.1.2" xref="S4.SS3.p1.3.m3.1.2.cmml"><mrow id="S4.SS3.p1.3.m3.1.2.2" xref="S4.SS3.p1.3.m3.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p1.3.m3.1.2.2.2" xref="S4.SS3.p1.3.m3.1.2.2.2.cmml">𝒢</mi><mo id="S4.SS3.p1.3.m3.1.2.2.1" xref="S4.SS3.p1.3.m3.1.2.2.1.cmml">⊕</mo><mi id="S4.SS3.p1.3.m3.1.2.2.3" xref="S4.SS3.p1.3.m3.1.2.2.3.cmml">z</mi></mrow><mo id="S4.SS3.p1.3.m3.1.2.1" xref="S4.SS3.p1.3.m3.1.2.1.cmml">≜</mo><mrow id="S4.SS3.p1.3.m3.1.2.3" xref="S4.SS3.p1.3.m3.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p1.3.m3.1.2.3.2" xref="S4.SS3.p1.3.m3.1.2.3.2.cmml">𝒢</mi><mo id="S4.SS3.p1.3.m3.1.2.3.1" xref="S4.SS3.p1.3.m3.1.2.3.1.cmml">∪</mo><mrow id="S4.SS3.p1.3.m3.1.2.3.3.2" xref="S4.SS3.p1.3.m3.1.2.3.3.1.cmml"><mo id="S4.SS3.p1.3.m3.1.2.3.3.2.1" stretchy="false" xref="S4.SS3.p1.3.m3.1.2.3.3.1.cmml">{</mo><mi id="S4.SS3.p1.3.m3.1.1" xref="S4.SS3.p1.3.m3.1.1.cmml">z</mi><mo id="S4.SS3.p1.3.m3.1.2.3.3.2.2" stretchy="false" xref="S4.SS3.p1.3.m3.1.2.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.3.m3.1b"><apply id="S4.SS3.p1.3.m3.1.2.cmml" xref="S4.SS3.p1.3.m3.1.2"><ci id="S4.SS3.p1.3.m3.1.2.1.cmml" xref="S4.SS3.p1.3.m3.1.2.1">≜</ci><apply id="S4.SS3.p1.3.m3.1.2.2.cmml" xref="S4.SS3.p1.3.m3.1.2.2"><csymbol cd="latexml" id="S4.SS3.p1.3.m3.1.2.2.1.cmml" xref="S4.SS3.p1.3.m3.1.2.2.1">direct-sum</csymbol><ci id="S4.SS3.p1.3.m3.1.2.2.2.cmml" xref="S4.SS3.p1.3.m3.1.2.2.2">𝒢</ci><ci id="S4.SS3.p1.3.m3.1.2.2.3.cmml" xref="S4.SS3.p1.3.m3.1.2.2.3">𝑧</ci></apply><apply id="S4.SS3.p1.3.m3.1.2.3.cmml" xref="S4.SS3.p1.3.m3.1.2.3"><union id="S4.SS3.p1.3.m3.1.2.3.1.cmml" xref="S4.SS3.p1.3.m3.1.2.3.1"></union><ci id="S4.SS3.p1.3.m3.1.2.3.2.cmml" xref="S4.SS3.p1.3.m3.1.2.3.2">𝒢</ci><set id="S4.SS3.p1.3.m3.1.2.3.3.1.cmml" xref="S4.SS3.p1.3.m3.1.2.3.3.2"><ci id="S4.SS3.p1.3.m3.1.1.cmml" xref="S4.SS3.p1.3.m3.1.1">𝑧</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.3.m3.1c">\mathcal{G}\oplus z\triangleq\mathcal{G}\cup\{z\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.3.m3.1d">caligraphic_G ⊕ italic_z ≜ caligraphic_G ∪ { italic_z }</annotation></semantics></math>; 2. <math alttext="\mathcal{G}\ominus z\triangleq{G}\setminus\{z\}" class="ltx_Math" display="inline" id="S4.SS3.p1.4.m4.1"><semantics id="S4.SS3.p1.4.m4.1a"><mrow id="S4.SS3.p1.4.m4.1.2" xref="S4.SS3.p1.4.m4.1.2.cmml"><mrow id="S4.SS3.p1.4.m4.1.2.2" xref="S4.SS3.p1.4.m4.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p1.4.m4.1.2.2.2" xref="S4.SS3.p1.4.m4.1.2.2.2.cmml">𝒢</mi><mo id="S4.SS3.p1.4.m4.1.2.2.1" xref="S4.SS3.p1.4.m4.1.2.2.1.cmml">⊖</mo><mi id="S4.SS3.p1.4.m4.1.2.2.3" xref="S4.SS3.p1.4.m4.1.2.2.3.cmml">z</mi></mrow><mo id="S4.SS3.p1.4.m4.1.2.1" xref="S4.SS3.p1.4.m4.1.2.1.cmml">≜</mo><mrow id="S4.SS3.p1.4.m4.1.2.3" xref="S4.SS3.p1.4.m4.1.2.3.cmml"><mi id="S4.SS3.p1.4.m4.1.2.3.2" xref="S4.SS3.p1.4.m4.1.2.3.2.cmml">G</mi><mo id="S4.SS3.p1.4.m4.1.2.3.1" xref="S4.SS3.p1.4.m4.1.2.3.1.cmml">∖</mo><mrow id="S4.SS3.p1.4.m4.1.2.3.3.2" xref="S4.SS3.p1.4.m4.1.2.3.3.1.cmml"><mo id="S4.SS3.p1.4.m4.1.2.3.3.2.1" stretchy="false" xref="S4.SS3.p1.4.m4.1.2.3.3.1.cmml">{</mo><mi id="S4.SS3.p1.4.m4.1.1" xref="S4.SS3.p1.4.m4.1.1.cmml">z</mi><mo id="S4.SS3.p1.4.m4.1.2.3.3.2.2" stretchy="false" xref="S4.SS3.p1.4.m4.1.2.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.4.m4.1b"><apply id="S4.SS3.p1.4.m4.1.2.cmml" xref="S4.SS3.p1.4.m4.1.2"><ci id="S4.SS3.p1.4.m4.1.2.1.cmml" xref="S4.SS3.p1.4.m4.1.2.1">≜</ci><apply id="S4.SS3.p1.4.m4.1.2.2.cmml" xref="S4.SS3.p1.4.m4.1.2.2"><csymbol cd="latexml" id="S4.SS3.p1.4.m4.1.2.2.1.cmml" xref="S4.SS3.p1.4.m4.1.2.2.1">symmetric-difference</csymbol><ci id="S4.SS3.p1.4.m4.1.2.2.2.cmml" xref="S4.SS3.p1.4.m4.1.2.2.2">𝒢</ci><ci id="S4.SS3.p1.4.m4.1.2.2.3.cmml" xref="S4.SS3.p1.4.m4.1.2.2.3">𝑧</ci></apply><apply id="S4.SS3.p1.4.m4.1.2.3.cmml" xref="S4.SS3.p1.4.m4.1.2.3"><setdiff id="S4.SS3.p1.4.m4.1.2.3.1.cmml" xref="S4.SS3.p1.4.m4.1.2.3.1"></setdiff><ci id="S4.SS3.p1.4.m4.1.2.3.2.cmml" xref="S4.SS3.p1.4.m4.1.2.3.2">𝐺</ci><set id="S4.SS3.p1.4.m4.1.2.3.3.1.cmml" xref="S4.SS3.p1.4.m4.1.2.3.3.2"><ci id="S4.SS3.p1.4.m4.1.1.cmml" xref="S4.SS3.p1.4.m4.1.1">𝑧</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.4.m4.1c">\mathcal{G}\ominus z\triangleq{G}\setminus\{z\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.4.m4.1d">caligraphic_G ⊖ italic_z ≜ italic_G ∖ { italic_z }</annotation></semantics></math>. Because we focus on real-time model personalization, we formulate the optimization in the online learning setting where the optimization algorithm only has access to the user’s current query <math alttext="q_{t}" class="ltx_Math" display="inline" id="S4.SS3.p1.5.m5.1"><semantics id="S4.SS3.p1.5.m5.1a"><msub id="S4.SS3.p1.5.m5.1.1" xref="S4.SS3.p1.5.m5.1.1.cmml"><mi id="S4.SS3.p1.5.m5.1.1.2" xref="S4.SS3.p1.5.m5.1.1.2.cmml">q</mi><mi id="S4.SS3.p1.5.m5.1.1.3" xref="S4.SS3.p1.5.m5.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.5.m5.1b"><apply id="S4.SS3.p1.5.m5.1.1.cmml" xref="S4.SS3.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.5.m5.1.1.1.cmml" xref="S4.SS3.p1.5.m5.1.1">subscript</csymbol><ci id="S4.SS3.p1.5.m5.1.1.2.cmml" xref="S4.SS3.p1.5.m5.1.1.2">𝑞</ci><ci id="S4.SS3.p1.5.m5.1.1.3.cmml" xref="S4.SS3.p1.5.m5.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.5.m5.1c">q_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.5.m5.1d">italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> at time <math alttext="t" class="ltx_Math" display="inline" id="S4.SS3.p1.6.m6.1"><semantics id="S4.SS3.p1.6.m6.1a"><mi id="S4.SS3.p1.6.m6.1.1" xref="S4.SS3.p1.6.m6.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.6.m6.1b"><ci id="S4.SS3.p1.6.m6.1.1.cmml" xref="S4.SS3.p1.6.m6.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.6.m6.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.6.m6.1d">italic_t</annotation></semantics></math> and feedback <math alttext="a_{t}" class="ltx_Math" display="inline" id="S4.SS3.p1.7.m7.1"><semantics id="S4.SS3.p1.7.m7.1a"><msub id="S4.SS3.p1.7.m7.1.1" xref="S4.SS3.p1.7.m7.1.1.cmml"><mi id="S4.SS3.p1.7.m7.1.1.2" xref="S4.SS3.p1.7.m7.1.1.2.cmml">a</mi><mi id="S4.SS3.p1.7.m7.1.1.3" xref="S4.SS3.p1.7.m7.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.7.m7.1b"><apply id="S4.SS3.p1.7.m7.1.1.cmml" xref="S4.SS3.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.7.m7.1.1.1.cmml" xref="S4.SS3.p1.7.m7.1.1">subscript</csymbol><ci id="S4.SS3.p1.7.m7.1.1.2.cmml" xref="S4.SS3.p1.7.m7.1.1.2">𝑎</ci><ci id="S4.SS3.p1.7.m7.1.1.3.cmml" xref="S4.SS3.p1.7.m7.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.7.m7.1c">a_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.7.m7.1d">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>. The trivial solution to this query is that we remove all the triples in <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S4.SS3.p1.8.m8.1"><semantics id="S4.SS3.p1.8.m8.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p1.8.m8.1.1" xref="S4.SS3.p1.8.m8.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.8.m8.1b"><ci id="S4.SS3.p1.8.m8.1.1.cmml" xref="S4.SS3.p1.8.m8.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.8.m8.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.8.m8.1d">caligraphic_G</annotation></semantics></math> that are probably retrieved based on <math alttext="q_{t}" class="ltx_Math" display="inline" id="S4.SS3.p1.9.m9.1"><semantics id="S4.SS3.p1.9.m9.1a"><msub id="S4.SS3.p1.9.m9.1.1" xref="S4.SS3.p1.9.m9.1.1.cmml"><mi id="S4.SS3.p1.9.m9.1.1.2" xref="S4.SS3.p1.9.m9.1.1.2.cmml">q</mi><mi id="S4.SS3.p1.9.m9.1.1.3" xref="S4.SS3.p1.9.m9.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.9.m9.1b"><apply id="S4.SS3.p1.9.m9.1.1.cmml" xref="S4.SS3.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.9.m9.1.1.1.cmml" xref="S4.SS3.p1.9.m9.1.1">subscript</csymbol><ci id="S4.SS3.p1.9.m9.1.1.2.cmml" xref="S4.SS3.p1.9.m9.1.1.2">𝑞</ci><ci id="S4.SS3.p1.9.m9.1.1.3.cmml" xref="S4.SS3.p1.9.m9.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.9.m9.1c">q_{t}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.9.m9.1d">italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, which are the triples starting with <math alttext="e_{q}" class="ltx_Math" display="inline" id="S4.SS3.p1.10.m10.1"><semantics id="S4.SS3.p1.10.m10.1a"><msub id="S4.SS3.p1.10.m10.1.1" xref="S4.SS3.p1.10.m10.1.1.cmml"><mi id="S4.SS3.p1.10.m10.1.1.2" xref="S4.SS3.p1.10.m10.1.1.2.cmml">e</mi><mi id="S4.SS3.p1.10.m10.1.1.3" xref="S4.SS3.p1.10.m10.1.1.3.cmml">q</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.10.m10.1b"><apply id="S4.SS3.p1.10.m10.1.1.cmml" xref="S4.SS3.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.10.m10.1.1.1.cmml" xref="S4.SS3.p1.10.m10.1.1">subscript</csymbol><ci id="S4.SS3.p1.10.m10.1.1.2.cmml" xref="S4.SS3.p1.10.m10.1.1.2">𝑒</ci><ci id="S4.SS3.p1.10.m10.1.1.3.cmml" xref="S4.SS3.p1.10.m10.1.1.3">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.10.m10.1c">e_{q}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.10.m10.1d">italic_e start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT</annotation></semantics></math>. Then we add the triple <math alttext="z^{*}" class="ltx_Math" display="inline" id="S4.SS3.p1.11.m11.1"><semantics id="S4.SS3.p1.11.m11.1a"><msup id="S4.SS3.p1.11.m11.1.1" xref="S4.SS3.p1.11.m11.1.1.cmml"><mi id="S4.SS3.p1.11.m11.1.1.2" xref="S4.SS3.p1.11.m11.1.1.2.cmml">z</mi><mo id="S4.SS3.p1.11.m11.1.1.3" xref="S4.SS3.p1.11.m11.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.11.m11.1b"><apply id="S4.SS3.p1.11.m11.1.1.cmml" xref="S4.SS3.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.11.m11.1.1.1.cmml" xref="S4.SS3.p1.11.m11.1.1">superscript</csymbol><ci id="S4.SS3.p1.11.m11.1.1.2.cmml" xref="S4.SS3.p1.11.m11.1.1.2">𝑧</ci><times id="S4.SS3.p1.11.m11.1.1.3.cmml" xref="S4.SS3.p1.11.m11.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.11.m11.1c">z^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.11.m11.1d">italic_z start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> from <math alttext="\mathcal{H}(q_{t},a_{t},K)" class="ltx_Math" display="inline" id="S4.SS3.p1.12.m12.3"><semantics id="S4.SS3.p1.12.m12.3a"><mrow id="S4.SS3.p1.12.m12.3.3" xref="S4.SS3.p1.12.m12.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p1.12.m12.3.3.4" xref="S4.SS3.p1.12.m12.3.3.4.cmml">ℋ</mi><mo id="S4.SS3.p1.12.m12.3.3.3" xref="S4.SS3.p1.12.m12.3.3.3.cmml">⁢</mo><mrow id="S4.SS3.p1.12.m12.3.3.2.2" xref="S4.SS3.p1.12.m12.3.3.2.3.cmml"><mo id="S4.SS3.p1.12.m12.3.3.2.2.3" stretchy="false" xref="S4.SS3.p1.12.m12.3.3.2.3.cmml">(</mo><msub id="S4.SS3.p1.12.m12.2.2.1.1.1" xref="S4.SS3.p1.12.m12.2.2.1.1.1.cmml"><mi id="S4.SS3.p1.12.m12.2.2.1.1.1.2" xref="S4.SS3.p1.12.m12.2.2.1.1.1.2.cmml">q</mi><mi id="S4.SS3.p1.12.m12.2.2.1.1.1.3" xref="S4.SS3.p1.12.m12.2.2.1.1.1.3.cmml">t</mi></msub><mo id="S4.SS3.p1.12.m12.3.3.2.2.4" xref="S4.SS3.p1.12.m12.3.3.2.3.cmml">,</mo><msub id="S4.SS3.p1.12.m12.3.3.2.2.2" xref="S4.SS3.p1.12.m12.3.3.2.2.2.cmml"><mi id="S4.SS3.p1.12.m12.3.3.2.2.2.2" xref="S4.SS3.p1.12.m12.3.3.2.2.2.2.cmml">a</mi><mi id="S4.SS3.p1.12.m12.3.3.2.2.2.3" xref="S4.SS3.p1.12.m12.3.3.2.2.2.3.cmml">t</mi></msub><mo id="S4.SS3.p1.12.m12.3.3.2.2.5" xref="S4.SS3.p1.12.m12.3.3.2.3.cmml">,</mo><mi id="S4.SS3.p1.12.m12.1.1" xref="S4.SS3.p1.12.m12.1.1.cmml">K</mi><mo id="S4.SS3.p1.12.m12.3.3.2.2.6" stretchy="false" xref="S4.SS3.p1.12.m12.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.12.m12.3b"><apply id="S4.SS3.p1.12.m12.3.3.cmml" xref="S4.SS3.p1.12.m12.3.3"><times id="S4.SS3.p1.12.m12.3.3.3.cmml" xref="S4.SS3.p1.12.m12.3.3.3"></times><ci id="S4.SS3.p1.12.m12.3.3.4.cmml" xref="S4.SS3.p1.12.m12.3.3.4">ℋ</ci><vector id="S4.SS3.p1.12.m12.3.3.2.3.cmml" xref="S4.SS3.p1.12.m12.3.3.2.2"><apply id="S4.SS3.p1.12.m12.2.2.1.1.1.cmml" xref="S4.SS3.p1.12.m12.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.12.m12.2.2.1.1.1.1.cmml" xref="S4.SS3.p1.12.m12.2.2.1.1.1">subscript</csymbol><ci id="S4.SS3.p1.12.m12.2.2.1.1.1.2.cmml" xref="S4.SS3.p1.12.m12.2.2.1.1.1.2">𝑞</ci><ci id="S4.SS3.p1.12.m12.2.2.1.1.1.3.cmml" xref="S4.SS3.p1.12.m12.2.2.1.1.1.3">𝑡</ci></apply><apply id="S4.SS3.p1.12.m12.3.3.2.2.2.cmml" xref="S4.SS3.p1.12.m12.3.3.2.2.2"><csymbol cd="ambiguous" id="S4.SS3.p1.12.m12.3.3.2.2.2.1.cmml" xref="S4.SS3.p1.12.m12.3.3.2.2.2">subscript</csymbol><ci id="S4.SS3.p1.12.m12.3.3.2.2.2.2.cmml" xref="S4.SS3.p1.12.m12.3.3.2.2.2.2">𝑎</ci><ci id="S4.SS3.p1.12.m12.3.3.2.2.2.3.cmml" xref="S4.SS3.p1.12.m12.3.3.2.2.2.3">𝑡</ci></apply><ci id="S4.SS3.p1.12.m12.1.1.cmml" xref="S4.SS3.p1.12.m12.1.1">𝐾</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.12.m12.3c">\mathcal{H}(q_{t},a_{t},K)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.12.m12.3d">caligraphic_H ( italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_K )</annotation></semantics></math> with the highest reasoning probability <math alttext="P_{\theta,\mathcal{G}}\left(a_{t}|q_{t},z^{*}\right)" class="ltx_Math" display="inline" id="S4.SS3.p1.13.m13.3"><semantics id="S4.SS3.p1.13.m13.3a"><mrow id="S4.SS3.p1.13.m13.3.3" xref="S4.SS3.p1.13.m13.3.3.cmml"><msub id="S4.SS3.p1.13.m13.3.3.3" xref="S4.SS3.p1.13.m13.3.3.3.cmml"><mi id="S4.SS3.p1.13.m13.3.3.3.2" xref="S4.SS3.p1.13.m13.3.3.3.2.cmml">P</mi><mrow id="S4.SS3.p1.13.m13.2.2.2.4" xref="S4.SS3.p1.13.m13.2.2.2.3.cmml"><mi id="S4.SS3.p1.13.m13.1.1.1.1" xref="S4.SS3.p1.13.m13.1.1.1.1.cmml">θ</mi><mo id="S4.SS3.p1.13.m13.2.2.2.4.1" xref="S4.SS3.p1.13.m13.2.2.2.3.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p1.13.m13.2.2.2.2" xref="S4.SS3.p1.13.m13.2.2.2.2.cmml">𝒢</mi></mrow></msub><mo id="S4.SS3.p1.13.m13.3.3.2" xref="S4.SS3.p1.13.m13.3.3.2.cmml">⁢</mo><mrow id="S4.SS3.p1.13.m13.3.3.1.1" xref="S4.SS3.p1.13.m13.3.3.1.1.1.cmml"><mo id="S4.SS3.p1.13.m13.3.3.1.1.2" xref="S4.SS3.p1.13.m13.3.3.1.1.1.cmml">(</mo><mrow id="S4.SS3.p1.13.m13.3.3.1.1.1" xref="S4.SS3.p1.13.m13.3.3.1.1.1.cmml"><msub id="S4.SS3.p1.13.m13.3.3.1.1.1.4" xref="S4.SS3.p1.13.m13.3.3.1.1.1.4.cmml"><mi id="S4.SS3.p1.13.m13.3.3.1.1.1.4.2" xref="S4.SS3.p1.13.m13.3.3.1.1.1.4.2.cmml">a</mi><mi id="S4.SS3.p1.13.m13.3.3.1.1.1.4.3" xref="S4.SS3.p1.13.m13.3.3.1.1.1.4.3.cmml">t</mi></msub><mo fence="false" id="S4.SS3.p1.13.m13.3.3.1.1.1.3" xref="S4.SS3.p1.13.m13.3.3.1.1.1.3.cmml">|</mo><mrow id="S4.SS3.p1.13.m13.3.3.1.1.1.2.2" xref="S4.SS3.p1.13.m13.3.3.1.1.1.2.3.cmml"><msub id="S4.SS3.p1.13.m13.3.3.1.1.1.1.1.1" xref="S4.SS3.p1.13.m13.3.3.1.1.1.1.1.1.cmml"><mi id="S4.SS3.p1.13.m13.3.3.1.1.1.1.1.1.2" xref="S4.SS3.p1.13.m13.3.3.1.1.1.1.1.1.2.cmml">q</mi><mi id="S4.SS3.p1.13.m13.3.3.1.1.1.1.1.1.3" xref="S4.SS3.p1.13.m13.3.3.1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="S4.SS3.p1.13.m13.3.3.1.1.1.2.2.3" xref="S4.SS3.p1.13.m13.3.3.1.1.1.2.3.cmml">,</mo><msup id="S4.SS3.p1.13.m13.3.3.1.1.1.2.2.2" xref="S4.SS3.p1.13.m13.3.3.1.1.1.2.2.2.cmml"><mi id="S4.SS3.p1.13.m13.3.3.1.1.1.2.2.2.2" xref="S4.SS3.p1.13.m13.3.3.1.1.1.2.2.2.2.cmml">z</mi><mo id="S4.SS3.p1.13.m13.3.3.1.1.1.2.2.2.3" xref="S4.SS3.p1.13.m13.3.3.1.1.1.2.2.2.3.cmml">∗</mo></msup></mrow></mrow><mo id="S4.SS3.p1.13.m13.3.3.1.1.3" xref="S4.SS3.p1.13.m13.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.13.m13.3b"><apply id="S4.SS3.p1.13.m13.3.3.cmml" xref="S4.SS3.p1.13.m13.3.3"><times id="S4.SS3.p1.13.m13.3.3.2.cmml" xref="S4.SS3.p1.13.m13.3.3.2"></times><apply id="S4.SS3.p1.13.m13.3.3.3.cmml" xref="S4.SS3.p1.13.m13.3.3.3"><csymbol cd="ambiguous" id="S4.SS3.p1.13.m13.3.3.3.1.cmml" xref="S4.SS3.p1.13.m13.3.3.3">subscript</csymbol><ci id="S4.SS3.p1.13.m13.3.3.3.2.cmml" xref="S4.SS3.p1.13.m13.3.3.3.2">𝑃</ci><list id="S4.SS3.p1.13.m13.2.2.2.3.cmml" xref="S4.SS3.p1.13.m13.2.2.2.4"><ci id="S4.SS3.p1.13.m13.1.1.1.1.cmml" xref="S4.SS3.p1.13.m13.1.1.1.1">𝜃</ci><ci id="S4.SS3.p1.13.m13.2.2.2.2.cmml" xref="S4.SS3.p1.13.m13.2.2.2.2">𝒢</ci></list></apply><apply id="S4.SS3.p1.13.m13.3.3.1.1.1.cmml" xref="S4.SS3.p1.13.m13.3.3.1.1"><csymbol cd="latexml" id="S4.SS3.p1.13.m13.3.3.1.1.1.3.cmml" xref="S4.SS3.p1.13.m13.3.3.1.1.1.3">conditional</csymbol><apply id="S4.SS3.p1.13.m13.3.3.1.1.1.4.cmml" xref="S4.SS3.p1.13.m13.3.3.1.1.1.4"><csymbol cd="ambiguous" id="S4.SS3.p1.13.m13.3.3.1.1.1.4.1.cmml" xref="S4.SS3.p1.13.m13.3.3.1.1.1.4">subscript</csymbol><ci id="S4.SS3.p1.13.m13.3.3.1.1.1.4.2.cmml" xref="S4.SS3.p1.13.m13.3.3.1.1.1.4.2">𝑎</ci><ci id="S4.SS3.p1.13.m13.3.3.1.1.1.4.3.cmml" xref="S4.SS3.p1.13.m13.3.3.1.1.1.4.3">𝑡</ci></apply><list id="S4.SS3.p1.13.m13.3.3.1.1.1.2.3.cmml" xref="S4.SS3.p1.13.m13.3.3.1.1.1.2.2"><apply id="S4.SS3.p1.13.m13.3.3.1.1.1.1.1.1.cmml" xref="S4.SS3.p1.13.m13.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.13.m13.3.3.1.1.1.1.1.1.1.cmml" xref="S4.SS3.p1.13.m13.3.3.1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS3.p1.13.m13.3.3.1.1.1.1.1.1.2.cmml" xref="S4.SS3.p1.13.m13.3.3.1.1.1.1.1.1.2">𝑞</ci><ci id="S4.SS3.p1.13.m13.3.3.1.1.1.1.1.1.3.cmml" xref="S4.SS3.p1.13.m13.3.3.1.1.1.1.1.1.3">𝑡</ci></apply><apply id="S4.SS3.p1.13.m13.3.3.1.1.1.2.2.2.cmml" xref="S4.SS3.p1.13.m13.3.3.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S4.SS3.p1.13.m13.3.3.1.1.1.2.2.2.1.cmml" xref="S4.SS3.p1.13.m13.3.3.1.1.1.2.2.2">superscript</csymbol><ci id="S4.SS3.p1.13.m13.3.3.1.1.1.2.2.2.2.cmml" xref="S4.SS3.p1.13.m13.3.3.1.1.1.2.2.2.2">𝑧</ci><times id="S4.SS3.p1.13.m13.3.3.1.1.1.2.2.2.3.cmml" xref="S4.SS3.p1.13.m13.3.3.1.1.1.2.2.2.3"></times></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.13.m13.3c">P_{\theta,\mathcal{G}}\left(a_{t}|q_{t},z^{*}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.13.m13.3d">italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT | italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_z start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math> into <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S4.SS3.p1.14.m14.1"><semantics id="S4.SS3.p1.14.m14.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p1.14.m14.1.1" xref="S4.SS3.p1.14.m14.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.14.m14.1b"><ci id="S4.SS3.p1.14.m14.1.1.cmml" xref="S4.SS3.p1.14.m14.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.14.m14.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.14.m14.1d">caligraphic_G</annotation></semantics></math>. In this case, <math alttext="P_{\theta,\mathcal{G}}\left(z^{*}|q_{t}\right)=1" class="ltx_Math" display="inline" id="S4.SS3.p1.15.m15.3"><semantics id="S4.SS3.p1.15.m15.3a"><mrow id="S4.SS3.p1.15.m15.3.3" xref="S4.SS3.p1.15.m15.3.3.cmml"><mrow id="S4.SS3.p1.15.m15.3.3.1" xref="S4.SS3.p1.15.m15.3.3.1.cmml"><msub id="S4.SS3.p1.15.m15.3.3.1.3" xref="S4.SS3.p1.15.m15.3.3.1.3.cmml"><mi id="S4.SS3.p1.15.m15.3.3.1.3.2" xref="S4.SS3.p1.15.m15.3.3.1.3.2.cmml">P</mi><mrow id="S4.SS3.p1.15.m15.2.2.2.4" xref="S4.SS3.p1.15.m15.2.2.2.3.cmml"><mi id="S4.SS3.p1.15.m15.1.1.1.1" xref="S4.SS3.p1.15.m15.1.1.1.1.cmml">θ</mi><mo id="S4.SS3.p1.15.m15.2.2.2.4.1" xref="S4.SS3.p1.15.m15.2.2.2.3.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p1.15.m15.2.2.2.2" xref="S4.SS3.p1.15.m15.2.2.2.2.cmml">𝒢</mi></mrow></msub><mo id="S4.SS3.p1.15.m15.3.3.1.2" xref="S4.SS3.p1.15.m15.3.3.1.2.cmml">⁢</mo><mrow id="S4.SS3.p1.15.m15.3.3.1.1.1" xref="S4.SS3.p1.15.m15.3.3.1.1.1.1.cmml"><mo id="S4.SS3.p1.15.m15.3.3.1.1.1.2" xref="S4.SS3.p1.15.m15.3.3.1.1.1.1.cmml">(</mo><mrow id="S4.SS3.p1.15.m15.3.3.1.1.1.1" xref="S4.SS3.p1.15.m15.3.3.1.1.1.1.cmml"><msup id="S4.SS3.p1.15.m15.3.3.1.1.1.1.2" xref="S4.SS3.p1.15.m15.3.3.1.1.1.1.2.cmml"><mi id="S4.SS3.p1.15.m15.3.3.1.1.1.1.2.2" xref="S4.SS3.p1.15.m15.3.3.1.1.1.1.2.2.cmml">z</mi><mo id="S4.SS3.p1.15.m15.3.3.1.1.1.1.2.3" xref="S4.SS3.p1.15.m15.3.3.1.1.1.1.2.3.cmml">∗</mo></msup><mo fence="false" id="S4.SS3.p1.15.m15.3.3.1.1.1.1.1" xref="S4.SS3.p1.15.m15.3.3.1.1.1.1.1.cmml">|</mo><msub id="S4.SS3.p1.15.m15.3.3.1.1.1.1.3" xref="S4.SS3.p1.15.m15.3.3.1.1.1.1.3.cmml"><mi id="S4.SS3.p1.15.m15.3.3.1.1.1.1.3.2" xref="S4.SS3.p1.15.m15.3.3.1.1.1.1.3.2.cmml">q</mi><mi id="S4.SS3.p1.15.m15.3.3.1.1.1.1.3.3" xref="S4.SS3.p1.15.m15.3.3.1.1.1.1.3.3.cmml">t</mi></msub></mrow><mo id="S4.SS3.p1.15.m15.3.3.1.1.1.3" xref="S4.SS3.p1.15.m15.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS3.p1.15.m15.3.3.2" xref="S4.SS3.p1.15.m15.3.3.2.cmml">=</mo><mn id="S4.SS3.p1.15.m15.3.3.3" xref="S4.SS3.p1.15.m15.3.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.15.m15.3b"><apply id="S4.SS3.p1.15.m15.3.3.cmml" xref="S4.SS3.p1.15.m15.3.3"><eq id="S4.SS3.p1.15.m15.3.3.2.cmml" xref="S4.SS3.p1.15.m15.3.3.2"></eq><apply id="S4.SS3.p1.15.m15.3.3.1.cmml" xref="S4.SS3.p1.15.m15.3.3.1"><times id="S4.SS3.p1.15.m15.3.3.1.2.cmml" xref="S4.SS3.p1.15.m15.3.3.1.2"></times><apply id="S4.SS3.p1.15.m15.3.3.1.3.cmml" xref="S4.SS3.p1.15.m15.3.3.1.3"><csymbol cd="ambiguous" id="S4.SS3.p1.15.m15.3.3.1.3.1.cmml" xref="S4.SS3.p1.15.m15.3.3.1.3">subscript</csymbol><ci id="S4.SS3.p1.15.m15.3.3.1.3.2.cmml" xref="S4.SS3.p1.15.m15.3.3.1.3.2">𝑃</ci><list id="S4.SS3.p1.15.m15.2.2.2.3.cmml" xref="S4.SS3.p1.15.m15.2.2.2.4"><ci id="S4.SS3.p1.15.m15.1.1.1.1.cmml" xref="S4.SS3.p1.15.m15.1.1.1.1">𝜃</ci><ci id="S4.SS3.p1.15.m15.2.2.2.2.cmml" xref="S4.SS3.p1.15.m15.2.2.2.2">𝒢</ci></list></apply><apply 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xref="S4.SS3.p1.15.m15.3.3.1.1.1.1.3.3">𝑡</ci></apply></apply></apply><cn id="S4.SS3.p1.15.m15.3.3.3.cmml" type="integer" xref="S4.SS3.p1.15.m15.3.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.15.m15.3c">P_{\theta,\mathcal{G}}\left(z^{*}|q_{t}\right)=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.15.m15.3d">italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_z start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) = 1</annotation></semantics></math> and the loss is minimized over <math alttext="(q_{t},a_{t})" class="ltx_Math" display="inline" id="S4.SS3.p1.16.m16.2"><semantics id="S4.SS3.p1.16.m16.2a"><mrow id="S4.SS3.p1.16.m16.2.2.2" xref="S4.SS3.p1.16.m16.2.2.3.cmml"><mo id="S4.SS3.p1.16.m16.2.2.2.3" stretchy="false" xref="S4.SS3.p1.16.m16.2.2.3.cmml">(</mo><msub id="S4.SS3.p1.16.m16.1.1.1.1" xref="S4.SS3.p1.16.m16.1.1.1.1.cmml"><mi id="S4.SS3.p1.16.m16.1.1.1.1.2" xref="S4.SS3.p1.16.m16.1.1.1.1.2.cmml">q</mi><mi id="S4.SS3.p1.16.m16.1.1.1.1.3" xref="S4.SS3.p1.16.m16.1.1.1.1.3.cmml">t</mi></msub><mo id="S4.SS3.p1.16.m16.2.2.2.4" xref="S4.SS3.p1.16.m16.2.2.3.cmml">,</mo><msub id="S4.SS3.p1.16.m16.2.2.2.2" xref="S4.SS3.p1.16.m16.2.2.2.2.cmml"><mi id="S4.SS3.p1.16.m16.2.2.2.2.2" xref="S4.SS3.p1.16.m16.2.2.2.2.2.cmml">a</mi><mi id="S4.SS3.p1.16.m16.2.2.2.2.3" xref="S4.SS3.p1.16.m16.2.2.2.2.3.cmml">t</mi></msub><mo id="S4.SS3.p1.16.m16.2.2.2.5" stretchy="false" xref="S4.SS3.p1.16.m16.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.16.m16.2b"><interval closure="open" id="S4.SS3.p1.16.m16.2.2.3.cmml" xref="S4.SS3.p1.16.m16.2.2.2"><apply id="S4.SS3.p1.16.m16.1.1.1.1.cmml" xref="S4.SS3.p1.16.m16.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.16.m16.1.1.1.1.1.cmml" xref="S4.SS3.p1.16.m16.1.1.1.1">subscript</csymbol><ci id="S4.SS3.p1.16.m16.1.1.1.1.2.cmml" xref="S4.SS3.p1.16.m16.1.1.1.1.2">𝑞</ci><ci id="S4.SS3.p1.16.m16.1.1.1.1.3.cmml" xref="S4.SS3.p1.16.m16.1.1.1.1.3">𝑡</ci></apply><apply id="S4.SS3.p1.16.m16.2.2.2.2.cmml" xref="S4.SS3.p1.16.m16.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS3.p1.16.m16.2.2.2.2.1.cmml" xref="S4.SS3.p1.16.m16.2.2.2.2">subscript</csymbol><ci id="S4.SS3.p1.16.m16.2.2.2.2.2.cmml" xref="S4.SS3.p1.16.m16.2.2.2.2.2">𝑎</ci><ci id="S4.SS3.p1.16.m16.2.2.2.2.3.cmml" xref="S4.SS3.p1.16.m16.2.2.2.2.3">𝑡</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.16.m16.2c">(q_{t},a_{t})</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.16.m16.2d">( italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math>. However, this greedy solution will potentially hurt the other queries due to removing too much knowledge from the KG. Following the example in <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S4.SS1" title="4.1 Knowledge Graph Tuning ‣ 4 Method ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4.1</span></a>, the greedy solution will not only remove the triple <math alttext="\left(Dog,Enjoy,Meat\right)" class="ltx_Math" display="inline" id="S4.SS3.p1.17.m17.3"><semantics id="S4.SS3.p1.17.m17.3a"><mrow id="S4.SS3.p1.17.m17.3.3.3" xref="S4.SS3.p1.17.m17.3.3.4.cmml"><mo id="S4.SS3.p1.17.m17.3.3.3.4" xref="S4.SS3.p1.17.m17.3.3.4.cmml">(</mo><mrow id="S4.SS3.p1.17.m17.1.1.1.1" xref="S4.SS3.p1.17.m17.1.1.1.1.cmml"><mi id="S4.SS3.p1.17.m17.1.1.1.1.2" xref="S4.SS3.p1.17.m17.1.1.1.1.2.cmml">D</mi><mo id="S4.SS3.p1.17.m17.1.1.1.1.1" xref="S4.SS3.p1.17.m17.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p1.17.m17.1.1.1.1.3" xref="S4.SS3.p1.17.m17.1.1.1.1.3.cmml">o</mi><mo 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xref="S4.SS3.p1.17.m17.2.2.2.2.6.cmml">y</mi></mrow><mo id="S4.SS3.p1.17.m17.3.3.3.6" xref="S4.SS3.p1.17.m17.3.3.4.cmml">,</mo><mrow id="S4.SS3.p1.17.m17.3.3.3.3" xref="S4.SS3.p1.17.m17.3.3.3.3.cmml"><mi id="S4.SS3.p1.17.m17.3.3.3.3.2" xref="S4.SS3.p1.17.m17.3.3.3.3.2.cmml">M</mi><mo id="S4.SS3.p1.17.m17.3.3.3.3.1" xref="S4.SS3.p1.17.m17.3.3.3.3.1.cmml">⁢</mo><mi id="S4.SS3.p1.17.m17.3.3.3.3.3" xref="S4.SS3.p1.17.m17.3.3.3.3.3.cmml">e</mi><mo id="S4.SS3.p1.17.m17.3.3.3.3.1a" xref="S4.SS3.p1.17.m17.3.3.3.3.1.cmml">⁢</mo><mi id="S4.SS3.p1.17.m17.3.3.3.3.4" xref="S4.SS3.p1.17.m17.3.3.3.3.4.cmml">a</mi><mo id="S4.SS3.p1.17.m17.3.3.3.3.1b" xref="S4.SS3.p1.17.m17.3.3.3.3.1.cmml">⁢</mo><mi id="S4.SS3.p1.17.m17.3.3.3.3.5" xref="S4.SS3.p1.17.m17.3.3.3.3.5.cmml">t</mi></mrow><mo id="S4.SS3.p1.17.m17.3.3.3.7" xref="S4.SS3.p1.17.m17.3.3.4.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.17.m17.3b"><vector id="S4.SS3.p1.17.m17.3.3.4.cmml" 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xref="S4.SS3.p1.17.m17.2.2.2.2.6">𝑦</ci></apply><apply id="S4.SS3.p1.17.m17.3.3.3.3.cmml" xref="S4.SS3.p1.17.m17.3.3.3.3"><times id="S4.SS3.p1.17.m17.3.3.3.3.1.cmml" xref="S4.SS3.p1.17.m17.3.3.3.3.1"></times><ci id="S4.SS3.p1.17.m17.3.3.3.3.2.cmml" xref="S4.SS3.p1.17.m17.3.3.3.3.2">𝑀</ci><ci id="S4.SS3.p1.17.m17.3.3.3.3.3.cmml" xref="S4.SS3.p1.17.m17.3.3.3.3.3">𝑒</ci><ci id="S4.SS3.p1.17.m17.3.3.3.3.4.cmml" xref="S4.SS3.p1.17.m17.3.3.3.3.4">𝑎</ci><ci id="S4.SS3.p1.17.m17.3.3.3.3.5.cmml" xref="S4.SS3.p1.17.m17.3.3.3.3.5">𝑡</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.17.m17.3c">\left(Dog,Enjoy,Meat\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.17.m17.3d">( italic_D italic_o italic_g , italic_E italic_n italic_j italic_o italic_y , italic_M italic_e italic_a italic_t )</annotation></semantics></math>, but also remove other triples starting from the entity <math alttext="Dog" class="ltx_Math" display="inline" id="S4.SS3.p1.18.m18.1"><semantics id="S4.SS3.p1.18.m18.1a"><mrow id="S4.SS3.p1.18.m18.1.1" xref="S4.SS3.p1.18.m18.1.1.cmml"><mi id="S4.SS3.p1.18.m18.1.1.2" xref="S4.SS3.p1.18.m18.1.1.2.cmml">D</mi><mo id="S4.SS3.p1.18.m18.1.1.1" xref="S4.SS3.p1.18.m18.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p1.18.m18.1.1.3" xref="S4.SS3.p1.18.m18.1.1.3.cmml">o</mi><mo id="S4.SS3.p1.18.m18.1.1.1a" xref="S4.SS3.p1.18.m18.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p1.18.m18.1.1.4" xref="S4.SS3.p1.18.m18.1.1.4.cmml">g</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.18.m18.1b"><apply id="S4.SS3.p1.18.m18.1.1.cmml" xref="S4.SS3.p1.18.m18.1.1"><times id="S4.SS3.p1.18.m18.1.1.1.cmml" xref="S4.SS3.p1.18.m18.1.1.1"></times><ci id="S4.SS3.p1.18.m18.1.1.2.cmml" xref="S4.SS3.p1.18.m18.1.1.2">𝐷</ci><ci id="S4.SS3.p1.18.m18.1.1.3.cmml" xref="S4.SS3.p1.18.m18.1.1.3">𝑜</ci><ci id="S4.SS3.p1.18.m18.1.1.4.cmml" xref="S4.SS3.p1.18.m18.1.1.4">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.18.m18.1c">Dog</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.18.m18.1d">italic_D italic_o italic_g</annotation></semantics></math> such as <math alttext="\left(Dog,Is,Animal\right)" class="ltx_Math" display="inline" id="S4.SS3.p1.19.m19.3"><semantics id="S4.SS3.p1.19.m19.3a"><mrow id="S4.SS3.p1.19.m19.3.3.3" xref="S4.SS3.p1.19.m19.3.3.4.cmml"><mo id="S4.SS3.p1.19.m19.3.3.3.4" xref="S4.SS3.p1.19.m19.3.3.4.cmml">(</mo><mrow id="S4.SS3.p1.19.m19.1.1.1.1" xref="S4.SS3.p1.19.m19.1.1.1.1.cmml"><mi id="S4.SS3.p1.19.m19.1.1.1.1.2" xref="S4.SS3.p1.19.m19.1.1.1.1.2.cmml">D</mi><mo id="S4.SS3.p1.19.m19.1.1.1.1.1" xref="S4.SS3.p1.19.m19.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p1.19.m19.1.1.1.1.3" xref="S4.SS3.p1.19.m19.1.1.1.1.3.cmml">o</mi><mo id="S4.SS3.p1.19.m19.1.1.1.1.1a" xref="S4.SS3.p1.19.m19.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p1.19.m19.1.1.1.1.4" xref="S4.SS3.p1.19.m19.1.1.1.1.4.cmml">g</mi></mrow><mo id="S4.SS3.p1.19.m19.3.3.3.5" xref="S4.SS3.p1.19.m19.3.3.4.cmml">,</mo><mrow id="S4.SS3.p1.19.m19.2.2.2.2" xref="S4.SS3.p1.19.m19.2.2.2.2.cmml"><mi id="S4.SS3.p1.19.m19.2.2.2.2.2" xref="S4.SS3.p1.19.m19.2.2.2.2.2.cmml">I</mi><mo id="S4.SS3.p1.19.m19.2.2.2.2.1" xref="S4.SS3.p1.19.m19.2.2.2.2.1.cmml">⁢</mo><mi id="S4.SS3.p1.19.m19.2.2.2.2.3" xref="S4.SS3.p1.19.m19.2.2.2.2.3.cmml">s</mi></mrow><mo id="S4.SS3.p1.19.m19.3.3.3.6" xref="S4.SS3.p1.19.m19.3.3.4.cmml">,</mo><mrow id="S4.SS3.p1.19.m19.3.3.3.3" xref="S4.SS3.p1.19.m19.3.3.3.3.cmml"><mi id="S4.SS3.p1.19.m19.3.3.3.3.2" xref="S4.SS3.p1.19.m19.3.3.3.3.2.cmml">A</mi><mo id="S4.SS3.p1.19.m19.3.3.3.3.1" xref="S4.SS3.p1.19.m19.3.3.3.3.1.cmml">⁢</mo><mi id="S4.SS3.p1.19.m19.3.3.3.3.3" xref="S4.SS3.p1.19.m19.3.3.3.3.3.cmml">n</mi><mo id="S4.SS3.p1.19.m19.3.3.3.3.1a" xref="S4.SS3.p1.19.m19.3.3.3.3.1.cmml">⁢</mo><mi id="S4.SS3.p1.19.m19.3.3.3.3.4" xref="S4.SS3.p1.19.m19.3.3.3.3.4.cmml">i</mi><mo id="S4.SS3.p1.19.m19.3.3.3.3.1b" xref="S4.SS3.p1.19.m19.3.3.3.3.1.cmml">⁢</mo><mi id="S4.SS3.p1.19.m19.3.3.3.3.5" xref="S4.SS3.p1.19.m19.3.3.3.3.5.cmml">m</mi><mo id="S4.SS3.p1.19.m19.3.3.3.3.1c" xref="S4.SS3.p1.19.m19.3.3.3.3.1.cmml">⁢</mo><mi id="S4.SS3.p1.19.m19.3.3.3.3.6" xref="S4.SS3.p1.19.m19.3.3.3.3.6.cmml">a</mi><mo id="S4.SS3.p1.19.m19.3.3.3.3.1d" xref="S4.SS3.p1.19.m19.3.3.3.3.1.cmml">⁢</mo><mi id="S4.SS3.p1.19.m19.3.3.3.3.7" xref="S4.SS3.p1.19.m19.3.3.3.3.7.cmml">l</mi></mrow><mo id="S4.SS3.p1.19.m19.3.3.3.7" xref="S4.SS3.p1.19.m19.3.3.4.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.19.m19.3b"><vector id="S4.SS3.p1.19.m19.3.3.4.cmml" xref="S4.SS3.p1.19.m19.3.3.3"><apply id="S4.SS3.p1.19.m19.1.1.1.1.cmml" xref="S4.SS3.p1.19.m19.1.1.1.1"><times id="S4.SS3.p1.19.m19.1.1.1.1.1.cmml" xref="S4.SS3.p1.19.m19.1.1.1.1.1"></times><ci id="S4.SS3.p1.19.m19.1.1.1.1.2.cmml" xref="S4.SS3.p1.19.m19.1.1.1.1.2">𝐷</ci><ci id="S4.SS3.p1.19.m19.1.1.1.1.3.cmml" xref="S4.SS3.p1.19.m19.1.1.1.1.3">𝑜</ci><ci id="S4.SS3.p1.19.m19.1.1.1.1.4.cmml" xref="S4.SS3.p1.19.m19.1.1.1.1.4">𝑔</ci></apply><apply id="S4.SS3.p1.19.m19.2.2.2.2.cmml" xref="S4.SS3.p1.19.m19.2.2.2.2"><times id="S4.SS3.p1.19.m19.2.2.2.2.1.cmml" xref="S4.SS3.p1.19.m19.2.2.2.2.1"></times><ci id="S4.SS3.p1.19.m19.2.2.2.2.2.cmml" xref="S4.SS3.p1.19.m19.2.2.2.2.2">𝐼</ci><ci id="S4.SS3.p1.19.m19.2.2.2.2.3.cmml" xref="S4.SS3.p1.19.m19.2.2.2.2.3">𝑠</ci></apply><apply id="S4.SS3.p1.19.m19.3.3.3.3.cmml" xref="S4.SS3.p1.19.m19.3.3.3.3"><times id="S4.SS3.p1.19.m19.3.3.3.3.1.cmml" xref="S4.SS3.p1.19.m19.3.3.3.3.1"></times><ci id="S4.SS3.p1.19.m19.3.3.3.3.2.cmml" xref="S4.SS3.p1.19.m19.3.3.3.3.2">𝐴</ci><ci id="S4.SS3.p1.19.m19.3.3.3.3.3.cmml" xref="S4.SS3.p1.19.m19.3.3.3.3.3">𝑛</ci><ci id="S4.SS3.p1.19.m19.3.3.3.3.4.cmml" xref="S4.SS3.p1.19.m19.3.3.3.3.4">𝑖</ci><ci id="S4.SS3.p1.19.m19.3.3.3.3.5.cmml" xref="S4.SS3.p1.19.m19.3.3.3.3.5">𝑚</ci><ci id="S4.SS3.p1.19.m19.3.3.3.3.6.cmml" xref="S4.SS3.p1.19.m19.3.3.3.3.6">𝑎</ci><ci id="S4.SS3.p1.19.m19.3.3.3.3.7.cmml" xref="S4.SS3.p1.19.m19.3.3.3.3.7">𝑙</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.19.m19.3c">\left(Dog,Is,Animal\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.19.m19.3d">( italic_D italic_o italic_g , italic_I italic_s , italic_A italic_n italic_i italic_m italic_a italic_l )</annotation></semantics></math>. The removed triples might be general or personalized factual knowledge that is essential to the other queries.</p> </div> <div class="ltx_para" id="S4.SS3.p2"> <p class="ltx_p" id="S4.SS3.p2.8">We propose a heuristic optimization algorithm based on two principles: 1. The computational cost should be low to achieve real-time personalization efficiency; 2. The KG should be less modified to preserve more knowledge. Both principles require us to modify as few triples as possible to achieve the optimization goal. The detailed algorithm is shown in <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#alg1" title="In 4.3 Optimization Algorithm ‣ 4 Method ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">1</span></a>. 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xref="S4.SS3.p2.1.m1.3.3.1.1.1.2.2.2.2.cmml">z</mi><mo id="S4.SS3.p2.1.m1.3.3.1.1.1.2.2.2.3" xref="S4.SS3.p2.1.m1.3.3.1.1.1.2.2.2.3.cmml">′</mo></msup></mrow></mrow><mo id="S4.SS3.p2.1.m1.3.3.1.1.3" xref="S4.SS3.p2.1.m1.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.1.m1.3b"><apply id="S4.SS3.p2.1.m1.3.3.cmml" xref="S4.SS3.p2.1.m1.3.3"><times id="S4.SS3.p2.1.m1.3.3.2.cmml" xref="S4.SS3.p2.1.m1.3.3.2"></times><apply id="S4.SS3.p2.1.m1.3.3.3.cmml" xref="S4.SS3.p2.1.m1.3.3.3"><csymbol cd="ambiguous" id="S4.SS3.p2.1.m1.3.3.3.1.cmml" xref="S4.SS3.p2.1.m1.3.3.3">subscript</csymbol><ci id="S4.SS3.p2.1.m1.3.3.3.2.cmml" xref="S4.SS3.p2.1.m1.3.3.3.2">𝑃</ci><list id="S4.SS3.p2.1.m1.2.2.2.3.cmml" xref="S4.SS3.p2.1.m1.2.2.2.4"><ci id="S4.SS3.p2.1.m1.1.1.1.1.cmml" xref="S4.SS3.p2.1.m1.1.1.1.1">𝜃</ci><ci id="S4.SS3.p2.1.m1.2.2.2.2.cmml" xref="S4.SS3.p2.1.m1.2.2.2.2">𝒢</ci></list></apply><apply id="S4.SS3.p2.1.m1.3.3.1.1.1.cmml" 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xref="S4.SS3.p2.1.m1.3.3.1.1.1.1.1.1.3">𝑡</ci></apply><apply id="S4.SS3.p2.1.m1.3.3.1.1.1.2.2.2.cmml" xref="S4.SS3.p2.1.m1.3.3.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S4.SS3.p2.1.m1.3.3.1.1.1.2.2.2.1.cmml" xref="S4.SS3.p2.1.m1.3.3.1.1.1.2.2.2">superscript</csymbol><ci id="S4.SS3.p2.1.m1.3.3.1.1.1.2.2.2.2.cmml" xref="S4.SS3.p2.1.m1.3.3.1.1.1.2.2.2.2">𝑧</ci><ci id="S4.SS3.p2.1.m1.3.3.1.1.1.2.2.2.3.cmml" xref="S4.SS3.p2.1.m1.3.3.1.1.1.2.2.2.3">′</ci></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.1.m1.3c">P_{\theta,\mathcal{G}}\left(a_{t}|q_{t},z^{\prime}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.1.m1.3d">italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT | italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> for the triples <math alttext="z^{\prime}" class="ltx_Math" display="inline" id="S4.SS3.p2.2.m2.1"><semantics id="S4.SS3.p2.2.m2.1a"><msup id="S4.SS3.p2.2.m2.1.1" xref="S4.SS3.p2.2.m2.1.1.cmml"><mi id="S4.SS3.p2.2.m2.1.1.2" xref="S4.SS3.p2.2.m2.1.1.2.cmml">z</mi><mo id="S4.SS3.p2.2.m2.1.1.3" xref="S4.SS3.p2.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.2.m2.1b"><apply id="S4.SS3.p2.2.m2.1.1.cmml" xref="S4.SS3.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.2.m2.1.1.1.cmml" xref="S4.SS3.p2.2.m2.1.1">superscript</csymbol><ci id="S4.SS3.p2.2.m2.1.1.2.cmml" xref="S4.SS3.p2.2.m2.1.1.2">𝑧</ci><ci id="S4.SS3.p2.2.m2.1.1.3.cmml" xref="S4.SS3.p2.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.2.m2.1c">z^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.2.m2.1d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> in <math alttext="\mathcal{H}(q,a,K)" class="ltx_Math" display="inline" id="S4.SS3.p2.3.m3.3"><semantics id="S4.SS3.p2.3.m3.3a"><mrow id="S4.SS3.p2.3.m3.3.4" xref="S4.SS3.p2.3.m3.3.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p2.3.m3.3.4.2" xref="S4.SS3.p2.3.m3.3.4.2.cmml">ℋ</mi><mo id="S4.SS3.p2.3.m3.3.4.1" xref="S4.SS3.p2.3.m3.3.4.1.cmml">⁢</mo><mrow id="S4.SS3.p2.3.m3.3.4.3.2" xref="S4.SS3.p2.3.m3.3.4.3.1.cmml"><mo id="S4.SS3.p2.3.m3.3.4.3.2.1" stretchy="false" xref="S4.SS3.p2.3.m3.3.4.3.1.cmml">(</mo><mi id="S4.SS3.p2.3.m3.1.1" xref="S4.SS3.p2.3.m3.1.1.cmml">q</mi><mo id="S4.SS3.p2.3.m3.3.4.3.2.2" xref="S4.SS3.p2.3.m3.3.4.3.1.cmml">,</mo><mi id="S4.SS3.p2.3.m3.2.2" xref="S4.SS3.p2.3.m3.2.2.cmml">a</mi><mo id="S4.SS3.p2.3.m3.3.4.3.2.3" xref="S4.SS3.p2.3.m3.3.4.3.1.cmml">,</mo><mi id="S4.SS3.p2.3.m3.3.3" xref="S4.SS3.p2.3.m3.3.3.cmml">K</mi><mo id="S4.SS3.p2.3.m3.3.4.3.2.4" stretchy="false" xref="S4.SS3.p2.3.m3.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.3.m3.3b"><apply id="S4.SS3.p2.3.m3.3.4.cmml" xref="S4.SS3.p2.3.m3.3.4"><times id="S4.SS3.p2.3.m3.3.4.1.cmml" xref="S4.SS3.p2.3.m3.3.4.1"></times><ci id="S4.SS3.p2.3.m3.3.4.2.cmml" xref="S4.SS3.p2.3.m3.3.4.2">ℋ</ci><vector id="S4.SS3.p2.3.m3.3.4.3.1.cmml" xref="S4.SS3.p2.3.m3.3.4.3.2"><ci id="S4.SS3.p2.3.m3.1.1.cmml" xref="S4.SS3.p2.3.m3.1.1">𝑞</ci><ci id="S4.SS3.p2.3.m3.2.2.cmml" xref="S4.SS3.p2.3.m3.2.2">𝑎</ci><ci id="S4.SS3.p2.3.m3.3.3.cmml" xref="S4.SS3.p2.3.m3.3.3">𝐾</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.3.m3.3c">\mathcal{H}(q,a,K)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.3.m3.3d">caligraphic_H ( italic_q , italic_a , italic_K )</annotation></semantics></math> and triples in <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S4.SS3.p2.4.m4.1"><semantics id="S4.SS3.p2.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p2.4.m4.1.1" xref="S4.SS3.p2.4.m4.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.4.m4.1b"><ci id="S4.SS3.p2.4.m4.1.1.cmml" xref="S4.SS3.p2.4.m4.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.4.m4.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.4.m4.1d">caligraphic_G</annotation></semantics></math> that starting from <math alttext="e_{q_{t}}" class="ltx_Math" display="inline" id="S4.SS3.p2.5.m5.1"><semantics id="S4.SS3.p2.5.m5.1a"><msub id="S4.SS3.p2.5.m5.1.1" xref="S4.SS3.p2.5.m5.1.1.cmml"><mi id="S4.SS3.p2.5.m5.1.1.2" xref="S4.SS3.p2.5.m5.1.1.2.cmml">e</mi><msub id="S4.SS3.p2.5.m5.1.1.3" xref="S4.SS3.p2.5.m5.1.1.3.cmml"><mi id="S4.SS3.p2.5.m5.1.1.3.2" xref="S4.SS3.p2.5.m5.1.1.3.2.cmml">q</mi><mi id="S4.SS3.p2.5.m5.1.1.3.3" xref="S4.SS3.p2.5.m5.1.1.3.3.cmml">t</mi></msub></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.5.m5.1b"><apply id="S4.SS3.p2.5.m5.1.1.cmml" xref="S4.SS3.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.5.m5.1.1.1.cmml" xref="S4.SS3.p2.5.m5.1.1">subscript</csymbol><ci id="S4.SS3.p2.5.m5.1.1.2.cmml" xref="S4.SS3.p2.5.m5.1.1.2">𝑒</ci><apply id="S4.SS3.p2.5.m5.1.1.3.cmml" xref="S4.SS3.p2.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p2.5.m5.1.1.3.1.cmml" xref="S4.SS3.p2.5.m5.1.1.3">subscript</csymbol><ci id="S4.SS3.p2.5.m5.1.1.3.2.cmml" xref="S4.SS3.p2.5.m5.1.1.3.2">𝑞</ci><ci id="S4.SS3.p2.5.m5.1.1.3.3.cmml" xref="S4.SS3.p2.5.m5.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.5.m5.1c">e_{q_{t}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.5.m5.1d">italic_e start_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Then, we add the triples with the highest reasoning probability (i.e., the triples most encourage the LLM to generate the correct answer) from <math alttext="\mathcal{H}(q,a,K)" class="ltx_Math" display="inline" id="S4.SS3.p2.6.m6.3"><semantics id="S4.SS3.p2.6.m6.3a"><mrow id="S4.SS3.p2.6.m6.3.4" xref="S4.SS3.p2.6.m6.3.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p2.6.m6.3.4.2" xref="S4.SS3.p2.6.m6.3.4.2.cmml">ℋ</mi><mo id="S4.SS3.p2.6.m6.3.4.1" xref="S4.SS3.p2.6.m6.3.4.1.cmml">⁢</mo><mrow id="S4.SS3.p2.6.m6.3.4.3.2" xref="S4.SS3.p2.6.m6.3.4.3.1.cmml"><mo id="S4.SS3.p2.6.m6.3.4.3.2.1" stretchy="false" xref="S4.SS3.p2.6.m6.3.4.3.1.cmml">(</mo><mi id="S4.SS3.p2.6.m6.1.1" xref="S4.SS3.p2.6.m6.1.1.cmml">q</mi><mo id="S4.SS3.p2.6.m6.3.4.3.2.2" xref="S4.SS3.p2.6.m6.3.4.3.1.cmml">,</mo><mi id="S4.SS3.p2.6.m6.2.2" xref="S4.SS3.p2.6.m6.2.2.cmml">a</mi><mo id="S4.SS3.p2.6.m6.3.4.3.2.3" xref="S4.SS3.p2.6.m6.3.4.3.1.cmml">,</mo><mi id="S4.SS3.p2.6.m6.3.3" xref="S4.SS3.p2.6.m6.3.3.cmml">K</mi><mo id="S4.SS3.p2.6.m6.3.4.3.2.4" stretchy="false" xref="S4.SS3.p2.6.m6.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.6.m6.3b"><apply id="S4.SS3.p2.6.m6.3.4.cmml" xref="S4.SS3.p2.6.m6.3.4"><times id="S4.SS3.p2.6.m6.3.4.1.cmml" xref="S4.SS3.p2.6.m6.3.4.1"></times><ci id="S4.SS3.p2.6.m6.3.4.2.cmml" xref="S4.SS3.p2.6.m6.3.4.2">ℋ</ci><vector id="S4.SS3.p2.6.m6.3.4.3.1.cmml" xref="S4.SS3.p2.6.m6.3.4.3.2"><ci id="S4.SS3.p2.6.m6.1.1.cmml" xref="S4.SS3.p2.6.m6.1.1">𝑞</ci><ci id="S4.SS3.p2.6.m6.2.2.cmml" xref="S4.SS3.p2.6.m6.2.2">𝑎</ci><ci id="S4.SS3.p2.6.m6.3.3.cmml" xref="S4.SS3.p2.6.m6.3.3">𝐾</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.6.m6.3c">\mathcal{H}(q,a,K)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.6.m6.3d">caligraphic_H ( italic_q , italic_a , italic_K )</annotation></semantics></math> to <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S4.SS3.p2.7.m7.1"><semantics id="S4.SS3.p2.7.m7.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p2.7.m7.1.1" xref="S4.SS3.p2.7.m7.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.7.m7.1b"><ci id="S4.SS3.p2.7.m7.1.1.cmml" xref="S4.SS3.p2.7.m7.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.7.m7.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.7.m7.1d">caligraphic_G</annotation></semantics></math> and remove the triples with the lowest reasoning probability from <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S4.SS3.p2.8.m8.1"><semantics id="S4.SS3.p2.8.m8.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p2.8.m8.1.1" xref="S4.SS3.p2.8.m8.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.8.m8.1b"><ci id="S4.SS3.p2.8.m8.1.1.cmml" xref="S4.SS3.p2.8.m8.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.8.m8.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.8.m8.1d">caligraphic_G</annotation></semantics></math> iteratively until the model generates the correct answer.</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed ltx_framed_top" id="alg1"> <figcaption class="ltx_caption" style="font-size:80%;"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg1.12.1.1">Algorithm 1</span> </span> Knowledge Graph Tuning based on User Feedback</figcaption><div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel" id="alg1.8"><span class="ltx_text ltx_font_bold" id="alg1.8.8" style="font-size:80%;">Input:<span class="ltx_text ltx_font_medium" id="alg1.7.7.7"> Knowledge graph <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="alg1.1.1.1.m1.1"><semantics id="alg1.1.1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="alg1.1.1.1.m1.1.1" xref="alg1.1.1.1.m1.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="alg1.1.1.1.m1.1b"><ci id="alg1.1.1.1.m1.1.1.cmml" xref="alg1.1.1.1.m1.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.1.1.1.m1.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="alg1.1.1.1.m1.1d">caligraphic_G</annotation></semantics></math>; KG-enhanced LLM <math alttext="f_{\theta,\mathcal{G}}" class="ltx_Math" display="inline" id="alg1.2.2.2.m2.2"><semantics id="alg1.2.2.2.m2.2a"><msub id="alg1.2.2.2.m2.2.3" xref="alg1.2.2.2.m2.2.3.cmml"><mi id="alg1.2.2.2.m2.2.3.2" xref="alg1.2.2.2.m2.2.3.2.cmml">f</mi><mrow id="alg1.2.2.2.m2.2.2.2.4" xref="alg1.2.2.2.m2.2.2.2.3.cmml"><mi id="alg1.2.2.2.m2.1.1.1.1" xref="alg1.2.2.2.m2.1.1.1.1.cmml">θ</mi><mo id="alg1.2.2.2.m2.2.2.2.4.1" xref="alg1.2.2.2.m2.2.2.2.3.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="alg1.2.2.2.m2.2.2.2.2" xref="alg1.2.2.2.m2.2.2.2.2.cmml">𝒢</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="alg1.2.2.2.m2.2b"><apply id="alg1.2.2.2.m2.2.3.cmml" xref="alg1.2.2.2.m2.2.3"><csymbol cd="ambiguous" id="alg1.2.2.2.m2.2.3.1.cmml" xref="alg1.2.2.2.m2.2.3">subscript</csymbol><ci id="alg1.2.2.2.m2.2.3.2.cmml" xref="alg1.2.2.2.m2.2.3.2">𝑓</ci><list id="alg1.2.2.2.m2.2.2.2.3.cmml" xref="alg1.2.2.2.m2.2.2.2.4"><ci id="alg1.2.2.2.m2.1.1.1.1.cmml" xref="alg1.2.2.2.m2.1.1.1.1">𝜃</ci><ci id="alg1.2.2.2.m2.2.2.2.2.cmml" xref="alg1.2.2.2.m2.2.2.2.2">𝒢</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.2.2.2.m2.2c">f_{\theta,\mathcal{G}}</annotation><annotation encoding="application/x-llamapun" id="alg1.2.2.2.m2.2d">italic_f start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT</annotation></semantics></math>; user’s query <math alttext="q_{t}" class="ltx_Math" display="inline" id="alg1.3.3.3.m3.1"><semantics id="alg1.3.3.3.m3.1a"><msub id="alg1.3.3.3.m3.1.1" xref="alg1.3.3.3.m3.1.1.cmml"><mi id="alg1.3.3.3.m3.1.1.2" xref="alg1.3.3.3.m3.1.1.2.cmml">q</mi><mi id="alg1.3.3.3.m3.1.1.3" xref="alg1.3.3.3.m3.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="alg1.3.3.3.m3.1b"><apply id="alg1.3.3.3.m3.1.1.cmml" xref="alg1.3.3.3.m3.1.1"><csymbol cd="ambiguous" id="alg1.3.3.3.m3.1.1.1.cmml" xref="alg1.3.3.3.m3.1.1">subscript</csymbol><ci id="alg1.3.3.3.m3.1.1.2.cmml" xref="alg1.3.3.3.m3.1.1.2">𝑞</ci><ci id="alg1.3.3.3.m3.1.1.3.cmml" xref="alg1.3.3.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.3.3.3.m3.1c">q_{t}</annotation><annotation encoding="application/x-llamapun" id="alg1.3.3.3.m3.1d">italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> at time <math alttext="t" class="ltx_Math" display="inline" id="alg1.4.4.4.m4.1"><semantics id="alg1.4.4.4.m4.1a"><mi id="alg1.4.4.4.m4.1.1" xref="alg1.4.4.4.m4.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="alg1.4.4.4.m4.1b"><ci id="alg1.4.4.4.m4.1.1.cmml" xref="alg1.4.4.4.m4.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.4.4.4.m4.1c">t</annotation><annotation encoding="application/x-llamapun" id="alg1.4.4.4.m4.1d">italic_t</annotation></semantics></math>; user feedback <math alttext="a_{t}" class="ltx_Math" display="inline" id="alg1.5.5.5.m5.1"><semantics id="alg1.5.5.5.m5.1a"><msub id="alg1.5.5.5.m5.1.1" xref="alg1.5.5.5.m5.1.1.cmml"><mi id="alg1.5.5.5.m5.1.1.2" xref="alg1.5.5.5.m5.1.1.2.cmml">a</mi><mi id="alg1.5.5.5.m5.1.1.3" xref="alg1.5.5.5.m5.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="alg1.5.5.5.m5.1b"><apply id="alg1.5.5.5.m5.1.1.cmml" xref="alg1.5.5.5.m5.1.1"><csymbol cd="ambiguous" id="alg1.5.5.5.m5.1.1.1.cmml" xref="alg1.5.5.5.m5.1.1">subscript</csymbol><ci id="alg1.5.5.5.m5.1.1.2.cmml" xref="alg1.5.5.5.m5.1.1.2">𝑎</ci><ci id="alg1.5.5.5.m5.1.1.3.cmml" xref="alg1.5.5.5.m5.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.5.5.5.m5.1c">a_{t}</annotation><annotation encoding="application/x-llamapun" id="alg1.5.5.5.m5.1d">italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>; size of personalized triple set for each query <math alttext="K" class="ltx_Math" display="inline" id="alg1.6.6.6.m6.1"><semantics id="alg1.6.6.6.m6.1a"><mi id="alg1.6.6.6.m6.1.1" xref="alg1.6.6.6.m6.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="alg1.6.6.6.m6.1b"><ci id="alg1.6.6.6.m6.1.1.cmml" xref="alg1.6.6.6.m6.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.6.6.6.m6.1c">K</annotation><annotation encoding="application/x-llamapun" id="alg1.6.6.6.m6.1d">italic_K</annotation></semantics></math>; loss threshold <math alttext="\epsilon" class="ltx_Math" display="inline" id="alg1.7.7.7.m7.1"><semantics id="alg1.7.7.7.m7.1a"><mi id="alg1.7.7.7.m7.1.1" xref="alg1.7.7.7.m7.1.1.cmml">ϵ</mi><annotation-xml encoding="MathML-Content" id="alg1.7.7.7.m7.1b"><ci id="alg1.7.7.7.m7.1.1.cmml" xref="alg1.7.7.7.m7.1.1">italic-ϵ</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.7.7.7.m7.1c">\epsilon</annotation><annotation encoding="application/x-llamapun" id="alg1.7.7.7.m7.1d">italic_ϵ</annotation></semantics></math>. <br class="ltx_break"/></span>Output:<span class="ltx_text ltx_font_medium" id="alg1.8.8.8"> Optimized KG <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="alg1.8.8.8.m1.1"><semantics id="alg1.8.8.8.m1.1a"><mi class="ltx_font_mathcaligraphic" id="alg1.8.8.8.m1.1.1" xref="alg1.8.8.8.m1.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="alg1.8.8.8.m1.1b"><ci id="alg1.8.8.8.m1.1.1.cmml" xref="alg1.8.8.8.m1.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.8.8.8.m1.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="alg1.8.8.8.m1.1d">caligraphic_G</annotation></semantics></math>.</span></span></p> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <div class="ltx_listing ltx_figure_panel ltx_listing" id="alg1.13"> <div class="ltx_listingline" id="alg1.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l1.1.1.1" style="font-size:80%;">1:</span></span><span class="ltx_text" id="alg1.l1.2" style="font-size:80%;">Construct personalized triple set </span><math alttext="\mathcal{H}(q_{t},a_{t},K)" class="ltx_Math" display="inline" id="alg1.l1.m1.3"><semantics id="alg1.l1.m1.3a"><mrow id="alg1.l1.m1.3.3" xref="alg1.l1.m1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.l1.m1.3.3.4" mathsize="80%" xref="alg1.l1.m1.3.3.4.cmml">ℋ</mi><mo id="alg1.l1.m1.3.3.3" xref="alg1.l1.m1.3.3.3.cmml">⁢</mo><mrow id="alg1.l1.m1.3.3.2.2" xref="alg1.l1.m1.3.3.2.3.cmml"><mo id="alg1.l1.m1.3.3.2.2.3" maxsize="80%" minsize="80%" xref="alg1.l1.m1.3.3.2.3.cmml">(</mo><msub id="alg1.l1.m1.2.2.1.1.1" xref="alg1.l1.m1.2.2.1.1.1.cmml"><mi id="alg1.l1.m1.2.2.1.1.1.2" mathsize="80%" xref="alg1.l1.m1.2.2.1.1.1.2.cmml">q</mi><mi id="alg1.l1.m1.2.2.1.1.1.3" mathsize="80%" xref="alg1.l1.m1.2.2.1.1.1.3.cmml">t</mi></msub><mo id="alg1.l1.m1.3.3.2.2.4" mathsize="80%" xref="alg1.l1.m1.3.3.2.3.cmml">,</mo><msub id="alg1.l1.m1.3.3.2.2.2" xref="alg1.l1.m1.3.3.2.2.2.cmml"><mi id="alg1.l1.m1.3.3.2.2.2.2" mathsize="80%" xref="alg1.l1.m1.3.3.2.2.2.2.cmml">a</mi><mi id="alg1.l1.m1.3.3.2.2.2.3" mathsize="80%" xref="alg1.l1.m1.3.3.2.2.2.3.cmml">t</mi></msub><mo id="alg1.l1.m1.3.3.2.2.5" mathsize="80%" xref="alg1.l1.m1.3.3.2.3.cmml">,</mo><mi id="alg1.l1.m1.1.1" mathsize="80%" xref="alg1.l1.m1.1.1.cmml">K</mi><mo id="alg1.l1.m1.3.3.2.2.6" maxsize="80%" minsize="80%" xref="alg1.l1.m1.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l1.m1.3b"><apply id="alg1.l1.m1.3.3.cmml" xref="alg1.l1.m1.3.3"><times id="alg1.l1.m1.3.3.3.cmml" xref="alg1.l1.m1.3.3.3"></times><ci id="alg1.l1.m1.3.3.4.cmml" xref="alg1.l1.m1.3.3.4">ℋ</ci><vector id="alg1.l1.m1.3.3.2.3.cmml" xref="alg1.l1.m1.3.3.2.2"><apply id="alg1.l1.m1.2.2.1.1.1.cmml" xref="alg1.l1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="alg1.l1.m1.2.2.1.1.1.1.cmml" xref="alg1.l1.m1.2.2.1.1.1">subscript</csymbol><ci id="alg1.l1.m1.2.2.1.1.1.2.cmml" xref="alg1.l1.m1.2.2.1.1.1.2">𝑞</ci><ci id="alg1.l1.m1.2.2.1.1.1.3.cmml" xref="alg1.l1.m1.2.2.1.1.1.3">𝑡</ci></apply><apply id="alg1.l1.m1.3.3.2.2.2.cmml" xref="alg1.l1.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="alg1.l1.m1.3.3.2.2.2.1.cmml" xref="alg1.l1.m1.3.3.2.2.2">subscript</csymbol><ci id="alg1.l1.m1.3.3.2.2.2.2.cmml" xref="alg1.l1.m1.3.3.2.2.2.2">𝑎</ci><ci id="alg1.l1.m1.3.3.2.2.2.3.cmml" xref="alg1.l1.m1.3.3.2.2.2.3">𝑡</ci></apply><ci id="alg1.l1.m1.1.1.cmml" xref="alg1.l1.m1.1.1">𝐾</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m1.3c">\mathcal{H}(q_{t},a_{t},K)</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m1.3d">caligraphic_H ( italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_K )</annotation></semantics></math><span class="ltx_text" id="alg1.l1.3" style="font-size:80%;"> </span> </div> <div class="ltx_listingline" id="alg1.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l2.1.1.1" style="font-size:80%;">2:</span></span><span class="ltx_text" id="alg1.l2.2" style="font-size:80%;">Extract the set of triples </span><math alttext="\mathcal{G}_{q_{t}}" class="ltx_Math" display="inline" id="alg1.l2.m1.1"><semantics id="alg1.l2.m1.1a"><msub id="alg1.l2.m1.1.1" xref="alg1.l2.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.l2.m1.1.1.2" mathsize="80%" xref="alg1.l2.m1.1.1.2.cmml">𝒢</mi><msub id="alg1.l2.m1.1.1.3" xref="alg1.l2.m1.1.1.3.cmml"><mi id="alg1.l2.m1.1.1.3.2" mathsize="80%" xref="alg1.l2.m1.1.1.3.2.cmml">q</mi><mi id="alg1.l2.m1.1.1.3.3" mathsize="80%" xref="alg1.l2.m1.1.1.3.3.cmml">t</mi></msub></msub><annotation-xml encoding="MathML-Content" id="alg1.l2.m1.1b"><apply id="alg1.l2.m1.1.1.cmml" xref="alg1.l2.m1.1.1"><csymbol cd="ambiguous" id="alg1.l2.m1.1.1.1.cmml" xref="alg1.l2.m1.1.1">subscript</csymbol><ci id="alg1.l2.m1.1.1.2.cmml" xref="alg1.l2.m1.1.1.2">𝒢</ci><apply id="alg1.l2.m1.1.1.3.cmml" xref="alg1.l2.m1.1.1.3"><csymbol cd="ambiguous" id="alg1.l2.m1.1.1.3.1.cmml" xref="alg1.l2.m1.1.1.3">subscript</csymbol><ci id="alg1.l2.m1.1.1.3.2.cmml" xref="alg1.l2.m1.1.1.3.2">𝑞</ci><ci id="alg1.l2.m1.1.1.3.3.cmml" xref="alg1.l2.m1.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l2.m1.1c">\mathcal{G}_{q_{t}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l2.m1.1d">caligraphic_G start_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text" id="alg1.l2.3" style="font-size:80%;"> in </span><math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="alg1.l2.m2.1"><semantics id="alg1.l2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="alg1.l2.m2.1.1" mathsize="80%" xref="alg1.l2.m2.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="alg1.l2.m2.1b"><ci id="alg1.l2.m2.1.1.cmml" xref="alg1.l2.m2.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l2.m2.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="alg1.l2.m2.1d">caligraphic_G</annotation></semantics></math><span class="ltx_text" id="alg1.l2.4" style="font-size:80%;"> starting from </span><math alttext="e_{q_{t}}" class="ltx_Math" display="inline" id="alg1.l2.m3.1"><semantics id="alg1.l2.m3.1a"><msub id="alg1.l2.m3.1.1" xref="alg1.l2.m3.1.1.cmml"><mi id="alg1.l2.m3.1.1.2" mathsize="80%" xref="alg1.l2.m3.1.1.2.cmml">e</mi><msub id="alg1.l2.m3.1.1.3" xref="alg1.l2.m3.1.1.3.cmml"><mi id="alg1.l2.m3.1.1.3.2" mathsize="80%" xref="alg1.l2.m3.1.1.3.2.cmml">q</mi><mi id="alg1.l2.m3.1.1.3.3" mathsize="80%" xref="alg1.l2.m3.1.1.3.3.cmml">t</mi></msub></msub><annotation-xml encoding="MathML-Content" id="alg1.l2.m3.1b"><apply id="alg1.l2.m3.1.1.cmml" xref="alg1.l2.m3.1.1"><csymbol cd="ambiguous" id="alg1.l2.m3.1.1.1.cmml" xref="alg1.l2.m3.1.1">subscript</csymbol><ci id="alg1.l2.m3.1.1.2.cmml" xref="alg1.l2.m3.1.1.2">𝑒</ci><apply id="alg1.l2.m3.1.1.3.cmml" xref="alg1.l2.m3.1.1.3"><csymbol cd="ambiguous" id="alg1.l2.m3.1.1.3.1.cmml" xref="alg1.l2.m3.1.1.3">subscript</csymbol><ci id="alg1.l2.m3.1.1.3.2.cmml" xref="alg1.l2.m3.1.1.3.2">𝑞</ci><ci id="alg1.l2.m3.1.1.3.3.cmml" xref="alg1.l2.m3.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l2.m3.1c">e_{q_{t}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l2.m3.1d">italic_e start_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text" id="alg1.l2.5" style="font-size:80%;">. </span> </div> <div class="ltx_listingline" id="alg1.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l3.1.1.1" style="font-size:80%;">3:</span></span><math alttext="N_{H}\leftarrow|\mathcal{H}(q_{t},a_{t},K)|" class="ltx_Math" display="inline" id="alg1.l3.m1.2"><semantics id="alg1.l3.m1.2a"><mrow id="alg1.l3.m1.2.2" xref="alg1.l3.m1.2.2.cmml"><msub id="alg1.l3.m1.2.2.3" xref="alg1.l3.m1.2.2.3.cmml"><mi id="alg1.l3.m1.2.2.3.2" mathsize="80%" xref="alg1.l3.m1.2.2.3.2.cmml">N</mi><mi id="alg1.l3.m1.2.2.3.3" mathsize="80%" xref="alg1.l3.m1.2.2.3.3.cmml">H</mi></msub><mo id="alg1.l3.m1.2.2.2" mathsize="80%" stretchy="false" xref="alg1.l3.m1.2.2.2.cmml">←</mo><mrow id="alg1.l3.m1.2.2.1.1" xref="alg1.l3.m1.2.2.1.2.cmml"><mo id="alg1.l3.m1.2.2.1.1.2" maxsize="80%" minsize="80%" xref="alg1.l3.m1.2.2.1.2.1.cmml">|</mo><mrow id="alg1.l3.m1.2.2.1.1.1" xref="alg1.l3.m1.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.l3.m1.2.2.1.1.1.4" mathsize="80%" xref="alg1.l3.m1.2.2.1.1.1.4.cmml">ℋ</mi><mo id="alg1.l3.m1.2.2.1.1.1.3" xref="alg1.l3.m1.2.2.1.1.1.3.cmml">⁢</mo><mrow id="alg1.l3.m1.2.2.1.1.1.2.2" xref="alg1.l3.m1.2.2.1.1.1.2.3.cmml"><mo id="alg1.l3.m1.2.2.1.1.1.2.2.3" maxsize="80%" minsize="80%" xref="alg1.l3.m1.2.2.1.1.1.2.3.cmml">(</mo><msub id="alg1.l3.m1.2.2.1.1.1.1.1.1" xref="alg1.l3.m1.2.2.1.1.1.1.1.1.cmml"><mi id="alg1.l3.m1.2.2.1.1.1.1.1.1.2" mathsize="80%" xref="alg1.l3.m1.2.2.1.1.1.1.1.1.2.cmml">q</mi><mi id="alg1.l3.m1.2.2.1.1.1.1.1.1.3" mathsize="80%" xref="alg1.l3.m1.2.2.1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="alg1.l3.m1.2.2.1.1.1.2.2.4" mathsize="80%" xref="alg1.l3.m1.2.2.1.1.1.2.3.cmml">,</mo><msub id="alg1.l3.m1.2.2.1.1.1.2.2.2" xref="alg1.l3.m1.2.2.1.1.1.2.2.2.cmml"><mi id="alg1.l3.m1.2.2.1.1.1.2.2.2.2" mathsize="80%" xref="alg1.l3.m1.2.2.1.1.1.2.2.2.2.cmml">a</mi><mi id="alg1.l3.m1.2.2.1.1.1.2.2.2.3" mathsize="80%" xref="alg1.l3.m1.2.2.1.1.1.2.2.2.3.cmml">t</mi></msub><mo id="alg1.l3.m1.2.2.1.1.1.2.2.5" mathsize="80%" xref="alg1.l3.m1.2.2.1.1.1.2.3.cmml">,</mo><mi id="alg1.l3.m1.1.1" mathsize="80%" xref="alg1.l3.m1.1.1.cmml">K</mi><mo id="alg1.l3.m1.2.2.1.1.1.2.2.6" maxsize="80%" minsize="80%" xref="alg1.l3.m1.2.2.1.1.1.2.3.cmml">)</mo></mrow></mrow><mo id="alg1.l3.m1.2.2.1.1.3" maxsize="80%" minsize="80%" xref="alg1.l3.m1.2.2.1.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l3.m1.2b"><apply id="alg1.l3.m1.2.2.cmml" xref="alg1.l3.m1.2.2"><ci id="alg1.l3.m1.2.2.2.cmml" xref="alg1.l3.m1.2.2.2">←</ci><apply id="alg1.l3.m1.2.2.3.cmml" xref="alg1.l3.m1.2.2.3"><csymbol cd="ambiguous" id="alg1.l3.m1.2.2.3.1.cmml" xref="alg1.l3.m1.2.2.3">subscript</csymbol><ci id="alg1.l3.m1.2.2.3.2.cmml" xref="alg1.l3.m1.2.2.3.2">𝑁</ci><ci id="alg1.l3.m1.2.2.3.3.cmml" xref="alg1.l3.m1.2.2.3.3">𝐻</ci></apply><apply id="alg1.l3.m1.2.2.1.2.cmml" xref="alg1.l3.m1.2.2.1.1"><abs id="alg1.l3.m1.2.2.1.2.1.cmml" xref="alg1.l3.m1.2.2.1.1.2"></abs><apply id="alg1.l3.m1.2.2.1.1.1.cmml" xref="alg1.l3.m1.2.2.1.1.1"><times id="alg1.l3.m1.2.2.1.1.1.3.cmml" xref="alg1.l3.m1.2.2.1.1.1.3"></times><ci id="alg1.l3.m1.2.2.1.1.1.4.cmml" xref="alg1.l3.m1.2.2.1.1.1.4">ℋ</ci><vector id="alg1.l3.m1.2.2.1.1.1.2.3.cmml" xref="alg1.l3.m1.2.2.1.1.1.2.2"><apply id="alg1.l3.m1.2.2.1.1.1.1.1.1.cmml" xref="alg1.l3.m1.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg1.l3.m1.2.2.1.1.1.1.1.1.1.cmml" xref="alg1.l3.m1.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="alg1.l3.m1.2.2.1.1.1.1.1.1.2.cmml" xref="alg1.l3.m1.2.2.1.1.1.1.1.1.2">𝑞</ci><ci id="alg1.l3.m1.2.2.1.1.1.1.1.1.3.cmml" xref="alg1.l3.m1.2.2.1.1.1.1.1.1.3">𝑡</ci></apply><apply id="alg1.l3.m1.2.2.1.1.1.2.2.2.cmml" xref="alg1.l3.m1.2.2.1.1.1.2.2.2"><csymbol cd="ambiguous" id="alg1.l3.m1.2.2.1.1.1.2.2.2.1.cmml" xref="alg1.l3.m1.2.2.1.1.1.2.2.2">subscript</csymbol><ci id="alg1.l3.m1.2.2.1.1.1.2.2.2.2.cmml" xref="alg1.l3.m1.2.2.1.1.1.2.2.2.2">𝑎</ci><ci id="alg1.l3.m1.2.2.1.1.1.2.2.2.3.cmml" xref="alg1.l3.m1.2.2.1.1.1.2.2.2.3">𝑡</ci></apply><ci id="alg1.l3.m1.1.1.cmml" xref="alg1.l3.m1.1.1">𝐾</ci></vector></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l3.m1.2c">N_{H}\leftarrow|\mathcal{H}(q_{t},a_{t},K)|</annotation><annotation encoding="application/x-llamapun" id="alg1.l3.m1.2d">italic_N start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ← | caligraphic_H ( italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_K ) |</annotation></semantics></math><span class="ltx_text" id="alg1.l3.2" style="font-size:80%;">, </span><math alttext="N_{\mathcal{G}}\leftarrow|\mathcal{G}_{q_{t}}|" class="ltx_Math" display="inline" id="alg1.l3.m2.1"><semantics id="alg1.l3.m2.1a"><mrow id="alg1.l3.m2.1.1" xref="alg1.l3.m2.1.1.cmml"><msub id="alg1.l3.m2.1.1.3" xref="alg1.l3.m2.1.1.3.cmml"><mi id="alg1.l3.m2.1.1.3.2" mathsize="80%" xref="alg1.l3.m2.1.1.3.2.cmml">N</mi><mi class="ltx_font_mathcaligraphic" id="alg1.l3.m2.1.1.3.3" mathsize="80%" xref="alg1.l3.m2.1.1.3.3.cmml">𝒢</mi></msub><mo id="alg1.l3.m2.1.1.2" mathsize="80%" stretchy="false" xref="alg1.l3.m2.1.1.2.cmml">←</mo><mrow id="alg1.l3.m2.1.1.1.1" xref="alg1.l3.m2.1.1.1.2.cmml"><mo id="alg1.l3.m2.1.1.1.1.2" maxsize="80%" minsize="80%" xref="alg1.l3.m2.1.1.1.2.1.cmml">|</mo><msub id="alg1.l3.m2.1.1.1.1.1" xref="alg1.l3.m2.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.l3.m2.1.1.1.1.1.2" mathsize="80%" xref="alg1.l3.m2.1.1.1.1.1.2.cmml">𝒢</mi><msub id="alg1.l3.m2.1.1.1.1.1.3" xref="alg1.l3.m2.1.1.1.1.1.3.cmml"><mi id="alg1.l3.m2.1.1.1.1.1.3.2" mathsize="80%" xref="alg1.l3.m2.1.1.1.1.1.3.2.cmml">q</mi><mi id="alg1.l3.m2.1.1.1.1.1.3.3" mathsize="80%" xref="alg1.l3.m2.1.1.1.1.1.3.3.cmml">t</mi></msub></msub><mo id="alg1.l3.m2.1.1.1.1.3" maxsize="80%" minsize="80%" xref="alg1.l3.m2.1.1.1.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l3.m2.1b"><apply id="alg1.l3.m2.1.1.cmml" xref="alg1.l3.m2.1.1"><ci id="alg1.l3.m2.1.1.2.cmml" xref="alg1.l3.m2.1.1.2">←</ci><apply id="alg1.l3.m2.1.1.3.cmml" xref="alg1.l3.m2.1.1.3"><csymbol cd="ambiguous" id="alg1.l3.m2.1.1.3.1.cmml" xref="alg1.l3.m2.1.1.3">subscript</csymbol><ci id="alg1.l3.m2.1.1.3.2.cmml" xref="alg1.l3.m2.1.1.3.2">𝑁</ci><ci id="alg1.l3.m2.1.1.3.3.cmml" xref="alg1.l3.m2.1.1.3.3">𝒢</ci></apply><apply id="alg1.l3.m2.1.1.1.2.cmml" xref="alg1.l3.m2.1.1.1.1"><abs id="alg1.l3.m2.1.1.1.2.1.cmml" xref="alg1.l3.m2.1.1.1.1.2"></abs><apply id="alg1.l3.m2.1.1.1.1.1.cmml" xref="alg1.l3.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="alg1.l3.m2.1.1.1.1.1.1.cmml" xref="alg1.l3.m2.1.1.1.1.1">subscript</csymbol><ci id="alg1.l3.m2.1.1.1.1.1.2.cmml" xref="alg1.l3.m2.1.1.1.1.1.2">𝒢</ci><apply id="alg1.l3.m2.1.1.1.1.1.3.cmml" xref="alg1.l3.m2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="alg1.l3.m2.1.1.1.1.1.3.1.cmml" xref="alg1.l3.m2.1.1.1.1.1.3">subscript</csymbol><ci id="alg1.l3.m2.1.1.1.1.1.3.2.cmml" xref="alg1.l3.m2.1.1.1.1.1.3.2">𝑞</ci><ci id="alg1.l3.m2.1.1.1.1.1.3.3.cmml" xref="alg1.l3.m2.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l3.m2.1c">N_{\mathcal{G}}\leftarrow|\mathcal{G}_{q_{t}}|</annotation><annotation encoding="application/x-llamapun" id="alg1.l3.m2.1d">italic_N start_POSTSUBSCRIPT caligraphic_G end_POSTSUBSCRIPT ← | caligraphic_G start_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT |</annotation></semantics></math><span class="ltx_text" id="alg1.l3.3" style="font-size:80%;"> </span> </div> <div class="ltx_listingline" id="alg1.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l4.1.1.1" style="font-size:80%;">4:</span></span><math alttext="\{z^{H}_{i},\lambda^{H}_{i}\}_{i\in[N_{H}]}\leftarrow\{z^{H}_{i},P_{\theta,% \mathcal{G}}\left(a_{t}|q_{t},z^{H}_{i}\right)\}_{z^{H}_{i}\in\mathcal{H}(q_{t% },a_{t},K)}" class="ltx_Math" display="inline" id="alg1.l4.m1.10"><semantics id="alg1.l4.m1.10a"><mrow id="alg1.l4.m1.10.10" xref="alg1.l4.m1.10.10.cmml"><msub id="alg1.l4.m1.8.8.2" xref="alg1.l4.m1.8.8.2.cmml"><mrow id="alg1.l4.m1.8.8.2.2.2" xref="alg1.l4.m1.8.8.2.2.3.cmml"><mo id="alg1.l4.m1.8.8.2.2.2.3" maxsize="80%" minsize="80%" xref="alg1.l4.m1.8.8.2.2.3.cmml">{</mo><msubsup id="alg1.l4.m1.7.7.1.1.1.1" xref="alg1.l4.m1.7.7.1.1.1.1.cmml"><mi id="alg1.l4.m1.7.7.1.1.1.1.2.2" mathsize="80%" xref="alg1.l4.m1.7.7.1.1.1.1.2.2.cmml">z</mi><mi id="alg1.l4.m1.7.7.1.1.1.1.3" mathsize="80%" xref="alg1.l4.m1.7.7.1.1.1.1.3.cmml">i</mi><mi id="alg1.l4.m1.7.7.1.1.1.1.2.3" mathsize="80%" xref="alg1.l4.m1.7.7.1.1.1.1.2.3.cmml">H</mi></msubsup><mo id="alg1.l4.m1.8.8.2.2.2.4" mathsize="80%" xref="alg1.l4.m1.8.8.2.2.3.cmml">,</mo><msubsup id="alg1.l4.m1.8.8.2.2.2.2" xref="alg1.l4.m1.8.8.2.2.2.2.cmml"><mi id="alg1.l4.m1.8.8.2.2.2.2.2.2" mathsize="80%" xref="alg1.l4.m1.8.8.2.2.2.2.2.2.cmml">λ</mi><mi id="alg1.l4.m1.8.8.2.2.2.2.3" mathsize="80%" xref="alg1.l4.m1.8.8.2.2.2.2.3.cmml">i</mi><mi id="alg1.l4.m1.8.8.2.2.2.2.2.3" mathsize="80%" xref="alg1.l4.m1.8.8.2.2.2.2.2.3.cmml">H</mi></msubsup><mo id="alg1.l4.m1.8.8.2.2.2.5" maxsize="80%" minsize="80%" xref="alg1.l4.m1.8.8.2.2.3.cmml">}</mo></mrow><mrow id="alg1.l4.m1.1.1.1" xref="alg1.l4.m1.1.1.1.cmml"><mi id="alg1.l4.m1.1.1.1.3" mathsize="80%" xref="alg1.l4.m1.1.1.1.3.cmml">i</mi><mo id="alg1.l4.m1.1.1.1.2" mathsize="80%" xref="alg1.l4.m1.1.1.1.2.cmml">∈</mo><mrow id="alg1.l4.m1.1.1.1.1.1" xref="alg1.l4.m1.1.1.1.1.2.cmml"><mo id="alg1.l4.m1.1.1.1.1.1.2" maxsize="80%" minsize="80%" xref="alg1.l4.m1.1.1.1.1.2.1.cmml">[</mo><msub id="alg1.l4.m1.1.1.1.1.1.1" xref="alg1.l4.m1.1.1.1.1.1.1.cmml"><mi id="alg1.l4.m1.1.1.1.1.1.1.2" mathsize="80%" xref="alg1.l4.m1.1.1.1.1.1.1.2.cmml">N</mi><mi id="alg1.l4.m1.1.1.1.1.1.1.3" mathsize="80%" xref="alg1.l4.m1.1.1.1.1.1.1.3.cmml">H</mi></msub><mo id="alg1.l4.m1.1.1.1.1.1.3" maxsize="80%" minsize="80%" xref="alg1.l4.m1.1.1.1.1.2.1.cmml">]</mo></mrow></mrow></msub><mo id="alg1.l4.m1.10.10.5" mathsize="80%" stretchy="false" xref="alg1.l4.m1.10.10.5.cmml">←</mo><msub id="alg1.l4.m1.10.10.4" xref="alg1.l4.m1.10.10.4.cmml"><mrow id="alg1.l4.m1.10.10.4.2.2" xref="alg1.l4.m1.10.10.4.2.3.cmml"><mo id="alg1.l4.m1.10.10.4.2.2.3" maxsize="80%" minsize="80%" xref="alg1.l4.m1.10.10.4.2.3.cmml">{</mo><msubsup id="alg1.l4.m1.9.9.3.1.1.1" xref="alg1.l4.m1.9.9.3.1.1.1.cmml"><mi id="alg1.l4.m1.9.9.3.1.1.1.2.2" mathsize="80%" xref="alg1.l4.m1.9.9.3.1.1.1.2.2.cmml">z</mi><mi id="alg1.l4.m1.9.9.3.1.1.1.3" mathsize="80%" xref="alg1.l4.m1.9.9.3.1.1.1.3.cmml">i</mi><mi id="alg1.l4.m1.9.9.3.1.1.1.2.3" mathsize="80%" xref="alg1.l4.m1.9.9.3.1.1.1.2.3.cmml">H</mi></msubsup><mo 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xref="alg1.l4.m1.6.6.3.3.2.2.2"><csymbol cd="ambiguous" id="alg1.l4.m1.6.6.3.3.2.2.2.1.cmml" xref="alg1.l4.m1.6.6.3.3.2.2.2">subscript</csymbol><ci id="alg1.l4.m1.6.6.3.3.2.2.2.2.cmml" xref="alg1.l4.m1.6.6.3.3.2.2.2.2">𝑎</ci><ci id="alg1.l4.m1.6.6.3.3.2.2.2.3.cmml" xref="alg1.l4.m1.6.6.3.3.2.2.2.3">𝑡</ci></apply><ci id="alg1.l4.m1.4.4.1.1.cmml" xref="alg1.l4.m1.4.4.1.1">𝐾</ci></vector></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.m1.10c">\{z^{H}_{i},\lambda^{H}_{i}\}_{i\in[N_{H}]}\leftarrow\{z^{H}_{i},P_{\theta,% \mathcal{G}}\left(a_{t}|q_{t},z^{H}_{i}\right)\}_{z^{H}_{i}\in\mathcal{H}(q_{t% },a_{t},K)}</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.m1.10d">{ italic_z start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_λ start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i ∈ [ italic_N start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ] end_POSTSUBSCRIPT ← { italic_z start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT | italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_z start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) } start_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_H ( italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_K ) end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text" id="alg1.l4.2" style="font-size:80%;"> </span> </div> <div class="ltx_listingline" id="alg1.l5"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" 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id="alg1.l5.m1.5d">italic_z start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 … italic_N start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT end_POSTSUBSCRIPT ← italic_z start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s ( 1 ) … italic_s ( italic_N start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT where italic_s ( italic_i ) = argsort ( italic_λ start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 … italic_N start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_i )</annotation></semantics></math><span class="ltx_text" id="alg1.l5.2" style="font-size:80%;"> </span> </div> <div class="ltx_listingline" id="alg1.l6"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l6.1.1.1" style="font-size:80%;">6:</span></span><math alttext="\{z^{\mathcal{G}}_{i},\lambda^{\mathcal{G}}_{i}\}_{i\in[N_{\mathcal{G}}]}% 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start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT | italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_z start_POSTSUPERSCRIPT caligraphic_G end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) } start_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT caligraphic_G end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_G start_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text" id="alg1.l6.2" style="font-size:80%;"> </span> </div> <div class="ltx_listingline" id="alg1.l7"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l7.1.1.1" style="font-size:80%;">7:</span></span><math alttext="z^{\mathcal{G}}_{1\ldots N_{\mathcal{G}}}\leftarrow z^{\mathcal{G}}_{s(1)% \ldots s(N_{\mathcal{G}})}\text{ where }s(i)=\text{argsort}(\lambda^{\mathcal{% G}}_{1\ldots N_{\mathcal{G}}},i)" class="ltx_Math" display="inline" 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end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 … italic_N start_POSTSUBSCRIPT caligraphic_G end_POSTSUBSCRIPT end_POSTSUBSCRIPT ← italic_z start_POSTSUPERSCRIPT caligraphic_G end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s ( 1 ) … italic_s ( italic_N start_POSTSUBSCRIPT caligraphic_G end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT where italic_s ( italic_i ) = argsort ( italic_λ start_POSTSUPERSCRIPT caligraphic_G end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 … italic_N start_POSTSUBSCRIPT caligraphic_G end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_i )</annotation></semantics></math><span class="ltx_text" id="alg1.l7.2" style="font-size:80%;"> </span> </div> <div class="ltx_listingline" id="alg1.l8"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l8.1.1.1" style="font-size:80%;">8:</span></span><math alttext="\mathcal{L}\leftarrow-\frac{1}{K}\sum_{z\in\mathcal{H}(q_{t},a_{t},K)}\log% \left[P_{\theta,\mathcal{G}}\left(a_{t}|q_{t},z\right)P_{\theta,\mathcal{G}}% 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encoding="application/x-llamapun" id="alg1.l8.m1.10d">caligraphic_L ← - divide start_ARG 1 end_ARG start_ARG italic_K end_ARG ∑ start_POSTSUBSCRIPT italic_z ∈ caligraphic_H ( italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_K ) end_POSTSUBSCRIPT roman_log [ italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT | italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_z ) italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_z | italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ]</annotation></semantics></math><span class="ltx_text" id="alg1.l8.2" style="font-size:80%;"> </span> </div> <div class="ltx_listingline" id="alg1.l9"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l9.1.1.1" style="font-size:80%;">9:</span></span><math alttext="count\_add\leftarrow 0" 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xref="alg1.l9.m1.1.1.2.7.cmml">_</mi><mo id="alg1.l9.m1.1.1.2.1e" xref="alg1.l9.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l9.m1.1.1.2.8" mathsize="80%" xref="alg1.l9.m1.1.1.2.8.cmml">a</mi><mo id="alg1.l9.m1.1.1.2.1f" xref="alg1.l9.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l9.m1.1.1.2.9" mathsize="80%" xref="alg1.l9.m1.1.1.2.9.cmml">d</mi><mo id="alg1.l9.m1.1.1.2.1g" xref="alg1.l9.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l9.m1.1.1.2.10" mathsize="80%" xref="alg1.l9.m1.1.1.2.10.cmml">d</mi></mrow><mo id="alg1.l9.m1.1.1.1" mathsize="80%" stretchy="false" xref="alg1.l9.m1.1.1.1.cmml">←</mo><mn id="alg1.l9.m1.1.1.3" mathsize="80%" xref="alg1.l9.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg1.l9.m1.1b"><apply id="alg1.l9.m1.1.1.cmml" xref="alg1.l9.m1.1.1"><ci id="alg1.l9.m1.1.1.1.cmml" xref="alg1.l9.m1.1.1.1">←</ci><apply id="alg1.l9.m1.1.1.2.cmml" xref="alg1.l9.m1.1.1.2"><times id="alg1.l9.m1.1.1.2.1.cmml" xref="alg1.l9.m1.1.1.2.1"></times><ci id="alg1.l9.m1.1.1.2.2.cmml" xref="alg1.l9.m1.1.1.2.2">𝑐</ci><ci id="alg1.l9.m1.1.1.2.3.cmml" xref="alg1.l9.m1.1.1.2.3">𝑜</ci><ci id="alg1.l9.m1.1.1.2.4.cmml" xref="alg1.l9.m1.1.1.2.4">𝑢</ci><ci id="alg1.l9.m1.1.1.2.5.cmml" xref="alg1.l9.m1.1.1.2.5">𝑛</ci><ci id="alg1.l9.m1.1.1.2.6.cmml" xref="alg1.l9.m1.1.1.2.6">𝑡</ci><ci id="alg1.l9.m1.1.1.2.7.cmml" xref="alg1.l9.m1.1.1.2.7">_</ci><ci id="alg1.l9.m1.1.1.2.8.cmml" xref="alg1.l9.m1.1.1.2.8">𝑎</ci><ci id="alg1.l9.m1.1.1.2.9.cmml" xref="alg1.l9.m1.1.1.2.9">𝑑</ci><ci id="alg1.l9.m1.1.1.2.10.cmml" xref="alg1.l9.m1.1.1.2.10">𝑑</ci></apply><cn id="alg1.l9.m1.1.1.3.cmml" type="integer" xref="alg1.l9.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l9.m1.1c">count\_add\leftarrow 0</annotation><annotation encoding="application/x-llamapun" id="alg1.l9.m1.1d">italic_c italic_o italic_u italic_n italic_t _ italic_a italic_d italic_d ← 0</annotation></semantics></math><span class="ltx_text" id="alg1.l9.2" style="font-size:80%;">, </span><math alttext="count\_remove\leftarrow 0" class="ltx_Math" display="inline" id="alg1.l9.m2.1"><semantics id="alg1.l9.m2.1a"><mrow id="alg1.l9.m2.1.1" xref="alg1.l9.m2.1.1.cmml"><mrow id="alg1.l9.m2.1.1.2" xref="alg1.l9.m2.1.1.2.cmml"><mi id="alg1.l9.m2.1.1.2.2" mathsize="80%" xref="alg1.l9.m2.1.1.2.2.cmml">c</mi><mo id="alg1.l9.m2.1.1.2.1" xref="alg1.l9.m2.1.1.2.1.cmml">⁢</mo><mi id="alg1.l9.m2.1.1.2.3" mathsize="80%" xref="alg1.l9.m2.1.1.2.3.cmml">o</mi><mo id="alg1.l9.m2.1.1.2.1a" xref="alg1.l9.m2.1.1.2.1.cmml">⁢</mo><mi id="alg1.l9.m2.1.1.2.4" mathsize="80%" xref="alg1.l9.m2.1.1.2.4.cmml">u</mi><mo id="alg1.l9.m2.1.1.2.1b" xref="alg1.l9.m2.1.1.2.1.cmml">⁢</mo><mi id="alg1.l9.m2.1.1.2.5" mathsize="80%" xref="alg1.l9.m2.1.1.2.5.cmml">n</mi><mo id="alg1.l9.m2.1.1.2.1c" xref="alg1.l9.m2.1.1.2.1.cmml">⁢</mo><mi id="alg1.l9.m2.1.1.2.6" mathsize="80%" xref="alg1.l9.m2.1.1.2.6.cmml">t</mi><mo id="alg1.l9.m2.1.1.2.1d" xref="alg1.l9.m2.1.1.2.1.cmml">⁢</mo><mi id="alg1.l9.m2.1.1.2.7" mathsize="80%" mathvariant="normal" xref="alg1.l9.m2.1.1.2.7.cmml">_</mi><mo id="alg1.l9.m2.1.1.2.1e" xref="alg1.l9.m2.1.1.2.1.cmml">⁢</mo><mi id="alg1.l9.m2.1.1.2.8" mathsize="80%" xref="alg1.l9.m2.1.1.2.8.cmml">r</mi><mo id="alg1.l9.m2.1.1.2.1f" xref="alg1.l9.m2.1.1.2.1.cmml">⁢</mo><mi id="alg1.l9.m2.1.1.2.9" mathsize="80%" xref="alg1.l9.m2.1.1.2.9.cmml">e</mi><mo id="alg1.l9.m2.1.1.2.1g" xref="alg1.l9.m2.1.1.2.1.cmml">⁢</mo><mi id="alg1.l9.m2.1.1.2.10" mathsize="80%" xref="alg1.l9.m2.1.1.2.10.cmml">m</mi><mo id="alg1.l9.m2.1.1.2.1h" xref="alg1.l9.m2.1.1.2.1.cmml">⁢</mo><mi id="alg1.l9.m2.1.1.2.11" mathsize="80%" xref="alg1.l9.m2.1.1.2.11.cmml">o</mi><mo id="alg1.l9.m2.1.1.2.1i" xref="alg1.l9.m2.1.1.2.1.cmml">⁢</mo><mi id="alg1.l9.m2.1.1.2.12" mathsize="80%" xref="alg1.l9.m2.1.1.2.12.cmml">v</mi><mo id="alg1.l9.m2.1.1.2.1j" xref="alg1.l9.m2.1.1.2.1.cmml">⁢</mo><mi id="alg1.l9.m2.1.1.2.13" mathsize="80%" xref="alg1.l9.m2.1.1.2.13.cmml">e</mi></mrow><mo id="alg1.l9.m2.1.1.1" mathsize="80%" stretchy="false" xref="alg1.l9.m2.1.1.1.cmml">←</mo><mn id="alg1.l9.m2.1.1.3" mathsize="80%" xref="alg1.l9.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg1.l9.m2.1b"><apply id="alg1.l9.m2.1.1.cmml" xref="alg1.l9.m2.1.1"><ci id="alg1.l9.m2.1.1.1.cmml" xref="alg1.l9.m2.1.1.1">←</ci><apply id="alg1.l9.m2.1.1.2.cmml" xref="alg1.l9.m2.1.1.2"><times id="alg1.l9.m2.1.1.2.1.cmml" xref="alg1.l9.m2.1.1.2.1"></times><ci id="alg1.l9.m2.1.1.2.2.cmml" xref="alg1.l9.m2.1.1.2.2">𝑐</ci><ci id="alg1.l9.m2.1.1.2.3.cmml" xref="alg1.l9.m2.1.1.2.3">𝑜</ci><ci id="alg1.l9.m2.1.1.2.4.cmml" xref="alg1.l9.m2.1.1.2.4">𝑢</ci><ci id="alg1.l9.m2.1.1.2.5.cmml" xref="alg1.l9.m2.1.1.2.5">𝑛</ci><ci id="alg1.l9.m2.1.1.2.6.cmml" xref="alg1.l9.m2.1.1.2.6">𝑡</ci><ci id="alg1.l9.m2.1.1.2.7.cmml" xref="alg1.l9.m2.1.1.2.7">_</ci><ci id="alg1.l9.m2.1.1.2.8.cmml" xref="alg1.l9.m2.1.1.2.8">𝑟</ci><ci id="alg1.l9.m2.1.1.2.9.cmml" xref="alg1.l9.m2.1.1.2.9">𝑒</ci><ci id="alg1.l9.m2.1.1.2.10.cmml" xref="alg1.l9.m2.1.1.2.10">𝑚</ci><ci id="alg1.l9.m2.1.1.2.11.cmml" xref="alg1.l9.m2.1.1.2.11">𝑜</ci><ci id="alg1.l9.m2.1.1.2.12.cmml" xref="alg1.l9.m2.1.1.2.12">𝑣</ci><ci id="alg1.l9.m2.1.1.2.13.cmml" xref="alg1.l9.m2.1.1.2.13">𝑒</ci></apply><cn id="alg1.l9.m2.1.1.3.cmml" type="integer" xref="alg1.l9.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l9.m2.1c">count\_remove\leftarrow 0</annotation><annotation encoding="application/x-llamapun" id="alg1.l9.m2.1d">italic_c italic_o italic_u italic_n italic_t _ italic_r italic_e italic_m italic_o italic_v italic_e ← 0</annotation></semantics></math><span class="ltx_text" id="alg1.l9.3" style="font-size:80%;"> </span> </div> <div class="ltx_listingline" id="alg1.l10"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l10.1.1.1" style="font-size:80%;">10:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l10.2" style="font-size:80%;">while</span><span class="ltx_text" id="alg1.l10.3" style="font-size:80%;"> </span><math alttext="count\_add<N_{H}" class="ltx_Math" display="inline" id="alg1.l10.m1.1"><semantics id="alg1.l10.m1.1a"><mrow id="alg1.l10.m1.1.1" xref="alg1.l10.m1.1.1.cmml"><mrow id="alg1.l10.m1.1.1.2" xref="alg1.l10.m1.1.1.2.cmml"><mi id="alg1.l10.m1.1.1.2.2" mathsize="80%" xref="alg1.l10.m1.1.1.2.2.cmml">c</mi><mo id="alg1.l10.m1.1.1.2.1" xref="alg1.l10.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l10.m1.1.1.2.3" mathsize="80%" xref="alg1.l10.m1.1.1.2.3.cmml">o</mi><mo id="alg1.l10.m1.1.1.2.1a" xref="alg1.l10.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l10.m1.1.1.2.4" mathsize="80%" xref="alg1.l10.m1.1.1.2.4.cmml">u</mi><mo id="alg1.l10.m1.1.1.2.1b" xref="alg1.l10.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l10.m1.1.1.2.5" mathsize="80%" xref="alg1.l10.m1.1.1.2.5.cmml">n</mi><mo id="alg1.l10.m1.1.1.2.1c" xref="alg1.l10.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l10.m1.1.1.2.6" mathsize="80%" xref="alg1.l10.m1.1.1.2.6.cmml">t</mi><mo id="alg1.l10.m1.1.1.2.1d" xref="alg1.l10.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l10.m1.1.1.2.7" mathsize="80%" mathvariant="normal" xref="alg1.l10.m1.1.1.2.7.cmml">_</mi><mo id="alg1.l10.m1.1.1.2.1e" xref="alg1.l10.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l10.m1.1.1.2.8" mathsize="80%" xref="alg1.l10.m1.1.1.2.8.cmml">a</mi><mo id="alg1.l10.m1.1.1.2.1f" xref="alg1.l10.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l10.m1.1.1.2.9" mathsize="80%" xref="alg1.l10.m1.1.1.2.9.cmml">d</mi><mo id="alg1.l10.m1.1.1.2.1g" xref="alg1.l10.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l10.m1.1.1.2.10" mathsize="80%" xref="alg1.l10.m1.1.1.2.10.cmml">d</mi></mrow><mo id="alg1.l10.m1.1.1.1" mathsize="80%" xref="alg1.l10.m1.1.1.1.cmml"><</mo><msub id="alg1.l10.m1.1.1.3" xref="alg1.l10.m1.1.1.3.cmml"><mi id="alg1.l10.m1.1.1.3.2" mathsize="80%" xref="alg1.l10.m1.1.1.3.2.cmml">N</mi><mi id="alg1.l10.m1.1.1.3.3" mathsize="80%" xref="alg1.l10.m1.1.1.3.3.cmml">H</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="alg1.l10.m1.1b"><apply id="alg1.l10.m1.1.1.cmml" xref="alg1.l10.m1.1.1"><lt id="alg1.l10.m1.1.1.1.cmml" xref="alg1.l10.m1.1.1.1"></lt><apply id="alg1.l10.m1.1.1.2.cmml" xref="alg1.l10.m1.1.1.2"><times id="alg1.l10.m1.1.1.2.1.cmml" xref="alg1.l10.m1.1.1.2.1"></times><ci id="alg1.l10.m1.1.1.2.2.cmml" xref="alg1.l10.m1.1.1.2.2">𝑐</ci><ci id="alg1.l10.m1.1.1.2.3.cmml" xref="alg1.l10.m1.1.1.2.3">𝑜</ci><ci id="alg1.l10.m1.1.1.2.4.cmml" xref="alg1.l10.m1.1.1.2.4">𝑢</ci><ci id="alg1.l10.m1.1.1.2.5.cmml" xref="alg1.l10.m1.1.1.2.5">𝑛</ci><ci id="alg1.l10.m1.1.1.2.6.cmml" xref="alg1.l10.m1.1.1.2.6">𝑡</ci><ci id="alg1.l10.m1.1.1.2.7.cmml" xref="alg1.l10.m1.1.1.2.7">_</ci><ci id="alg1.l10.m1.1.1.2.8.cmml" xref="alg1.l10.m1.1.1.2.8">𝑎</ci><ci id="alg1.l10.m1.1.1.2.9.cmml" xref="alg1.l10.m1.1.1.2.9">𝑑</ci><ci id="alg1.l10.m1.1.1.2.10.cmml" xref="alg1.l10.m1.1.1.2.10">𝑑</ci></apply><apply id="alg1.l10.m1.1.1.3.cmml" xref="alg1.l10.m1.1.1.3"><csymbol cd="ambiguous" id="alg1.l10.m1.1.1.3.1.cmml" xref="alg1.l10.m1.1.1.3">subscript</csymbol><ci id="alg1.l10.m1.1.1.3.2.cmml" xref="alg1.l10.m1.1.1.3.2">𝑁</ci><ci id="alg1.l10.m1.1.1.3.3.cmml" xref="alg1.l10.m1.1.1.3.3">𝐻</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l10.m1.1c">count\_add<N_{H}</annotation><annotation encoding="application/x-llamapun" id="alg1.l10.m1.1d">italic_c italic_o italic_u italic_n italic_t _ italic_a italic_d italic_d < italic_N start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text" id="alg1.l10.4" style="font-size:80%;"> or </span><math alttext="count\_remove<N_{\mathcal{G}}" class="ltx_Math" display="inline" id="alg1.l10.m2.1"><semantics id="alg1.l10.m2.1a"><mrow id="alg1.l10.m2.1.1" xref="alg1.l10.m2.1.1.cmml"><mrow id="alg1.l10.m2.1.1.2" xref="alg1.l10.m2.1.1.2.cmml"><mi id="alg1.l10.m2.1.1.2.2" mathsize="80%" xref="alg1.l10.m2.1.1.2.2.cmml">c</mi><mo id="alg1.l10.m2.1.1.2.1" xref="alg1.l10.m2.1.1.2.1.cmml">⁢</mo><mi id="alg1.l10.m2.1.1.2.3" mathsize="80%" xref="alg1.l10.m2.1.1.2.3.cmml">o</mi><mo id="alg1.l10.m2.1.1.2.1a" xref="alg1.l10.m2.1.1.2.1.cmml">⁢</mo><mi id="alg1.l10.m2.1.1.2.4" mathsize="80%" xref="alg1.l10.m2.1.1.2.4.cmml">u</mi><mo id="alg1.l10.m2.1.1.2.1b" xref="alg1.l10.m2.1.1.2.1.cmml">⁢</mo><mi id="alg1.l10.m2.1.1.2.5" mathsize="80%" xref="alg1.l10.m2.1.1.2.5.cmml">n</mi><mo id="alg1.l10.m2.1.1.2.1c" xref="alg1.l10.m2.1.1.2.1.cmml">⁢</mo><mi id="alg1.l10.m2.1.1.2.6" mathsize="80%" xref="alg1.l10.m2.1.1.2.6.cmml">t</mi><mo id="alg1.l10.m2.1.1.2.1d" xref="alg1.l10.m2.1.1.2.1.cmml">⁢</mo><mi id="alg1.l10.m2.1.1.2.7" mathsize="80%" mathvariant="normal" xref="alg1.l10.m2.1.1.2.7.cmml">_</mi><mo id="alg1.l10.m2.1.1.2.1e" xref="alg1.l10.m2.1.1.2.1.cmml">⁢</mo><mi id="alg1.l10.m2.1.1.2.8" mathsize="80%" xref="alg1.l10.m2.1.1.2.8.cmml">r</mi><mo id="alg1.l10.m2.1.1.2.1f" xref="alg1.l10.m2.1.1.2.1.cmml">⁢</mo><mi id="alg1.l10.m2.1.1.2.9" mathsize="80%" xref="alg1.l10.m2.1.1.2.9.cmml">e</mi><mo id="alg1.l10.m2.1.1.2.1g" xref="alg1.l10.m2.1.1.2.1.cmml">⁢</mo><mi id="alg1.l10.m2.1.1.2.10" mathsize="80%" xref="alg1.l10.m2.1.1.2.10.cmml">m</mi><mo id="alg1.l10.m2.1.1.2.1h" xref="alg1.l10.m2.1.1.2.1.cmml">⁢</mo><mi id="alg1.l10.m2.1.1.2.11" mathsize="80%" xref="alg1.l10.m2.1.1.2.11.cmml">o</mi><mo id="alg1.l10.m2.1.1.2.1i" xref="alg1.l10.m2.1.1.2.1.cmml">⁢</mo><mi id="alg1.l10.m2.1.1.2.12" mathsize="80%" xref="alg1.l10.m2.1.1.2.12.cmml">v</mi><mo id="alg1.l10.m2.1.1.2.1j" xref="alg1.l10.m2.1.1.2.1.cmml">⁢</mo><mi id="alg1.l10.m2.1.1.2.13" mathsize="80%" xref="alg1.l10.m2.1.1.2.13.cmml">e</mi></mrow><mo id="alg1.l10.m2.1.1.1" mathsize="80%" xref="alg1.l10.m2.1.1.1.cmml"><</mo><msub id="alg1.l10.m2.1.1.3" xref="alg1.l10.m2.1.1.3.cmml"><mi id="alg1.l10.m2.1.1.3.2" mathsize="80%" xref="alg1.l10.m2.1.1.3.2.cmml">N</mi><mi class="ltx_font_mathcaligraphic" id="alg1.l10.m2.1.1.3.3" mathsize="80%" xref="alg1.l10.m2.1.1.3.3.cmml">𝒢</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="alg1.l10.m2.1b"><apply id="alg1.l10.m2.1.1.cmml" xref="alg1.l10.m2.1.1"><lt id="alg1.l10.m2.1.1.1.cmml" xref="alg1.l10.m2.1.1.1"></lt><apply id="alg1.l10.m2.1.1.2.cmml" xref="alg1.l10.m2.1.1.2"><times id="alg1.l10.m2.1.1.2.1.cmml" xref="alg1.l10.m2.1.1.2.1"></times><ci id="alg1.l10.m2.1.1.2.2.cmml" xref="alg1.l10.m2.1.1.2.2">𝑐</ci><ci id="alg1.l10.m2.1.1.2.3.cmml" xref="alg1.l10.m2.1.1.2.3">𝑜</ci><ci id="alg1.l10.m2.1.1.2.4.cmml" xref="alg1.l10.m2.1.1.2.4">𝑢</ci><ci id="alg1.l10.m2.1.1.2.5.cmml" xref="alg1.l10.m2.1.1.2.5">𝑛</ci><ci id="alg1.l10.m2.1.1.2.6.cmml" xref="alg1.l10.m2.1.1.2.6">𝑡</ci><ci id="alg1.l10.m2.1.1.2.7.cmml" xref="alg1.l10.m2.1.1.2.7">_</ci><ci id="alg1.l10.m2.1.1.2.8.cmml" xref="alg1.l10.m2.1.1.2.8">𝑟</ci><ci id="alg1.l10.m2.1.1.2.9.cmml" xref="alg1.l10.m2.1.1.2.9">𝑒</ci><ci id="alg1.l10.m2.1.1.2.10.cmml" xref="alg1.l10.m2.1.1.2.10">𝑚</ci><ci id="alg1.l10.m2.1.1.2.11.cmml" xref="alg1.l10.m2.1.1.2.11">𝑜</ci><ci id="alg1.l10.m2.1.1.2.12.cmml" xref="alg1.l10.m2.1.1.2.12">𝑣</ci><ci id="alg1.l10.m2.1.1.2.13.cmml" xref="alg1.l10.m2.1.1.2.13">𝑒</ci></apply><apply id="alg1.l10.m2.1.1.3.cmml" xref="alg1.l10.m2.1.1.3"><csymbol cd="ambiguous" id="alg1.l10.m2.1.1.3.1.cmml" xref="alg1.l10.m2.1.1.3">subscript</csymbol><ci id="alg1.l10.m2.1.1.3.2.cmml" xref="alg1.l10.m2.1.1.3.2">𝑁</ci><ci id="alg1.l10.m2.1.1.3.3.cmml" xref="alg1.l10.m2.1.1.3.3">𝒢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l10.m2.1c">count\_remove<N_{\mathcal{G}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l10.m2.1d">italic_c italic_o italic_u italic_n italic_t _ italic_r italic_e italic_m italic_o italic_v italic_e < italic_N start_POSTSUBSCRIPT caligraphic_G end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text" id="alg1.l10.5" style="font-size:80%;"> </span><span class="ltx_text ltx_font_bold" id="alg1.l10.6" style="font-size:80%;">do</span><span class="ltx_text" id="alg1.l10.7" style="font-size:80%;"> </span> </div> <div class="ltx_listingline" id="alg1.l11"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l11.1.1.1" style="font-size:80%;">11:</span></span><span class="ltx_text" id="alg1.l11.2" style="font-size:80%;"> </span><span class="ltx_text ltx_font_bold" id="alg1.l11.3" style="font-size:80%;">if</span><span class="ltx_text" id="alg1.l11.4" style="font-size:80%;"> </span><math alttext="count\_add<N_{H}" class="ltx_Math" display="inline" id="alg1.l11.m1.1"><semantics id="alg1.l11.m1.1a"><mrow id="alg1.l11.m1.1.1" xref="alg1.l11.m1.1.1.cmml"><mrow id="alg1.l11.m1.1.1.2" xref="alg1.l11.m1.1.1.2.cmml"><mi id="alg1.l11.m1.1.1.2.2" mathsize="80%" xref="alg1.l11.m1.1.1.2.2.cmml">c</mi><mo id="alg1.l11.m1.1.1.2.1" xref="alg1.l11.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l11.m1.1.1.2.3" mathsize="80%" xref="alg1.l11.m1.1.1.2.3.cmml">o</mi><mo id="alg1.l11.m1.1.1.2.1a" xref="alg1.l11.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l11.m1.1.1.2.4" mathsize="80%" xref="alg1.l11.m1.1.1.2.4.cmml">u</mi><mo id="alg1.l11.m1.1.1.2.1b" xref="alg1.l11.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l11.m1.1.1.2.5" mathsize="80%" xref="alg1.l11.m1.1.1.2.5.cmml">n</mi><mo id="alg1.l11.m1.1.1.2.1c" xref="alg1.l11.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l11.m1.1.1.2.6" mathsize="80%" xref="alg1.l11.m1.1.1.2.6.cmml">t</mi><mo id="alg1.l11.m1.1.1.2.1d" xref="alg1.l11.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l11.m1.1.1.2.7" mathsize="80%" mathvariant="normal" xref="alg1.l11.m1.1.1.2.7.cmml">_</mi><mo id="alg1.l11.m1.1.1.2.1e" xref="alg1.l11.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l11.m1.1.1.2.8" mathsize="80%" xref="alg1.l11.m1.1.1.2.8.cmml">a</mi><mo id="alg1.l11.m1.1.1.2.1f" xref="alg1.l11.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l11.m1.1.1.2.9" mathsize="80%" xref="alg1.l11.m1.1.1.2.9.cmml">d</mi><mo id="alg1.l11.m1.1.1.2.1g" xref="alg1.l11.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l11.m1.1.1.2.10" mathsize="80%" xref="alg1.l11.m1.1.1.2.10.cmml">d</mi></mrow><mo id="alg1.l11.m1.1.1.1" mathsize="80%" xref="alg1.l11.m1.1.1.1.cmml"><</mo><msub id="alg1.l11.m1.1.1.3" xref="alg1.l11.m1.1.1.3.cmml"><mi id="alg1.l11.m1.1.1.3.2" mathsize="80%" xref="alg1.l11.m1.1.1.3.2.cmml">N</mi><mi id="alg1.l11.m1.1.1.3.3" mathsize="80%" xref="alg1.l11.m1.1.1.3.3.cmml">H</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="alg1.l11.m1.1b"><apply id="alg1.l11.m1.1.1.cmml" xref="alg1.l11.m1.1.1"><lt id="alg1.l11.m1.1.1.1.cmml" xref="alg1.l11.m1.1.1.1"></lt><apply id="alg1.l11.m1.1.1.2.cmml" xref="alg1.l11.m1.1.1.2"><times id="alg1.l11.m1.1.1.2.1.cmml" xref="alg1.l11.m1.1.1.2.1"></times><ci id="alg1.l11.m1.1.1.2.2.cmml" xref="alg1.l11.m1.1.1.2.2">𝑐</ci><ci id="alg1.l11.m1.1.1.2.3.cmml" xref="alg1.l11.m1.1.1.2.3">𝑜</ci><ci id="alg1.l11.m1.1.1.2.4.cmml" xref="alg1.l11.m1.1.1.2.4">𝑢</ci><ci id="alg1.l11.m1.1.1.2.5.cmml" xref="alg1.l11.m1.1.1.2.5">𝑛</ci><ci id="alg1.l11.m1.1.1.2.6.cmml" xref="alg1.l11.m1.1.1.2.6">𝑡</ci><ci id="alg1.l11.m1.1.1.2.7.cmml" xref="alg1.l11.m1.1.1.2.7">_</ci><ci id="alg1.l11.m1.1.1.2.8.cmml" xref="alg1.l11.m1.1.1.2.8">𝑎</ci><ci id="alg1.l11.m1.1.1.2.9.cmml" xref="alg1.l11.m1.1.1.2.9">𝑑</ci><ci id="alg1.l11.m1.1.1.2.10.cmml" xref="alg1.l11.m1.1.1.2.10">𝑑</ci></apply><apply id="alg1.l11.m1.1.1.3.cmml" xref="alg1.l11.m1.1.1.3"><csymbol cd="ambiguous" id="alg1.l11.m1.1.1.3.1.cmml" xref="alg1.l11.m1.1.1.3">subscript</csymbol><ci id="alg1.l11.m1.1.1.3.2.cmml" xref="alg1.l11.m1.1.1.3.2">𝑁</ci><ci id="alg1.l11.m1.1.1.3.3.cmml" xref="alg1.l11.m1.1.1.3.3">𝐻</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l11.m1.1c">count\_add<N_{H}</annotation><annotation encoding="application/x-llamapun" id="alg1.l11.m1.1d">italic_c italic_o italic_u italic_n italic_t _ italic_a italic_d italic_d < italic_N start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text" id="alg1.l11.5" style="font-size:80%;"> </span><span class="ltx_text ltx_font_bold" id="alg1.l11.6" style="font-size:80%;">then</span><span class="ltx_text" id="alg1.l11.7" style="font-size:80%;"> </span> </div> <div class="ltx_listingline" id="alg1.l12"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l12.1.1.1" style="font-size:80%;">12:</span></span><span class="ltx_text" id="alg1.l12.2" style="font-size:80%;"> </span><math alttext="count\_add\leftarrow count\_add+1" class="ltx_Math" display="inline" id="alg1.l12.m1.1"><semantics id="alg1.l12.m1.1a"><mrow id="alg1.l12.m1.1.1" xref="alg1.l12.m1.1.1.cmml"><mrow id="alg1.l12.m1.1.1.2" xref="alg1.l12.m1.1.1.2.cmml"><mi id="alg1.l12.m1.1.1.2.2" mathsize="80%" xref="alg1.l12.m1.1.1.2.2.cmml">c</mi><mo id="alg1.l12.m1.1.1.2.1" xref="alg1.l12.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l12.m1.1.1.2.3" mathsize="80%" xref="alg1.l12.m1.1.1.2.3.cmml">o</mi><mo id="alg1.l12.m1.1.1.2.1a" 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id="alg1.l12.m1.1.1.3.2.1d" xref="alg1.l12.m1.1.1.3.2.1.cmml">⁢</mo><mi id="alg1.l12.m1.1.1.3.2.7" mathsize="80%" mathvariant="normal" xref="alg1.l12.m1.1.1.3.2.7.cmml">_</mi><mo id="alg1.l12.m1.1.1.3.2.1e" xref="alg1.l12.m1.1.1.3.2.1.cmml">⁢</mo><mi id="alg1.l12.m1.1.1.3.2.8" mathsize="80%" xref="alg1.l12.m1.1.1.3.2.8.cmml">a</mi><mo id="alg1.l12.m1.1.1.3.2.1f" xref="alg1.l12.m1.1.1.3.2.1.cmml">⁢</mo><mi id="alg1.l12.m1.1.1.3.2.9" mathsize="80%" xref="alg1.l12.m1.1.1.3.2.9.cmml">d</mi><mo id="alg1.l12.m1.1.1.3.2.1g" xref="alg1.l12.m1.1.1.3.2.1.cmml">⁢</mo><mi id="alg1.l12.m1.1.1.3.2.10" mathsize="80%" xref="alg1.l12.m1.1.1.3.2.10.cmml">d</mi></mrow><mo id="alg1.l12.m1.1.1.3.1" mathsize="80%" xref="alg1.l12.m1.1.1.3.1.cmml">+</mo><mn id="alg1.l12.m1.1.1.3.3" mathsize="80%" xref="alg1.l12.m1.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l12.m1.1b"><apply id="alg1.l12.m1.1.1.cmml" xref="alg1.l12.m1.1.1"><ci id="alg1.l12.m1.1.1.1.cmml" 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id="alg1.l12.m1.1d">italic_c italic_o italic_u italic_n italic_t _ italic_a italic_d italic_d ← italic_c italic_o italic_u italic_n italic_t _ italic_a italic_d italic_d + 1</annotation></semantics></math><span class="ltx_text" id="alg1.l12.3" style="font-size:80%;"> </span> </div> <div class="ltx_listingline" id="alg1.l13"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l13.1.1.1" style="font-size:80%;">13:</span></span><span class="ltx_text" id="alg1.l13.2" style="font-size:80%;"> </span><math alttext="\mathcal{G}\leftarrow\mathcal{G}\cup{z^{H}_{count\_add}}" class="ltx_Math" display="inline" id="alg1.l13.m1.1"><semantics id="alg1.l13.m1.1a"><mrow id="alg1.l13.m1.1.1" xref="alg1.l13.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.l13.m1.1.1.2" mathsize="80%" xref="alg1.l13.m1.1.1.2.cmml">𝒢</mi><mo id="alg1.l13.m1.1.1.1" mathsize="80%" stretchy="false" xref="alg1.l13.m1.1.1.1.cmml">←</mo><mrow id="alg1.l13.m1.1.1.3" xref="alg1.l13.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.l13.m1.1.1.3.2" mathsize="80%" xref="alg1.l13.m1.1.1.3.2.cmml">𝒢</mi><mo id="alg1.l13.m1.1.1.3.1" mathsize="80%" xref="alg1.l13.m1.1.1.3.1.cmml">∪</mo><msubsup id="alg1.l13.m1.1.1.3.3" xref="alg1.l13.m1.1.1.3.3.cmml"><mi id="alg1.l13.m1.1.1.3.3.2.2" mathsize="80%" xref="alg1.l13.m1.1.1.3.3.2.2.cmml">z</mi><mrow id="alg1.l13.m1.1.1.3.3.3" xref="alg1.l13.m1.1.1.3.3.3.cmml"><mi id="alg1.l13.m1.1.1.3.3.3.2" mathsize="80%" xref="alg1.l13.m1.1.1.3.3.3.2.cmml">c</mi><mo id="alg1.l13.m1.1.1.3.3.3.1" xref="alg1.l13.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="alg1.l13.m1.1.1.3.3.3.3" mathsize="80%" xref="alg1.l13.m1.1.1.3.3.3.3.cmml">o</mi><mo id="alg1.l13.m1.1.1.3.3.3.1a" xref="alg1.l13.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="alg1.l13.m1.1.1.3.3.3.4" mathsize="80%" xref="alg1.l13.m1.1.1.3.3.3.4.cmml">u</mi><mo id="alg1.l13.m1.1.1.3.3.3.1b" xref="alg1.l13.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="alg1.l13.m1.1.1.3.3.3.5" mathsize="80%" xref="alg1.l13.m1.1.1.3.3.3.5.cmml">n</mi><mo id="alg1.l13.m1.1.1.3.3.3.1c" xref="alg1.l13.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="alg1.l13.m1.1.1.3.3.3.6" mathsize="80%" xref="alg1.l13.m1.1.1.3.3.3.6.cmml">t</mi><mo id="alg1.l13.m1.1.1.3.3.3.1d" xref="alg1.l13.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="alg1.l13.m1.1.1.3.3.3.7" mathsize="80%" mathvariant="normal" xref="alg1.l13.m1.1.1.3.3.3.7.cmml">_</mi><mo id="alg1.l13.m1.1.1.3.3.3.1e" xref="alg1.l13.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="alg1.l13.m1.1.1.3.3.3.8" mathsize="80%" xref="alg1.l13.m1.1.1.3.3.3.8.cmml">a</mi><mo id="alg1.l13.m1.1.1.3.3.3.1f" xref="alg1.l13.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="alg1.l13.m1.1.1.3.3.3.9" mathsize="80%" xref="alg1.l13.m1.1.1.3.3.3.9.cmml">d</mi><mo id="alg1.l13.m1.1.1.3.3.3.1g" xref="alg1.l13.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="alg1.l13.m1.1.1.3.3.3.10" mathsize="80%" xref="alg1.l13.m1.1.1.3.3.3.10.cmml">d</mi></mrow><mi id="alg1.l13.m1.1.1.3.3.2.3" mathsize="80%" xref="alg1.l13.m1.1.1.3.3.2.3.cmml">H</mi></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l13.m1.1b"><apply id="alg1.l13.m1.1.1.cmml" xref="alg1.l13.m1.1.1"><ci id="alg1.l13.m1.1.1.1.cmml" xref="alg1.l13.m1.1.1.1">←</ci><ci id="alg1.l13.m1.1.1.2.cmml" xref="alg1.l13.m1.1.1.2">𝒢</ci><apply id="alg1.l13.m1.1.1.3.cmml" xref="alg1.l13.m1.1.1.3"><union id="alg1.l13.m1.1.1.3.1.cmml" xref="alg1.l13.m1.1.1.3.1"></union><ci id="alg1.l13.m1.1.1.3.2.cmml" xref="alg1.l13.m1.1.1.3.2">𝒢</ci><apply id="alg1.l13.m1.1.1.3.3.cmml" xref="alg1.l13.m1.1.1.3.3"><csymbol cd="ambiguous" id="alg1.l13.m1.1.1.3.3.1.cmml" xref="alg1.l13.m1.1.1.3.3">subscript</csymbol><apply id="alg1.l13.m1.1.1.3.3.2.cmml" xref="alg1.l13.m1.1.1.3.3"><csymbol cd="ambiguous" id="alg1.l13.m1.1.1.3.3.2.1.cmml" xref="alg1.l13.m1.1.1.3.3">superscript</csymbol><ci id="alg1.l13.m1.1.1.3.3.2.2.cmml" xref="alg1.l13.m1.1.1.3.3.2.2">𝑧</ci><ci id="alg1.l13.m1.1.1.3.3.2.3.cmml" xref="alg1.l13.m1.1.1.3.3.2.3">𝐻</ci></apply><apply id="alg1.l13.m1.1.1.3.3.3.cmml" xref="alg1.l13.m1.1.1.3.3.3"><times id="alg1.l13.m1.1.1.3.3.3.1.cmml" xref="alg1.l13.m1.1.1.3.3.3.1"></times><ci id="alg1.l13.m1.1.1.3.3.3.2.cmml" xref="alg1.l13.m1.1.1.3.3.3.2">𝑐</ci><ci id="alg1.l13.m1.1.1.3.3.3.3.cmml" xref="alg1.l13.m1.1.1.3.3.3.3">𝑜</ci><ci id="alg1.l13.m1.1.1.3.3.3.4.cmml" xref="alg1.l13.m1.1.1.3.3.3.4">𝑢</ci><ci id="alg1.l13.m1.1.1.3.3.3.5.cmml" xref="alg1.l13.m1.1.1.3.3.3.5">𝑛</ci><ci id="alg1.l13.m1.1.1.3.3.3.6.cmml" xref="alg1.l13.m1.1.1.3.3.3.6">𝑡</ci><ci id="alg1.l13.m1.1.1.3.3.3.7.cmml" xref="alg1.l13.m1.1.1.3.3.3.7">_</ci><ci id="alg1.l13.m1.1.1.3.3.3.8.cmml" xref="alg1.l13.m1.1.1.3.3.3.8">𝑎</ci><ci id="alg1.l13.m1.1.1.3.3.3.9.cmml" xref="alg1.l13.m1.1.1.3.3.3.9">𝑑</ci><ci id="alg1.l13.m1.1.1.3.3.3.10.cmml" xref="alg1.l13.m1.1.1.3.3.3.10">𝑑</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l13.m1.1c">\mathcal{G}\leftarrow\mathcal{G}\cup{z^{H}_{count\_add}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l13.m1.1d">caligraphic_G ← caligraphic_G ∪ italic_z start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c italic_o italic_u italic_n italic_t _ italic_a italic_d italic_d end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text" id="alg1.l13.3" style="font-size:80%;"> </span> </div> <div class="ltx_listingline" id="alg1.l14"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l14.1.1.1" style="font-size:80%;">14:</span></span><span class="ltx_text" id="alg1.l14.2" style="font-size:80%;"> </span><math alttext="\mathcal{L}\leftarrow-\frac{1}{K}\sum_{z\in\mathcal{H}(q_{t},a_{t},K)}\log% \left[P_{\theta,\mathcal{G}}\left(a_{t}|q_{t},z\right)P_{\theta,\mathcal{G}}% \left(z|q_{t}\right)\right]" class="ltx_Math" display="inline" id="alg1.l14.m1.10"><semantics id="alg1.l14.m1.10a"><mrow 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id="alg1.l14.m1.10.10.1.1.1.2.2" maxsize="80%" minsize="80%" stretchy="true" xref="alg1.l14.m1.10.10.1.1.1.2.2.cmml">∑</mo><mrow id="alg1.l14.m1.3.3.3" xref="alg1.l14.m1.3.3.3.cmml"><mi id="alg1.l14.m1.3.3.3.5" mathsize="80%" xref="alg1.l14.m1.3.3.3.5.cmml">z</mi><mo id="alg1.l14.m1.3.3.3.4" mathsize="80%" xref="alg1.l14.m1.3.3.3.4.cmml">∈</mo><mrow id="alg1.l14.m1.3.3.3.3" xref="alg1.l14.m1.3.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.l14.m1.3.3.3.3.4" mathsize="80%" xref="alg1.l14.m1.3.3.3.3.4.cmml">ℋ</mi><mo id="alg1.l14.m1.3.3.3.3.3" xref="alg1.l14.m1.3.3.3.3.3.cmml">⁢</mo><mrow id="alg1.l14.m1.3.3.3.3.2.2" xref="alg1.l14.m1.3.3.3.3.2.3.cmml"><mo id="alg1.l14.m1.3.3.3.3.2.2.3" maxsize="80%" minsize="80%" xref="alg1.l14.m1.3.3.3.3.2.3.cmml">(</mo><msub id="alg1.l14.m1.2.2.2.2.1.1.1" xref="alg1.l14.m1.2.2.2.2.1.1.1.cmml"><mi id="alg1.l14.m1.2.2.2.2.1.1.1.2" mathsize="80%" xref="alg1.l14.m1.2.2.2.2.1.1.1.2.cmml">q</mi><mi id="alg1.l14.m1.2.2.2.2.1.1.1.3" mathsize="80%" xref="alg1.l14.m1.2.2.2.2.1.1.1.3.cmml">t</mi></msub><mo id="alg1.l14.m1.3.3.3.3.2.2.4" mathsize="80%" xref="alg1.l14.m1.3.3.3.3.2.3.cmml">,</mo><msub id="alg1.l14.m1.3.3.3.3.2.2.2" xref="alg1.l14.m1.3.3.3.3.2.2.2.cmml"><mi id="alg1.l14.m1.3.3.3.3.2.2.2.2" mathsize="80%" xref="alg1.l14.m1.3.3.3.3.2.2.2.2.cmml">a</mi><mi id="alg1.l14.m1.3.3.3.3.2.2.2.3" mathsize="80%" xref="alg1.l14.m1.3.3.3.3.2.2.2.3.cmml">t</mi></msub><mo id="alg1.l14.m1.3.3.3.3.2.2.5" mathsize="80%" xref="alg1.l14.m1.3.3.3.3.2.3.cmml">,</mo><mi id="alg1.l14.m1.1.1.1.1" mathsize="80%" xref="alg1.l14.m1.1.1.1.1.cmml">K</mi><mo id="alg1.l14.m1.3.3.3.3.2.2.6" maxsize="80%" minsize="80%" xref="alg1.l14.m1.3.3.3.3.2.3.cmml">)</mo></mrow></mrow></mrow></msub><mrow id="alg1.l14.m1.10.10.1.1.1.1.1" xref="alg1.l14.m1.10.10.1.1.1.1.2.cmml"><mi id="alg1.l14.m1.9.9" mathsize="80%" xref="alg1.l14.m1.9.9.cmml">log</mi><mo id="alg1.l14.m1.10.10.1.1.1.1.1a" xref="alg1.l14.m1.10.10.1.1.1.1.2.cmml">⁡</mo><mrow 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id="alg1.l14.m1.10c">\mathcal{L}\leftarrow-\frac{1}{K}\sum_{z\in\mathcal{H}(q_{t},a_{t},K)}\log% \left[P_{\theta,\mathcal{G}}\left(a_{t}|q_{t},z\right)P_{\theta,\mathcal{G}}% \left(z|q_{t}\right)\right]</annotation><annotation encoding="application/x-llamapun" id="alg1.l14.m1.10d">caligraphic_L ← - divide start_ARG 1 end_ARG start_ARG italic_K end_ARG ∑ start_POSTSUBSCRIPT italic_z ∈ caligraphic_H ( italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_K ) end_POSTSUBSCRIPT roman_log [ italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT | italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_z ) italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_z | italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ]</annotation></semantics></math><span class="ltx_text" id="alg1.l14.3" style="font-size:80%;"> </span> </div> <div class="ltx_listingline" id="alg1.l15"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l15.1.1.1" style="font-size:80%;">15:</span></span><span class="ltx_text" id="alg1.l15.2" style="font-size:80%;"> </span><span class="ltx_text ltx_font_bold" id="alg1.l15.3" style="font-size:80%;">if</span><span class="ltx_text" id="alg1.l15.4" style="font-size:80%;"> </span><math alttext="\mathcal{L}\leq\epsilon" class="ltx_Math" display="inline" id="alg1.l15.m1.1"><semantics id="alg1.l15.m1.1a"><mrow id="alg1.l15.m1.1.1" xref="alg1.l15.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.l15.m1.1.1.2" mathsize="80%" xref="alg1.l15.m1.1.1.2.cmml">ℒ</mi><mo id="alg1.l15.m1.1.1.1" mathsize="80%" xref="alg1.l15.m1.1.1.1.cmml">≤</mo><mi id="alg1.l15.m1.1.1.3" mathsize="80%" xref="alg1.l15.m1.1.1.3.cmml">ϵ</mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l15.m1.1b"><apply id="alg1.l15.m1.1.1.cmml" xref="alg1.l15.m1.1.1"><leq id="alg1.l15.m1.1.1.1.cmml" xref="alg1.l15.m1.1.1.1"></leq><ci id="alg1.l15.m1.1.1.2.cmml" xref="alg1.l15.m1.1.1.2">ℒ</ci><ci id="alg1.l15.m1.1.1.3.cmml" xref="alg1.l15.m1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l15.m1.1c">\mathcal{L}\leq\epsilon</annotation><annotation encoding="application/x-llamapun" id="alg1.l15.m1.1d">caligraphic_L ≤ italic_ϵ</annotation></semantics></math><span class="ltx_text" id="alg1.l15.5" style="font-size:80%;"> </span><span class="ltx_text ltx_font_bold" id="alg1.l15.6" style="font-size:80%;">then Break</span><span class="ltx_text" id="alg1.l15.7" style="font-size:80%;"> </span> </div> <div class="ltx_listingline" id="alg1.l16"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l16.1.1.1" style="font-size:80%;">16:</span></span><span class="ltx_text" id="alg1.l16.2" style="font-size:80%;"> </span><span class="ltx_text ltx_font_bold" id="alg1.l16.3" style="font-size:80%;">end</span><span class="ltx_text" id="alg1.l16.4" style="font-size:80%;"> </span><span class="ltx_text ltx_font_bold" id="alg1.l16.5" style="font-size:80%;">if</span> </div> <div class="ltx_listingline" id="alg1.l17"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l17.1.1.1" style="font-size:80%;">17:</span></span><span class="ltx_text" id="alg1.l17.2" style="font-size:80%;"> </span><span class="ltx_text ltx_font_bold" id="alg1.l17.3" style="font-size:80%;">if</span><span class="ltx_text" id="alg1.l17.4" style="font-size:80%;"> </span><math alttext="count\_remove<N_{\mathcal{G}}" class="ltx_Math" display="inline" id="alg1.l17.m1.1"><semantics id="alg1.l17.m1.1a"><mrow id="alg1.l17.m1.1.1" xref="alg1.l17.m1.1.1.cmml"><mrow id="alg1.l17.m1.1.1.2" xref="alg1.l17.m1.1.1.2.cmml"><mi id="alg1.l17.m1.1.1.2.2" mathsize="80%" xref="alg1.l17.m1.1.1.2.2.cmml">c</mi><mo id="alg1.l17.m1.1.1.2.1" xref="alg1.l17.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l17.m1.1.1.2.3" mathsize="80%" xref="alg1.l17.m1.1.1.2.3.cmml">o</mi><mo id="alg1.l17.m1.1.1.2.1a" xref="alg1.l17.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l17.m1.1.1.2.4" mathsize="80%" xref="alg1.l17.m1.1.1.2.4.cmml">u</mi><mo id="alg1.l17.m1.1.1.2.1b" xref="alg1.l17.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l17.m1.1.1.2.5" mathsize="80%" xref="alg1.l17.m1.1.1.2.5.cmml">n</mi><mo id="alg1.l17.m1.1.1.2.1c" xref="alg1.l17.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l17.m1.1.1.2.6" mathsize="80%" xref="alg1.l17.m1.1.1.2.6.cmml">t</mi><mo id="alg1.l17.m1.1.1.2.1d" xref="alg1.l17.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l17.m1.1.1.2.7" mathsize="80%" mathvariant="normal" xref="alg1.l17.m1.1.1.2.7.cmml">_</mi><mo id="alg1.l17.m1.1.1.2.1e" xref="alg1.l17.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l17.m1.1.1.2.8" mathsize="80%" xref="alg1.l17.m1.1.1.2.8.cmml">r</mi><mo id="alg1.l17.m1.1.1.2.1f" xref="alg1.l17.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l17.m1.1.1.2.9" mathsize="80%" xref="alg1.l17.m1.1.1.2.9.cmml">e</mi><mo id="alg1.l17.m1.1.1.2.1g" xref="alg1.l17.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l17.m1.1.1.2.10" mathsize="80%" xref="alg1.l17.m1.1.1.2.10.cmml">m</mi><mo id="alg1.l17.m1.1.1.2.1h" xref="alg1.l17.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l17.m1.1.1.2.11" mathsize="80%" xref="alg1.l17.m1.1.1.2.11.cmml">o</mi><mo id="alg1.l17.m1.1.1.2.1i" xref="alg1.l17.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l17.m1.1.1.2.12" mathsize="80%" xref="alg1.l17.m1.1.1.2.12.cmml">v</mi><mo id="alg1.l17.m1.1.1.2.1j" xref="alg1.l17.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l17.m1.1.1.2.13" mathsize="80%" xref="alg1.l17.m1.1.1.2.13.cmml">e</mi></mrow><mo id="alg1.l17.m1.1.1.1" mathsize="80%" xref="alg1.l17.m1.1.1.1.cmml"><</mo><msub id="alg1.l17.m1.1.1.3" xref="alg1.l17.m1.1.1.3.cmml"><mi id="alg1.l17.m1.1.1.3.2" mathsize="80%" xref="alg1.l17.m1.1.1.3.2.cmml">N</mi><mi class="ltx_font_mathcaligraphic" id="alg1.l17.m1.1.1.3.3" mathsize="80%" xref="alg1.l17.m1.1.1.3.3.cmml">𝒢</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="alg1.l17.m1.1b"><apply id="alg1.l17.m1.1.1.cmml" xref="alg1.l17.m1.1.1"><lt id="alg1.l17.m1.1.1.1.cmml" xref="alg1.l17.m1.1.1.1"></lt><apply id="alg1.l17.m1.1.1.2.cmml" xref="alg1.l17.m1.1.1.2"><times id="alg1.l17.m1.1.1.2.1.cmml" xref="alg1.l17.m1.1.1.2.1"></times><ci id="alg1.l17.m1.1.1.2.2.cmml" xref="alg1.l17.m1.1.1.2.2">𝑐</ci><ci id="alg1.l17.m1.1.1.2.3.cmml" xref="alg1.l17.m1.1.1.2.3">𝑜</ci><ci id="alg1.l17.m1.1.1.2.4.cmml" xref="alg1.l17.m1.1.1.2.4">𝑢</ci><ci id="alg1.l17.m1.1.1.2.5.cmml" xref="alg1.l17.m1.1.1.2.5">𝑛</ci><ci id="alg1.l17.m1.1.1.2.6.cmml" xref="alg1.l17.m1.1.1.2.6">𝑡</ci><ci id="alg1.l17.m1.1.1.2.7.cmml" xref="alg1.l17.m1.1.1.2.7">_</ci><ci id="alg1.l17.m1.1.1.2.8.cmml" xref="alg1.l17.m1.1.1.2.8">𝑟</ci><ci id="alg1.l17.m1.1.1.2.9.cmml" xref="alg1.l17.m1.1.1.2.9">𝑒</ci><ci id="alg1.l17.m1.1.1.2.10.cmml" xref="alg1.l17.m1.1.1.2.10">𝑚</ci><ci id="alg1.l17.m1.1.1.2.11.cmml" xref="alg1.l17.m1.1.1.2.11">𝑜</ci><ci id="alg1.l17.m1.1.1.2.12.cmml" xref="alg1.l17.m1.1.1.2.12">𝑣</ci><ci id="alg1.l17.m1.1.1.2.13.cmml" xref="alg1.l17.m1.1.1.2.13">𝑒</ci></apply><apply id="alg1.l17.m1.1.1.3.cmml" xref="alg1.l17.m1.1.1.3"><csymbol cd="ambiguous" id="alg1.l17.m1.1.1.3.1.cmml" xref="alg1.l17.m1.1.1.3">subscript</csymbol><ci id="alg1.l17.m1.1.1.3.2.cmml" xref="alg1.l17.m1.1.1.3.2">𝑁</ci><ci id="alg1.l17.m1.1.1.3.3.cmml" xref="alg1.l17.m1.1.1.3.3">𝒢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l17.m1.1c">count\_remove<N_{\mathcal{G}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l17.m1.1d">italic_c italic_o italic_u italic_n italic_t _ italic_r italic_e italic_m italic_o italic_v italic_e < italic_N start_POSTSUBSCRIPT caligraphic_G end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text" id="alg1.l17.5" style="font-size:80%;"> </span><span class="ltx_text ltx_font_bold" id="alg1.l17.6" style="font-size:80%;">then</span><span class="ltx_text" id="alg1.l17.7" style="font-size:80%;"> </span> </div> <div class="ltx_listingline" id="alg1.l18"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l18.1.1.1" style="font-size:80%;">18:</span></span><span class="ltx_text" id="alg1.l18.2" style="font-size:80%;"> </span><math alttext="count\_remove\leftarrow count\_remove+1" class="ltx_Math" display="inline" id="alg1.l18.m1.1"><semantics id="alg1.l18.m1.1a"><mrow id="alg1.l18.m1.1.1" xref="alg1.l18.m1.1.1.cmml"><mrow id="alg1.l18.m1.1.1.2" xref="alg1.l18.m1.1.1.2.cmml"><mi id="alg1.l18.m1.1.1.2.2" mathsize="80%" xref="alg1.l18.m1.1.1.2.2.cmml">c</mi><mo id="alg1.l18.m1.1.1.2.1" xref="alg1.l18.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l18.m1.1.1.2.3" mathsize="80%" xref="alg1.l18.m1.1.1.2.3.cmml">o</mi><mo id="alg1.l18.m1.1.1.2.1a" xref="alg1.l18.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l18.m1.1.1.2.4" mathsize="80%" xref="alg1.l18.m1.1.1.2.4.cmml">u</mi><mo id="alg1.l18.m1.1.1.2.1b" xref="alg1.l18.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l18.m1.1.1.2.5" mathsize="80%" xref="alg1.l18.m1.1.1.2.5.cmml">n</mi><mo id="alg1.l18.m1.1.1.2.1c" xref="alg1.l18.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l18.m1.1.1.2.6" mathsize="80%" xref="alg1.l18.m1.1.1.2.6.cmml">t</mi><mo id="alg1.l18.m1.1.1.2.1d" xref="alg1.l18.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l18.m1.1.1.2.7" mathsize="80%" mathvariant="normal" xref="alg1.l18.m1.1.1.2.7.cmml">_</mi><mo id="alg1.l18.m1.1.1.2.1e" xref="alg1.l18.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l18.m1.1.1.2.8" mathsize="80%" xref="alg1.l18.m1.1.1.2.8.cmml">r</mi><mo id="alg1.l18.m1.1.1.2.1f" xref="alg1.l18.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l18.m1.1.1.2.9" mathsize="80%" xref="alg1.l18.m1.1.1.2.9.cmml">e</mi><mo id="alg1.l18.m1.1.1.2.1g" xref="alg1.l18.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l18.m1.1.1.2.10" mathsize="80%" xref="alg1.l18.m1.1.1.2.10.cmml">m</mi><mo id="alg1.l18.m1.1.1.2.1h" xref="alg1.l18.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l18.m1.1.1.2.11" mathsize="80%" xref="alg1.l18.m1.1.1.2.11.cmml">o</mi><mo id="alg1.l18.m1.1.1.2.1i" xref="alg1.l18.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l18.m1.1.1.2.12" mathsize="80%" xref="alg1.l18.m1.1.1.2.12.cmml">v</mi><mo id="alg1.l18.m1.1.1.2.1j" xref="alg1.l18.m1.1.1.2.1.cmml">⁢</mo><mi id="alg1.l18.m1.1.1.2.13" mathsize="80%" xref="alg1.l18.m1.1.1.2.13.cmml">e</mi></mrow><mo id="alg1.l18.m1.1.1.1" mathsize="80%" stretchy="false" xref="alg1.l18.m1.1.1.1.cmml">←</mo><mrow id="alg1.l18.m1.1.1.3" xref="alg1.l18.m1.1.1.3.cmml"><mrow id="alg1.l18.m1.1.1.3.2" xref="alg1.l18.m1.1.1.3.2.cmml"><mi id="alg1.l18.m1.1.1.3.2.2" mathsize="80%" xref="alg1.l18.m1.1.1.3.2.2.cmml">c</mi><mo id="alg1.l18.m1.1.1.3.2.1" xref="alg1.l18.m1.1.1.3.2.1.cmml">⁢</mo><mi id="alg1.l18.m1.1.1.3.2.3" mathsize="80%" xref="alg1.l18.m1.1.1.3.2.3.cmml">o</mi><mo id="alg1.l18.m1.1.1.3.2.1a" xref="alg1.l18.m1.1.1.3.2.1.cmml">⁢</mo><mi id="alg1.l18.m1.1.1.3.2.4" mathsize="80%" xref="alg1.l18.m1.1.1.3.2.4.cmml">u</mi><mo id="alg1.l18.m1.1.1.3.2.1b" xref="alg1.l18.m1.1.1.3.2.1.cmml">⁢</mo><mi id="alg1.l18.m1.1.1.3.2.5" mathsize="80%" xref="alg1.l18.m1.1.1.3.2.5.cmml">n</mi><mo id="alg1.l18.m1.1.1.3.2.1c" xref="alg1.l18.m1.1.1.3.2.1.cmml">⁢</mo><mi id="alg1.l18.m1.1.1.3.2.6" mathsize="80%" xref="alg1.l18.m1.1.1.3.2.6.cmml">t</mi><mo id="alg1.l18.m1.1.1.3.2.1d" xref="alg1.l18.m1.1.1.3.2.1.cmml">⁢</mo><mi id="alg1.l18.m1.1.1.3.2.7" mathsize="80%" mathvariant="normal" xref="alg1.l18.m1.1.1.3.2.7.cmml">_</mi><mo id="alg1.l18.m1.1.1.3.2.1e" xref="alg1.l18.m1.1.1.3.2.1.cmml">⁢</mo><mi id="alg1.l18.m1.1.1.3.2.8" mathsize="80%" xref="alg1.l18.m1.1.1.3.2.8.cmml">r</mi><mo id="alg1.l18.m1.1.1.3.2.1f" xref="alg1.l18.m1.1.1.3.2.1.cmml">⁢</mo><mi id="alg1.l18.m1.1.1.3.2.9" mathsize="80%" xref="alg1.l18.m1.1.1.3.2.9.cmml">e</mi><mo id="alg1.l18.m1.1.1.3.2.1g" xref="alg1.l18.m1.1.1.3.2.1.cmml">⁢</mo><mi id="alg1.l18.m1.1.1.3.2.10" mathsize="80%" xref="alg1.l18.m1.1.1.3.2.10.cmml">m</mi><mo id="alg1.l18.m1.1.1.3.2.1h" xref="alg1.l18.m1.1.1.3.2.1.cmml">⁢</mo><mi id="alg1.l18.m1.1.1.3.2.11" mathsize="80%" xref="alg1.l18.m1.1.1.3.2.11.cmml">o</mi><mo id="alg1.l18.m1.1.1.3.2.1i" xref="alg1.l18.m1.1.1.3.2.1.cmml">⁢</mo><mi id="alg1.l18.m1.1.1.3.2.12" mathsize="80%" xref="alg1.l18.m1.1.1.3.2.12.cmml">v</mi><mo id="alg1.l18.m1.1.1.3.2.1j" xref="alg1.l18.m1.1.1.3.2.1.cmml">⁢</mo><mi id="alg1.l18.m1.1.1.3.2.13" mathsize="80%" xref="alg1.l18.m1.1.1.3.2.13.cmml">e</mi></mrow><mo id="alg1.l18.m1.1.1.3.1" mathsize="80%" xref="alg1.l18.m1.1.1.3.1.cmml">+</mo><mn id="alg1.l18.m1.1.1.3.3" mathsize="80%" xref="alg1.l18.m1.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l18.m1.1b"><apply id="alg1.l18.m1.1.1.cmml" xref="alg1.l18.m1.1.1"><ci id="alg1.l18.m1.1.1.1.cmml" xref="alg1.l18.m1.1.1.1">←</ci><apply id="alg1.l18.m1.1.1.2.cmml" xref="alg1.l18.m1.1.1.2"><times id="alg1.l18.m1.1.1.2.1.cmml" xref="alg1.l18.m1.1.1.2.1"></times><ci id="alg1.l18.m1.1.1.2.2.cmml" xref="alg1.l18.m1.1.1.2.2">𝑐</ci><ci id="alg1.l18.m1.1.1.2.3.cmml" xref="alg1.l18.m1.1.1.2.3">𝑜</ci><ci id="alg1.l18.m1.1.1.2.4.cmml" xref="alg1.l18.m1.1.1.2.4">𝑢</ci><ci id="alg1.l18.m1.1.1.2.5.cmml" xref="alg1.l18.m1.1.1.2.5">𝑛</ci><ci id="alg1.l18.m1.1.1.2.6.cmml" xref="alg1.l18.m1.1.1.2.6">𝑡</ci><ci id="alg1.l18.m1.1.1.2.7.cmml" xref="alg1.l18.m1.1.1.2.7">_</ci><ci id="alg1.l18.m1.1.1.2.8.cmml" xref="alg1.l18.m1.1.1.2.8">𝑟</ci><ci id="alg1.l18.m1.1.1.2.9.cmml" xref="alg1.l18.m1.1.1.2.9">𝑒</ci><ci id="alg1.l18.m1.1.1.2.10.cmml" xref="alg1.l18.m1.1.1.2.10">𝑚</ci><ci id="alg1.l18.m1.1.1.2.11.cmml" xref="alg1.l18.m1.1.1.2.11">𝑜</ci><ci id="alg1.l18.m1.1.1.2.12.cmml" xref="alg1.l18.m1.1.1.2.12">𝑣</ci><ci id="alg1.l18.m1.1.1.2.13.cmml" xref="alg1.l18.m1.1.1.2.13">𝑒</ci></apply><apply id="alg1.l18.m1.1.1.3.cmml" xref="alg1.l18.m1.1.1.3"><plus id="alg1.l18.m1.1.1.3.1.cmml" xref="alg1.l18.m1.1.1.3.1"></plus><apply id="alg1.l18.m1.1.1.3.2.cmml" xref="alg1.l18.m1.1.1.3.2"><times id="alg1.l18.m1.1.1.3.2.1.cmml" xref="alg1.l18.m1.1.1.3.2.1"></times><ci id="alg1.l18.m1.1.1.3.2.2.cmml" xref="alg1.l18.m1.1.1.3.2.2">𝑐</ci><ci id="alg1.l18.m1.1.1.3.2.3.cmml" xref="alg1.l18.m1.1.1.3.2.3">𝑜</ci><ci id="alg1.l18.m1.1.1.3.2.4.cmml" xref="alg1.l18.m1.1.1.3.2.4">𝑢</ci><ci id="alg1.l18.m1.1.1.3.2.5.cmml" xref="alg1.l18.m1.1.1.3.2.5">𝑛</ci><ci id="alg1.l18.m1.1.1.3.2.6.cmml" xref="alg1.l18.m1.1.1.3.2.6">𝑡</ci><ci id="alg1.l18.m1.1.1.3.2.7.cmml" xref="alg1.l18.m1.1.1.3.2.7">_</ci><ci id="alg1.l18.m1.1.1.3.2.8.cmml" xref="alg1.l18.m1.1.1.3.2.8">𝑟</ci><ci id="alg1.l18.m1.1.1.3.2.9.cmml" xref="alg1.l18.m1.1.1.3.2.9">𝑒</ci><ci id="alg1.l18.m1.1.1.3.2.10.cmml" xref="alg1.l18.m1.1.1.3.2.10">𝑚</ci><ci id="alg1.l18.m1.1.1.3.2.11.cmml" xref="alg1.l18.m1.1.1.3.2.11">𝑜</ci><ci id="alg1.l18.m1.1.1.3.2.12.cmml" xref="alg1.l18.m1.1.1.3.2.12">𝑣</ci><ci id="alg1.l18.m1.1.1.3.2.13.cmml" xref="alg1.l18.m1.1.1.3.2.13">𝑒</ci></apply><cn id="alg1.l18.m1.1.1.3.3.cmml" type="integer" xref="alg1.l18.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l18.m1.1c">count\_remove\leftarrow count\_remove+1</annotation><annotation encoding="application/x-llamapun" id="alg1.l18.m1.1d">italic_c italic_o italic_u italic_n italic_t _ italic_r italic_e italic_m italic_o italic_v italic_e ← italic_c italic_o italic_u italic_n italic_t _ italic_r italic_e italic_m italic_o italic_v italic_e + 1</annotation></semantics></math><span class="ltx_text" id="alg1.l18.3" style="font-size:80%;"> </span> </div> <div class="ltx_listingline" id="alg1.l19"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l19.1.1.1" style="font-size:80%;">19:</span></span><span class="ltx_text" id="alg1.l19.2" style="font-size:80%;"> </span><math alttext="\mathcal{G}\leftarrow\mathcal{G}\setminus{z^{\mathcal{G}}_{count\_remove}}" class="ltx_Math" display="inline" id="alg1.l19.m1.1"><semantics id="alg1.l19.m1.1a"><mrow id="alg1.l19.m1.1.1" xref="alg1.l19.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.l19.m1.1.1.2" mathsize="80%" xref="alg1.l19.m1.1.1.2.cmml">𝒢</mi><mo id="alg1.l19.m1.1.1.1" mathsize="80%" stretchy="false" xref="alg1.l19.m1.1.1.1.cmml">←</mo><mrow id="alg1.l19.m1.1.1.3" xref="alg1.l19.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.l19.m1.1.1.3.2" mathsize="80%" xref="alg1.l19.m1.1.1.3.2.cmml">𝒢</mi><mo id="alg1.l19.m1.1.1.3.1" mathsize="80%" xref="alg1.l19.m1.1.1.3.1.cmml">∖</mo><msubsup id="alg1.l19.m1.1.1.3.3" xref="alg1.l19.m1.1.1.3.3.cmml"><mi id="alg1.l19.m1.1.1.3.3.2.2" mathsize="80%" xref="alg1.l19.m1.1.1.3.3.2.2.cmml">z</mi><mrow id="alg1.l19.m1.1.1.3.3.3" xref="alg1.l19.m1.1.1.3.3.3.cmml"><mi id="alg1.l19.m1.1.1.3.3.3.2" mathsize="80%" xref="alg1.l19.m1.1.1.3.3.3.2.cmml">c</mi><mo id="alg1.l19.m1.1.1.3.3.3.1" xref="alg1.l19.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="alg1.l19.m1.1.1.3.3.3.3" mathsize="80%" xref="alg1.l19.m1.1.1.3.3.3.3.cmml">o</mi><mo id="alg1.l19.m1.1.1.3.3.3.1a" xref="alg1.l19.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="alg1.l19.m1.1.1.3.3.3.4" mathsize="80%" xref="alg1.l19.m1.1.1.3.3.3.4.cmml">u</mi><mo id="alg1.l19.m1.1.1.3.3.3.1b" xref="alg1.l19.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="alg1.l19.m1.1.1.3.3.3.5" mathsize="80%" xref="alg1.l19.m1.1.1.3.3.3.5.cmml">n</mi><mo id="alg1.l19.m1.1.1.3.3.3.1c" xref="alg1.l19.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="alg1.l19.m1.1.1.3.3.3.6" mathsize="80%" xref="alg1.l19.m1.1.1.3.3.3.6.cmml">t</mi><mo id="alg1.l19.m1.1.1.3.3.3.1d" xref="alg1.l19.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="alg1.l19.m1.1.1.3.3.3.7" mathsize="80%" mathvariant="normal" xref="alg1.l19.m1.1.1.3.3.3.7.cmml">_</mi><mo id="alg1.l19.m1.1.1.3.3.3.1e" xref="alg1.l19.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="alg1.l19.m1.1.1.3.3.3.8" mathsize="80%" xref="alg1.l19.m1.1.1.3.3.3.8.cmml">r</mi><mo id="alg1.l19.m1.1.1.3.3.3.1f" xref="alg1.l19.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="alg1.l19.m1.1.1.3.3.3.9" mathsize="80%" xref="alg1.l19.m1.1.1.3.3.3.9.cmml">e</mi><mo id="alg1.l19.m1.1.1.3.3.3.1g" xref="alg1.l19.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="alg1.l19.m1.1.1.3.3.3.10" mathsize="80%" xref="alg1.l19.m1.1.1.3.3.3.10.cmml">m</mi><mo id="alg1.l19.m1.1.1.3.3.3.1h" xref="alg1.l19.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="alg1.l19.m1.1.1.3.3.3.11" mathsize="80%" xref="alg1.l19.m1.1.1.3.3.3.11.cmml">o</mi><mo id="alg1.l19.m1.1.1.3.3.3.1i" xref="alg1.l19.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="alg1.l19.m1.1.1.3.3.3.12" mathsize="80%" xref="alg1.l19.m1.1.1.3.3.3.12.cmml">v</mi><mo id="alg1.l19.m1.1.1.3.3.3.1j" xref="alg1.l19.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="alg1.l19.m1.1.1.3.3.3.13" mathsize="80%" xref="alg1.l19.m1.1.1.3.3.3.13.cmml">e</mi></mrow><mi class="ltx_font_mathcaligraphic" id="alg1.l19.m1.1.1.3.3.2.3" mathsize="80%" xref="alg1.l19.m1.1.1.3.3.2.3.cmml">𝒢</mi></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l19.m1.1b"><apply id="alg1.l19.m1.1.1.cmml" xref="alg1.l19.m1.1.1"><ci id="alg1.l19.m1.1.1.1.cmml" xref="alg1.l19.m1.1.1.1">←</ci><ci id="alg1.l19.m1.1.1.2.cmml" xref="alg1.l19.m1.1.1.2">𝒢</ci><apply id="alg1.l19.m1.1.1.3.cmml" xref="alg1.l19.m1.1.1.3"><setdiff id="alg1.l19.m1.1.1.3.1.cmml" xref="alg1.l19.m1.1.1.3.1"></setdiff><ci id="alg1.l19.m1.1.1.3.2.cmml" xref="alg1.l19.m1.1.1.3.2">𝒢</ci><apply id="alg1.l19.m1.1.1.3.3.cmml" xref="alg1.l19.m1.1.1.3.3"><csymbol cd="ambiguous" id="alg1.l19.m1.1.1.3.3.1.cmml" xref="alg1.l19.m1.1.1.3.3">subscript</csymbol><apply id="alg1.l19.m1.1.1.3.3.2.cmml" xref="alg1.l19.m1.1.1.3.3"><csymbol cd="ambiguous" id="alg1.l19.m1.1.1.3.3.2.1.cmml" xref="alg1.l19.m1.1.1.3.3">superscript</csymbol><ci id="alg1.l19.m1.1.1.3.3.2.2.cmml" xref="alg1.l19.m1.1.1.3.3.2.2">𝑧</ci><ci id="alg1.l19.m1.1.1.3.3.2.3.cmml" xref="alg1.l19.m1.1.1.3.3.2.3">𝒢</ci></apply><apply id="alg1.l19.m1.1.1.3.3.3.cmml" xref="alg1.l19.m1.1.1.3.3.3"><times id="alg1.l19.m1.1.1.3.3.3.1.cmml" xref="alg1.l19.m1.1.1.3.3.3.1"></times><ci id="alg1.l19.m1.1.1.3.3.3.2.cmml" xref="alg1.l19.m1.1.1.3.3.3.2">𝑐</ci><ci id="alg1.l19.m1.1.1.3.3.3.3.cmml" xref="alg1.l19.m1.1.1.3.3.3.3">𝑜</ci><ci id="alg1.l19.m1.1.1.3.3.3.4.cmml" xref="alg1.l19.m1.1.1.3.3.3.4">𝑢</ci><ci id="alg1.l19.m1.1.1.3.3.3.5.cmml" xref="alg1.l19.m1.1.1.3.3.3.5">𝑛</ci><ci id="alg1.l19.m1.1.1.3.3.3.6.cmml" xref="alg1.l19.m1.1.1.3.3.3.6">𝑡</ci><ci id="alg1.l19.m1.1.1.3.3.3.7.cmml" xref="alg1.l19.m1.1.1.3.3.3.7">_</ci><ci id="alg1.l19.m1.1.1.3.3.3.8.cmml" xref="alg1.l19.m1.1.1.3.3.3.8">𝑟</ci><ci id="alg1.l19.m1.1.1.3.3.3.9.cmml" xref="alg1.l19.m1.1.1.3.3.3.9">𝑒</ci><ci id="alg1.l19.m1.1.1.3.3.3.10.cmml" xref="alg1.l19.m1.1.1.3.3.3.10">𝑚</ci><ci id="alg1.l19.m1.1.1.3.3.3.11.cmml" xref="alg1.l19.m1.1.1.3.3.3.11">𝑜</ci><ci id="alg1.l19.m1.1.1.3.3.3.12.cmml" xref="alg1.l19.m1.1.1.3.3.3.12">𝑣</ci><ci id="alg1.l19.m1.1.1.3.3.3.13.cmml" xref="alg1.l19.m1.1.1.3.3.3.13">𝑒</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l19.m1.1c">\mathcal{G}\leftarrow\mathcal{G}\setminus{z^{\mathcal{G}}_{count\_remove}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l19.m1.1d">caligraphic_G ← caligraphic_G ∖ italic_z start_POSTSUPERSCRIPT caligraphic_G end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c italic_o italic_u italic_n italic_t _ italic_r italic_e italic_m italic_o italic_v italic_e end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text" id="alg1.l19.3" style="font-size:80%;"> </span> </div> <div class="ltx_listingline" id="alg1.l20"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l20.1.1.1" style="font-size:80%;">20:</span></span><span class="ltx_text" id="alg1.l20.2" style="font-size:80%;"> </span><math alttext="\mathcal{L}\leftarrow-\frac{1}{K}\sum_{z\in\mathcal{H}(q_{t},a_{t},K)}\log% 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xref="alg1.l20.m1.10.10.1.1.2.cmml">⁢</mo><mrow id="alg1.l20.m1.10.10.1.1.1" xref="alg1.l20.m1.10.10.1.1.1.cmml"><msub id="alg1.l20.m1.10.10.1.1.1.2" xref="alg1.l20.m1.10.10.1.1.1.2.cmml"><mo id="alg1.l20.m1.10.10.1.1.1.2.2" maxsize="80%" minsize="80%" stretchy="true" xref="alg1.l20.m1.10.10.1.1.1.2.2.cmml">∑</mo><mrow id="alg1.l20.m1.3.3.3" xref="alg1.l20.m1.3.3.3.cmml"><mi id="alg1.l20.m1.3.3.3.5" mathsize="80%" xref="alg1.l20.m1.3.3.3.5.cmml">z</mi><mo id="alg1.l20.m1.3.3.3.4" mathsize="80%" xref="alg1.l20.m1.3.3.3.4.cmml">∈</mo><mrow id="alg1.l20.m1.3.3.3.3" xref="alg1.l20.m1.3.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.l20.m1.3.3.3.3.4" mathsize="80%" xref="alg1.l20.m1.3.3.3.3.4.cmml">ℋ</mi><mo id="alg1.l20.m1.3.3.3.3.3" xref="alg1.l20.m1.3.3.3.3.3.cmml">⁢</mo><mrow id="alg1.l20.m1.3.3.3.3.2.2" xref="alg1.l20.m1.3.3.3.3.2.3.cmml"><mo id="alg1.l20.m1.3.3.3.3.2.2.3" maxsize="80%" minsize="80%" xref="alg1.l20.m1.3.3.3.3.2.3.cmml">(</mo><msub 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xref="alg1.l20.m1.10.10.1.1.1.1.1.1.1.2.1.1.3.3">𝑡</ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l20.m1.10c">\mathcal{L}\leftarrow-\frac{1}{K}\sum_{z\in\mathcal{H}(q_{t},a_{t},K)}\log% \left[P_{\theta,\mathcal{G}}\left(a_{t}|q_{t},z\right)P_{\theta,\mathcal{G}}% \left(z|q_{t}\right)\right]</annotation><annotation encoding="application/x-llamapun" id="alg1.l20.m1.10d">caligraphic_L ← - divide start_ARG 1 end_ARG start_ARG italic_K end_ARG ∑ start_POSTSUBSCRIPT italic_z ∈ caligraphic_H ( italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_K ) end_POSTSUBSCRIPT roman_log [ italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT | italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT , italic_z ) italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_z | italic_q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) ]</annotation></semantics></math><span class="ltx_text" id="alg1.l20.3" style="font-size:80%;"> </span> </div> <div class="ltx_listingline" id="alg1.l21"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l21.1.1.1" style="font-size:80%;">21:</span></span><span class="ltx_text" id="alg1.l21.2" style="font-size:80%;"> </span><span class="ltx_text ltx_font_bold" id="alg1.l21.3" style="font-size:80%;">if</span><span class="ltx_text" id="alg1.l21.4" style="font-size:80%;"> </span><math alttext="\mathcal{L}\leq\epsilon" class="ltx_Math" display="inline" id="alg1.l21.m1.1"><semantics id="alg1.l21.m1.1a"><mrow id="alg1.l21.m1.1.1" xref="alg1.l21.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.l21.m1.1.1.2" mathsize="80%" xref="alg1.l21.m1.1.1.2.cmml">ℒ</mi><mo id="alg1.l21.m1.1.1.1" mathsize="80%" xref="alg1.l21.m1.1.1.1.cmml">≤</mo><mi id="alg1.l21.m1.1.1.3" mathsize="80%" xref="alg1.l21.m1.1.1.3.cmml">ϵ</mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l21.m1.1b"><apply id="alg1.l21.m1.1.1.cmml" xref="alg1.l21.m1.1.1"><leq id="alg1.l21.m1.1.1.1.cmml" xref="alg1.l21.m1.1.1.1"></leq><ci id="alg1.l21.m1.1.1.2.cmml" xref="alg1.l21.m1.1.1.2">ℒ</ci><ci id="alg1.l21.m1.1.1.3.cmml" xref="alg1.l21.m1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l21.m1.1c">\mathcal{L}\leq\epsilon</annotation><annotation encoding="application/x-llamapun" id="alg1.l21.m1.1d">caligraphic_L ≤ italic_ϵ</annotation></semantics></math><span class="ltx_text" id="alg1.l21.5" style="font-size:80%;"> </span><span class="ltx_text ltx_font_bold" id="alg1.l21.6" style="font-size:80%;">then Break</span><span class="ltx_text" id="alg1.l21.7" style="font-size:80%;"> </span> </div> <div class="ltx_listingline" id="alg1.l22"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l22.1.1.1" style="font-size:80%;">22:</span></span><span class="ltx_text" id="alg1.l22.2" style="font-size:80%;"> </span><span class="ltx_text ltx_font_bold" id="alg1.l22.3" style="font-size:80%;">end</span><span class="ltx_text" id="alg1.l22.4" style="font-size:80%;"> </span><span class="ltx_text ltx_font_bold" id="alg1.l22.5" style="font-size:80%;">if</span> </div> <div class="ltx_listingline" id="alg1.l23"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l23.1.1.1" style="font-size:80%;">23:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l23.2" style="font-size:80%;">end</span><span class="ltx_text" id="alg1.l23.3" style="font-size:80%;"> </span><span class="ltx_text ltx_font_bold" id="alg1.l23.4" style="font-size:80%;">while</span> </div> <div class="ltx_listingline" id="alg1.l24"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l24.1.1.1" style="font-size:80%;">24:</span></span><span class="ltx_text ltx_font_bold" id="alg1.l24.2" style="font-size:80%;">return</span><span class="ltx_text" id="alg1.l24.3" style="font-size:80%;"> </span><math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="alg1.l24.m1.1"><semantics id="alg1.l24.m1.1a"><mi class="ltx_font_mathcaligraphic" id="alg1.l24.m1.1.1" mathsize="80%" xref="alg1.l24.m1.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="alg1.l24.m1.1b"><ci id="alg1.l24.m1.1.1.cmml" xref="alg1.l24.m1.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l24.m1.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="alg1.l24.m1.1d">caligraphic_G</annotation></semantics></math><span class="ltx_text" id="alg1.l24.4" style="font-size:80%;"> </span> </div> </div> </div> </div> </figure> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5 </span>Experiments</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">We evaluate KGT in terms of personalization performance and efficiency compared with the baselines. The experiments are conducted on a server with eight A100 GPUs.</p> </div> <section class="ltx_subsection" id="S5.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">5.1 </span>Experimental Setup</h3> <section class="ltx_paragraph" id="S5.SS1.SSS0.Px1"> <h5 class="ltx_title ltx_title_paragraph">Datasets</h5> <div class="ltx_para" id="S5.SS1.SSS0.Px1.p1"> <p class="ltx_p" id="S5.SS1.SSS0.Px1.p1.1">To evaluate the effectiveness of KGT on personalizing the LLM, we conduct experiments in the setting where the user provides answers that conflict with the common factual knowledge that LLM learned from the pre-training dataset. We evaluate KGT on two datasets: <span class="ltx_text ltx_font_italic" id="S5.SS1.SSS0.Px1.p1.1.1">CounterFact</span> <cite class="ltx_cite ltx_citemacro_citep">(Meng et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib19" title="">2022</a>)</cite> dataset and <span class="ltx_text ltx_font_italic" id="S5.SS1.SSS0.Px1.p1.1.2">CounterFactExtension</span> dataset we create based on <span class="ltx_text ltx_font_italic" id="S5.SS1.SSS0.Px1.p1.1.3">PARALLEL</span> dataset <cite class="ltx_cite ltx_citemacro_citep">(Elazar et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib9" title="">2021</a>)</cite> utilizing GPT-4 <cite class="ltx_cite ltx_citemacro_citep">(OpenAI, <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib21" title="">2024</a>)</cite>. The details about our dataset can be found in <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#A2" title="Appendix B Details on the CounterFactExtension Dataset ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_tag">Appendix</span> <span class="ltx_text ltx_ref_tag">B</span></a> Both datasets consist of query-answer pairs involving factual knowledge that conflicts with reality. To evaluate the real-time model personalization in practice, we sequentially input query-answer pairs into the model, ensuring that each pair is accessed only once during training. Once all the pairs in the dataset have been processed by the model, we assess the personalization effectiveness across the entire dataset.</p> </div> </section> <section class="ltx_paragraph" id="S5.SS1.SSS0.Px2"> <h5 class="ltx_title ltx_title_paragraph">Baselines and Configurations</h5> <div class="ltx_para" id="S5.SS1.SSS0.Px2.p1"> <p class="ltx_p" id="S5.SS1.SSS0.Px2.p1.1">We compare KGT with fine-tunning (FT), ROME <cite class="ltx_cite ltx_citemacro_citep">(Meng et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib19" title="">2022</a>)</cite>, Knowledge editing (KE) <cite class="ltx_cite ltx_citemacro_citep">(De Cao et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib6" title="">2021</a>)</cite>, Knowledge neurons (KN) <cite class="ltx_cite ltx_citemacro_citep">(Dai et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib5" title="">2022</a>)</cite>, and MEND <cite class="ltx_cite ltx_citemacro_citep">(Mitchell et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib20" title="">2022</a>)</cite>. For all baselines, we test several specifications of layers to edit and report the best results for each baseline. For FT, we follow previous arts <cite class="ltx_cite ltx_citemacro_citep">(Meng et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib19" title="">2022</a>)</cite> to execute full fine-tuning on a single layer. For KN, we specify a subset of all layers as candidates of knowledge neurons to reduce the search space for large models. And baseline results where no knowledge is edited, are referred to as ‘no edit’. We conduct experiments on GPT2-xl <cite class="ltx_cite ltx_citemacro_citep">(Radford et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib23" title="">2019</a>)</cite>, Llama2-7B <cite class="ltx_cite ltx_citemacro_citep">(Touvron et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib31" title="">2023</a>)</cite>, and Llama3-8B <cite class="ltx_cite ltx_citemacro_citep">(AI, <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib1" title="">2024</a>)</cite>. We equip each model with a KG, ConceptNet <cite class="ltx_cite ltx_citemacro_citep">(Speer et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib29" title="">2017</a>)</cite>, to enhance the inference. <math alttext="K" class="ltx_Math" display="inline" id="S5.SS1.SSS0.Px2.p1.1.m1.1"><semantics id="S5.SS1.SSS0.Px2.p1.1.m1.1a"><mi id="S5.SS1.SSS0.Px2.p1.1.m1.1.1" xref="S5.SS1.SSS0.Px2.p1.1.m1.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS0.Px2.p1.1.m1.1b"><ci id="S5.SS1.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S5.SS1.SSS0.Px2.p1.1.m1.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS0.Px2.p1.1.m1.1c">K</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS0.Px2.p1.1.m1.1d">italic_K</annotation></semantics></math> in <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S4.E3" title="In Knowledge Retrieval ‣ 4.2 Objective Formulation ‣ 4 Method ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_tag">Equation</span> <span class="ltx_text ltx_ref_tag">3</span></a> is set as five for experiments.</p> </div> </section> <section class="ltx_paragraph" id="S5.SS1.SSS0.Px3"> <h5 class="ltx_title ltx_title_paragraph">Metrics</h5> <div class="ltx_para" id="S5.SS1.SSS0.Px3.p1"> <p class="ltx_p" id="S5.SS1.SSS0.Px3.p1.1">We utilize two metrics to evaluate the performance of personalization: (1) <span class="ltx_text ltx_font_italic" id="S5.SS1.SSS0.Px3.p1.1.1">Efficacy Score</span> <cite class="ltx_cite ltx_citemacro_citep">(Meng et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib19" title="">2022</a>)</cite>, measuring the success rate of personalization using the training query-answer pairs directly. If the model generates the user’s personalized answer with a higher probability than the answer before tuning, the model is considered successful for that pair. (2) <span class="ltx_text ltx_font_italic" id="S5.SS1.SSS0.Px3.p1.1.2">Paraphrase Score</span> <cite class="ltx_cite ltx_citemacro_citep">(Meng et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib19" title="">2022</a>)</cite>, indicating the model’s ability to accurately recall personalized knowledge in <span class="ltx_text ltx_font_bold" id="S5.SS1.SSS0.Px3.p1.1.3">paraphrased</span> forms. This assesses its personalization capacity while mitigating the impact of overfitting to specific contexts within the training dataset.</p> </div> </section> </section> <section class="ltx_subsection" id="S5.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">5.2 </span>Personalization Performance</h3> <div class="ltx_para" id="S5.SS2.p1"> <p class="ltx_p" id="S5.SS2.p1.2">We evaluate the setting that the user only provides the answer <math alttext="a" class="ltx_Math" display="inline" id="S5.SS2.p1.1.m1.1"><semantics id="S5.SS2.p1.1.m1.1a"><mi id="S5.SS2.p1.1.m1.1.1" xref="S5.SS2.p1.1.m1.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.1.m1.1b"><ci id="S5.SS2.p1.1.m1.1.1.cmml" xref="S5.SS2.p1.1.m1.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.1.m1.1c">a</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.1.m1.1d">italic_a</annotation></semantics></math> as the feedback, and the model will extract the relations and construct <math alttext="\mathcal{H}\left(q,a,K\right)" class="ltx_Math" display="inline" id="S5.SS2.p1.2.m2.3"><semantics id="S5.SS2.p1.2.m2.3a"><mrow id="S5.SS2.p1.2.m2.3.4" xref="S5.SS2.p1.2.m2.3.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS2.p1.2.m2.3.4.2" xref="S5.SS2.p1.2.m2.3.4.2.cmml">ℋ</mi><mo id="S5.SS2.p1.2.m2.3.4.1" xref="S5.SS2.p1.2.m2.3.4.1.cmml">⁢</mo><mrow id="S5.SS2.p1.2.m2.3.4.3.2" xref="S5.SS2.p1.2.m2.3.4.3.1.cmml"><mo id="S5.SS2.p1.2.m2.3.4.3.2.1" xref="S5.SS2.p1.2.m2.3.4.3.1.cmml">(</mo><mi id="S5.SS2.p1.2.m2.1.1" xref="S5.SS2.p1.2.m2.1.1.cmml">q</mi><mo id="S5.SS2.p1.2.m2.3.4.3.2.2" xref="S5.SS2.p1.2.m2.3.4.3.1.cmml">,</mo><mi id="S5.SS2.p1.2.m2.2.2" xref="S5.SS2.p1.2.m2.2.2.cmml">a</mi><mo id="S5.SS2.p1.2.m2.3.4.3.2.3" xref="S5.SS2.p1.2.m2.3.4.3.1.cmml">,</mo><mi id="S5.SS2.p1.2.m2.3.3" xref="S5.SS2.p1.2.m2.3.3.cmml">K</mi><mo id="S5.SS2.p1.2.m2.3.4.3.2.4" xref="S5.SS2.p1.2.m2.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.2.m2.3b"><apply id="S5.SS2.p1.2.m2.3.4.cmml" xref="S5.SS2.p1.2.m2.3.4"><times id="S5.SS2.p1.2.m2.3.4.1.cmml" xref="S5.SS2.p1.2.m2.3.4.1"></times><ci id="S5.SS2.p1.2.m2.3.4.2.cmml" xref="S5.SS2.p1.2.m2.3.4.2">ℋ</ci><vector id="S5.SS2.p1.2.m2.3.4.3.1.cmml" xref="S5.SS2.p1.2.m2.3.4.3.2"><ci id="S5.SS2.p1.2.m2.1.1.cmml" xref="S5.SS2.p1.2.m2.1.1">𝑞</ci><ci id="S5.SS2.p1.2.m2.2.2.cmml" xref="S5.SS2.p1.2.m2.2.2">𝑎</ci><ci id="S5.SS2.p1.2.m2.3.3.cmml" xref="S5.SS2.p1.2.m2.3.3">𝐾</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.2.m2.3c">\mathcal{H}\left(q,a,K\right)</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.2.m2.3d">caligraphic_H ( italic_q , italic_a , italic_K )</annotation></semantics></math>. The results on <span class="ltx_text ltx_font_italic" id="S5.SS2.p1.2.1">CounterFact</span> dataset are shown in <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S5.T1" title="In 5.2 Personalization Performance ‣ 5 Experiments ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_tag">Table</span> <span class="ltx_text ltx_ref_tag">1</span></a>. It is shown that KGT outperforms the baselines significantly in both efficacy and paraphrase scores. Specifically, using Llama3-8B, KGT improves efficacy by more than 39%, 41%, 55%, 45%, 43%, and 61% on efficacy compared with FT, KE, KN, MEND, and no-edit, respectively. For paraphrase score, KGT outperforms by more than 36%, 32%, 46%, 37%, 33%, 34% compared with FT, KE, KN, MEND, and no-edit, respectively, on Llama3-8B. It is also observed that the results of KGT on Llama3 are better than Llama2. Our analysis is that Llama3 is more powerful in understanding and following instructions, which makes knowledge enhancement from the KG more effective. Such an improvement suggests that KGT will achieve better performance when the pre-trained LLMs are more and more powerful.</p> </div> <figure class="ltx_table" id="S5.T1"> <figcaption class="ltx_caption ltx_centering" style="font-size:90%;"><span class="ltx_tag ltx_tag_table">Table 1: </span>Results on <span class="ltx_text ltx_font_italic" id="S5.T1.41.1">CounterFact</span> dataset when the user only provides the answers to the queries as feedback.</figcaption> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S5.T1.36" style="width:433.6pt;height:165.6pt;vertical-align:-1.0pt;"><span class="ltx_transformed_inner" style="transform:translate(-3.7pt,1.4pt) scale(0.983118514620529,0.983118514620529) ;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S5.T1.36.36"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S5.T1.36.36.37.1"> <th class="ltx_td ltx_th ltx_th_row ltx_border_tt" id="S5.T1.36.36.37.1.1"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="2" id="S5.T1.36.36.37.1.2"><span class="ltx_text ltx_font_bold" id="S5.T1.36.36.37.1.2.1" style="font-size:90%;">GPT2</span></th> <th class="ltx_td ltx_th ltx_th_column ltx_border_tt" id="S5.T1.36.36.37.1.3"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="2" id="S5.T1.36.36.37.1.4"><span class="ltx_text ltx_font_bold" id="S5.T1.36.36.37.1.4.1" style="font-size:90%;">Llama2-7B</span></th> <th class="ltx_td ltx_th ltx_th_column ltx_border_tt" id="S5.T1.36.36.37.1.5"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="2" id="S5.T1.36.36.37.1.6"><span class="ltx_text ltx_font_bold" id="S5.T1.36.36.37.1.6.1" style="font-size:90%;">Llama3-8B</span></th> </tr> <tr class="ltx_tr" id="S5.T1.36.36.38.2"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row" id="S5.T1.36.36.38.2.1"><span class="ltx_text ltx_font_bold" id="S5.T1.36.36.38.2.1.1" style="font-size:90%;">Method</span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T1.36.36.38.2.2"> <table class="ltx_tabular ltx_align_middle" id="S5.T1.36.36.38.2.2.1"> <tr class="ltx_tr" id="S5.T1.36.36.38.2.2.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T1.36.36.38.2.2.1.1.1"><span class="ltx_text" id="S5.T1.36.36.38.2.2.1.1.1.1" style="font-size:90%;">Efficacy</span></td> </tr> </table> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T1.36.36.38.2.3"> <table class="ltx_tabular ltx_align_middle" id="S5.T1.36.36.38.2.3.1"> <tr class="ltx_tr" id="S5.T1.36.36.38.2.3.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T1.36.36.38.2.3.1.1.1"><span class="ltx_text" id="S5.T1.36.36.38.2.3.1.1.1.1" style="font-size:90%;">Paraphrase</span></td> </tr> </table> </th> <th class="ltx_td ltx_th ltx_th_column" id="S5.T1.36.36.38.2.4"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T1.36.36.38.2.5"> <table class="ltx_tabular ltx_align_middle" id="S5.T1.36.36.38.2.5.1"> <tr class="ltx_tr" id="S5.T1.36.36.38.2.5.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T1.36.36.38.2.5.1.1.1"><span class="ltx_text" id="S5.T1.36.36.38.2.5.1.1.1.1" style="font-size:90%;">Efficacy</span></td> </tr> </table> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T1.36.36.38.2.6"> <table class="ltx_tabular ltx_align_middle" id="S5.T1.36.36.38.2.6.1"> <tr class="ltx_tr" id="S5.T1.36.36.38.2.6.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T1.36.36.38.2.6.1.1.1"><span class="ltx_text" id="S5.T1.36.36.38.2.6.1.1.1.1" style="font-size:90%;">Paraphrase</span></td> </tr> </table> </th> <th class="ltx_td ltx_th ltx_th_column" id="S5.T1.36.36.38.2.7"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T1.36.36.38.2.8"> <table class="ltx_tabular ltx_align_middle" id="S5.T1.36.36.38.2.8.1"> <tr class="ltx_tr" id="S5.T1.36.36.38.2.8.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T1.36.36.38.2.8.1.1.1"><span class="ltx_text" id="S5.T1.36.36.38.2.8.1.1.1.1" style="font-size:90%;">Efficacy</span></td> </tr> </table> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T1.36.36.38.2.9"> <table class="ltx_tabular ltx_align_middle" id="S5.T1.36.36.38.2.9.1"> <tr class="ltx_tr" id="S5.T1.36.36.38.2.9.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T1.36.36.38.2.9.1.1.1"><span class="ltx_text" id="S5.T1.36.36.38.2.9.1.1.1.1" style="font-size:90%;">Paraphrase</span></td> </tr> </table> </th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S5.T1.6.6.6"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_tt" id="S5.T1.6.6.6.7"><span class="ltx_text" id="S5.T1.6.6.6.7.1" style="font-size:90%;">FT</span></th> <td class="ltx_td ltx_align_center ltx_border_tt" 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id="S5.T1.1.1.1.1.m1.1c">58.43\%_{\pm 0.15\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.1.1.1.1.m1.1d">58.43 % start_POSTSUBSCRIPT ± 0.15 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T1.2.2.2.2"><math alttext="55.77\%_{\pm 0.20\%}" class="ltx_Math" display="inline" id="S5.T1.2.2.2.2.m1.1"><semantics id="S5.T1.2.2.2.2.m1.1a"><msub id="S5.T1.2.2.2.2.m1.1.1" xref="S5.T1.2.2.2.2.m1.1.1.cmml"><mrow id="S5.T1.2.2.2.2.m1.1.1.2" xref="S5.T1.2.2.2.2.m1.1.1.2.cmml"><mn id="S5.T1.2.2.2.2.m1.1.1.2.2" mathsize="90%" xref="S5.T1.2.2.2.2.m1.1.1.2.2.cmml">55.77</mn><mo id="S5.T1.2.2.2.2.m1.1.1.2.1" mathsize="90%" xref="S5.T1.2.2.2.2.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.2.2.2.2.m1.1.1.3" xref="S5.T1.2.2.2.2.m1.1.1.3.cmml"><mo id="S5.T1.2.2.2.2.m1.1.1.3a" mathsize="90%" xref="S5.T1.2.2.2.2.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.2.2.2.2.m1.1.1.3.2" xref="S5.T1.2.2.2.2.m1.1.1.3.2.cmml"><mn id="S5.T1.2.2.2.2.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.2.2.2.2.m1.1.1.3.2.2.cmml">0.20</mn><mo id="S5.T1.2.2.2.2.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.2.2.2.2.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.2.2.2.2.m1.1b"><apply id="S5.T1.2.2.2.2.m1.1.1.cmml" xref="S5.T1.2.2.2.2.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.2.2.2.2.m1.1.1.1.cmml" xref="S5.T1.2.2.2.2.m1.1.1">subscript</csymbol><apply id="S5.T1.2.2.2.2.m1.1.1.2.cmml" xref="S5.T1.2.2.2.2.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.2.2.2.2.m1.1.1.2.1.cmml" xref="S5.T1.2.2.2.2.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.2.2.2.2.m1.1.1.2.2.cmml" type="float" xref="S5.T1.2.2.2.2.m1.1.1.2.2">55.77</cn></apply><apply id="S5.T1.2.2.2.2.m1.1.1.3.cmml" xref="S5.T1.2.2.2.2.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.2.2.2.2.m1.1.1.3.1.cmml" xref="S5.T1.2.2.2.2.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.2.2.2.2.m1.1.1.3.2.cmml" xref="S5.T1.2.2.2.2.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.2.2.2.2.m1.1.1.3.2.1.cmml" xref="S5.T1.2.2.2.2.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.2.2.2.2.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.2.2.2.2.m1.1.1.3.2.2">0.20</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.2.2.2.2.m1.1c">55.77\%_{\pm 0.20\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.2.2.2.2.m1.1d">55.77 % start_POSTSUBSCRIPT ± 0.20 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_border_tt" id="S5.T1.6.6.6.8"></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T1.3.3.3.3"><math alttext="62.47\%_{\pm 0.11\%}" class="ltx_Math" display="inline" id="S5.T1.3.3.3.3.m1.1"><semantics id="S5.T1.3.3.3.3.m1.1a"><msub id="S5.T1.3.3.3.3.m1.1.1" xref="S5.T1.3.3.3.3.m1.1.1.cmml"><mrow id="S5.T1.3.3.3.3.m1.1.1.2" xref="S5.T1.3.3.3.3.m1.1.1.2.cmml"><mn id="S5.T1.3.3.3.3.m1.1.1.2.2" mathsize="90%" xref="S5.T1.3.3.3.3.m1.1.1.2.2.cmml">62.47</mn><mo id="S5.T1.3.3.3.3.m1.1.1.2.1" mathsize="90%" xref="S5.T1.3.3.3.3.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.3.3.3.3.m1.1.1.3" xref="S5.T1.3.3.3.3.m1.1.1.3.cmml"><mo id="S5.T1.3.3.3.3.m1.1.1.3a" mathsize="90%" xref="S5.T1.3.3.3.3.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.3.3.3.3.m1.1.1.3.2" xref="S5.T1.3.3.3.3.m1.1.1.3.2.cmml"><mn id="S5.T1.3.3.3.3.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.3.3.3.3.m1.1.1.3.2.2.cmml">0.11</mn><mo id="S5.T1.3.3.3.3.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.3.3.3.3.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.3.3.3.3.m1.1b"><apply id="S5.T1.3.3.3.3.m1.1.1.cmml" xref="S5.T1.3.3.3.3.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.3.3.3.3.m1.1.1.1.cmml" xref="S5.T1.3.3.3.3.m1.1.1">subscript</csymbol><apply id="S5.T1.3.3.3.3.m1.1.1.2.cmml" xref="S5.T1.3.3.3.3.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.3.3.3.3.m1.1.1.2.1.cmml" xref="S5.T1.3.3.3.3.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.3.3.3.3.m1.1.1.2.2.cmml" type="float" xref="S5.T1.3.3.3.3.m1.1.1.2.2">62.47</cn></apply><apply id="S5.T1.3.3.3.3.m1.1.1.3.cmml" xref="S5.T1.3.3.3.3.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.3.3.3.3.m1.1.1.3.1.cmml" xref="S5.T1.3.3.3.3.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.3.3.3.3.m1.1.1.3.2.cmml" xref="S5.T1.3.3.3.3.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.3.3.3.3.m1.1.1.3.2.1.cmml" xref="S5.T1.3.3.3.3.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.3.3.3.3.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.3.3.3.3.m1.1.1.3.2.2">0.11</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.3.3.3.3.m1.1c">62.47\%_{\pm 0.11\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.3.3.3.3.m1.1d">62.47 % start_POSTSUBSCRIPT ± 0.11 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T1.4.4.4.4"><math alttext="63.09\%_{\pm 0.09\%}" class="ltx_Math" display="inline" id="S5.T1.4.4.4.4.m1.1"><semantics id="S5.T1.4.4.4.4.m1.1a"><msub id="S5.T1.4.4.4.4.m1.1.1" xref="S5.T1.4.4.4.4.m1.1.1.cmml"><mrow id="S5.T1.4.4.4.4.m1.1.1.2" xref="S5.T1.4.4.4.4.m1.1.1.2.cmml"><mn id="S5.T1.4.4.4.4.m1.1.1.2.2" mathsize="90%" xref="S5.T1.4.4.4.4.m1.1.1.2.2.cmml">63.09</mn><mo id="S5.T1.4.4.4.4.m1.1.1.2.1" mathsize="90%" xref="S5.T1.4.4.4.4.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.4.4.4.4.m1.1.1.3" xref="S5.T1.4.4.4.4.m1.1.1.3.cmml"><mo id="S5.T1.4.4.4.4.m1.1.1.3a" mathsize="90%" xref="S5.T1.4.4.4.4.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.4.4.4.4.m1.1.1.3.2" xref="S5.T1.4.4.4.4.m1.1.1.3.2.cmml"><mn id="S5.T1.4.4.4.4.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.4.4.4.4.m1.1.1.3.2.2.cmml">0.09</mn><mo id="S5.T1.4.4.4.4.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.4.4.4.4.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.4.4.4.4.m1.1b"><apply id="S5.T1.4.4.4.4.m1.1.1.cmml" xref="S5.T1.4.4.4.4.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.4.4.4.4.m1.1.1.1.cmml" xref="S5.T1.4.4.4.4.m1.1.1">subscript</csymbol><apply id="S5.T1.4.4.4.4.m1.1.1.2.cmml" xref="S5.T1.4.4.4.4.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.4.4.4.4.m1.1.1.2.1.cmml" xref="S5.T1.4.4.4.4.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.4.4.4.4.m1.1.1.2.2.cmml" type="float" xref="S5.T1.4.4.4.4.m1.1.1.2.2">63.09</cn></apply><apply id="S5.T1.4.4.4.4.m1.1.1.3.cmml" xref="S5.T1.4.4.4.4.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.4.4.4.4.m1.1.1.3.1.cmml" xref="S5.T1.4.4.4.4.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.4.4.4.4.m1.1.1.3.2.cmml" xref="S5.T1.4.4.4.4.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.4.4.4.4.m1.1.1.3.2.1.cmml" xref="S5.T1.4.4.4.4.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.4.4.4.4.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.4.4.4.4.m1.1.1.3.2.2">0.09</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.4.4.4.4.m1.1c">63.09\%_{\pm 0.09\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.4.4.4.4.m1.1d">63.09 % start_POSTSUBSCRIPT ± 0.09 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_border_tt" id="S5.T1.6.6.6.9"></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T1.5.5.5.5"><math alttext="54.44\%_{\pm 0.35\%}" class="ltx_Math" display="inline" id="S5.T1.5.5.5.5.m1.1"><semantics id="S5.T1.5.5.5.5.m1.1a"><msub id="S5.T1.5.5.5.5.m1.1.1" xref="S5.T1.5.5.5.5.m1.1.1.cmml"><mrow id="S5.T1.5.5.5.5.m1.1.1.2" xref="S5.T1.5.5.5.5.m1.1.1.2.cmml"><mn id="S5.T1.5.5.5.5.m1.1.1.2.2" mathsize="90%" xref="S5.T1.5.5.5.5.m1.1.1.2.2.cmml">54.44</mn><mo id="S5.T1.5.5.5.5.m1.1.1.2.1" mathsize="90%" xref="S5.T1.5.5.5.5.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.5.5.5.5.m1.1.1.3" xref="S5.T1.5.5.5.5.m1.1.1.3.cmml"><mo id="S5.T1.5.5.5.5.m1.1.1.3a" mathsize="90%" xref="S5.T1.5.5.5.5.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.5.5.5.5.m1.1.1.3.2" xref="S5.T1.5.5.5.5.m1.1.1.3.2.cmml"><mn id="S5.T1.5.5.5.5.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.5.5.5.5.m1.1.1.3.2.2.cmml">0.35</mn><mo id="S5.T1.5.5.5.5.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.5.5.5.5.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.5.5.5.5.m1.1b"><apply id="S5.T1.5.5.5.5.m1.1.1.cmml" xref="S5.T1.5.5.5.5.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.5.5.5.5.m1.1.1.1.cmml" xref="S5.T1.5.5.5.5.m1.1.1">subscript</csymbol><apply id="S5.T1.5.5.5.5.m1.1.1.2.cmml" xref="S5.T1.5.5.5.5.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.5.5.5.5.m1.1.1.2.1.cmml" xref="S5.T1.5.5.5.5.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.5.5.5.5.m1.1.1.2.2.cmml" type="float" xref="S5.T1.5.5.5.5.m1.1.1.2.2">54.44</cn></apply><apply id="S5.T1.5.5.5.5.m1.1.1.3.cmml" xref="S5.T1.5.5.5.5.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.5.5.5.5.m1.1.1.3.1.cmml" xref="S5.T1.5.5.5.5.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.5.5.5.5.m1.1.1.3.2.cmml" xref="S5.T1.5.5.5.5.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.5.5.5.5.m1.1.1.3.2.1.cmml" xref="S5.T1.5.5.5.5.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.5.5.5.5.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.5.5.5.5.m1.1.1.3.2.2">0.35</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.5.5.5.5.m1.1c">54.44\%_{\pm 0.35\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.5.5.5.5.m1.1d">54.44 % start_POSTSUBSCRIPT ± 0.35 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T1.6.6.6.6"><math alttext="50.52\%_{\pm 0.05\%}" class="ltx_Math" display="inline" id="S5.T1.6.6.6.6.m1.1"><semantics id="S5.T1.6.6.6.6.m1.1a"><msub id="S5.T1.6.6.6.6.m1.1.1" xref="S5.T1.6.6.6.6.m1.1.1.cmml"><mrow id="S5.T1.6.6.6.6.m1.1.1.2" xref="S5.T1.6.6.6.6.m1.1.1.2.cmml"><mn id="S5.T1.6.6.6.6.m1.1.1.2.2" mathsize="90%" xref="S5.T1.6.6.6.6.m1.1.1.2.2.cmml">50.52</mn><mo id="S5.T1.6.6.6.6.m1.1.1.2.1" mathsize="90%" xref="S5.T1.6.6.6.6.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.6.6.6.6.m1.1.1.3" xref="S5.T1.6.6.6.6.m1.1.1.3.cmml"><mo id="S5.T1.6.6.6.6.m1.1.1.3a" mathsize="90%" xref="S5.T1.6.6.6.6.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.6.6.6.6.m1.1.1.3.2" xref="S5.T1.6.6.6.6.m1.1.1.3.2.cmml"><mn id="S5.T1.6.6.6.6.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.6.6.6.6.m1.1.1.3.2.2.cmml">0.05</mn><mo id="S5.T1.6.6.6.6.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.6.6.6.6.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.6.6.6.6.m1.1b"><apply id="S5.T1.6.6.6.6.m1.1.1.cmml" xref="S5.T1.6.6.6.6.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.6.6.6.6.m1.1.1.1.cmml" xref="S5.T1.6.6.6.6.m1.1.1">subscript</csymbol><apply id="S5.T1.6.6.6.6.m1.1.1.2.cmml" xref="S5.T1.6.6.6.6.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.6.6.6.6.m1.1.1.2.1.cmml" xref="S5.T1.6.6.6.6.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.6.6.6.6.m1.1.1.2.2.cmml" type="float" xref="S5.T1.6.6.6.6.m1.1.1.2.2">50.52</cn></apply><apply id="S5.T1.6.6.6.6.m1.1.1.3.cmml" xref="S5.T1.6.6.6.6.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.6.6.6.6.m1.1.1.3.1.cmml" xref="S5.T1.6.6.6.6.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.6.6.6.6.m1.1.1.3.2.cmml" xref="S5.T1.6.6.6.6.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.6.6.6.6.m1.1.1.3.2.1.cmml" xref="S5.T1.6.6.6.6.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.6.6.6.6.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.6.6.6.6.m1.1.1.3.2.2">0.05</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.6.6.6.6.m1.1c">50.52\%_{\pm 0.05\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.6.6.6.6.m1.1d">50.52 % start_POSTSUBSCRIPT ± 0.05 % end_POSTSUBSCRIPT</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T1.12.12.12"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S5.T1.12.12.12.7"><span class="ltx_text" id="S5.T1.12.12.12.7.1" style="font-size:90%;">ROME</span></th> <td class="ltx_td ltx_align_center" id="S5.T1.7.7.7.1"><math alttext="49.38\%_{\pm 1.20\%}" class="ltx_Math" display="inline" id="S5.T1.7.7.7.1.m1.1"><semantics id="S5.T1.7.7.7.1.m1.1a"><msub id="S5.T1.7.7.7.1.m1.1.1" xref="S5.T1.7.7.7.1.m1.1.1.cmml"><mrow id="S5.T1.7.7.7.1.m1.1.1.2" xref="S5.T1.7.7.7.1.m1.1.1.2.cmml"><mn id="S5.T1.7.7.7.1.m1.1.1.2.2" mathsize="90%" xref="S5.T1.7.7.7.1.m1.1.1.2.2.cmml">49.38</mn><mo id="S5.T1.7.7.7.1.m1.1.1.2.1" mathsize="90%" xref="S5.T1.7.7.7.1.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.7.7.7.1.m1.1.1.3" xref="S5.T1.7.7.7.1.m1.1.1.3.cmml"><mo id="S5.T1.7.7.7.1.m1.1.1.3a" mathsize="90%" xref="S5.T1.7.7.7.1.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.7.7.7.1.m1.1.1.3.2" xref="S5.T1.7.7.7.1.m1.1.1.3.2.cmml"><mn id="S5.T1.7.7.7.1.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.7.7.7.1.m1.1.1.3.2.2.cmml">1.20</mn><mo id="S5.T1.7.7.7.1.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.7.7.7.1.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.7.7.7.1.m1.1b"><apply id="S5.T1.7.7.7.1.m1.1.1.cmml" xref="S5.T1.7.7.7.1.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.7.7.7.1.m1.1.1.1.cmml" xref="S5.T1.7.7.7.1.m1.1.1">subscript</csymbol><apply id="S5.T1.7.7.7.1.m1.1.1.2.cmml" xref="S5.T1.7.7.7.1.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.7.7.7.1.m1.1.1.2.1.cmml" xref="S5.T1.7.7.7.1.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.7.7.7.1.m1.1.1.2.2.cmml" type="float" xref="S5.T1.7.7.7.1.m1.1.1.2.2">49.38</cn></apply><apply id="S5.T1.7.7.7.1.m1.1.1.3.cmml" xref="S5.T1.7.7.7.1.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.7.7.7.1.m1.1.1.3.1.cmml" xref="S5.T1.7.7.7.1.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.7.7.7.1.m1.1.1.3.2.cmml" xref="S5.T1.7.7.7.1.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.7.7.7.1.m1.1.1.3.2.1.cmml" xref="S5.T1.7.7.7.1.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.7.7.7.1.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.7.7.7.1.m1.1.1.3.2.2">1.20</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.7.7.7.1.m1.1c">49.38\%_{\pm 1.20\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.7.7.7.1.m1.1d">49.38 % start_POSTSUBSCRIPT ± 1.20 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T1.8.8.8.2"><math alttext="48.22\%_{\pm 1.36\%}" class="ltx_Math" display="inline" id="S5.T1.8.8.8.2.m1.1"><semantics id="S5.T1.8.8.8.2.m1.1a"><msub id="S5.T1.8.8.8.2.m1.1.1" xref="S5.T1.8.8.8.2.m1.1.1.cmml"><mrow id="S5.T1.8.8.8.2.m1.1.1.2" xref="S5.T1.8.8.8.2.m1.1.1.2.cmml"><mn id="S5.T1.8.8.8.2.m1.1.1.2.2" mathsize="90%" xref="S5.T1.8.8.8.2.m1.1.1.2.2.cmml">48.22</mn><mo id="S5.T1.8.8.8.2.m1.1.1.2.1" mathsize="90%" xref="S5.T1.8.8.8.2.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.8.8.8.2.m1.1.1.3" xref="S5.T1.8.8.8.2.m1.1.1.3.cmml"><mo id="S5.T1.8.8.8.2.m1.1.1.3a" mathsize="90%" xref="S5.T1.8.8.8.2.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.8.8.8.2.m1.1.1.3.2" xref="S5.T1.8.8.8.2.m1.1.1.3.2.cmml"><mn id="S5.T1.8.8.8.2.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.8.8.8.2.m1.1.1.3.2.2.cmml">1.36</mn><mo id="S5.T1.8.8.8.2.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.8.8.8.2.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.8.8.8.2.m1.1b"><apply id="S5.T1.8.8.8.2.m1.1.1.cmml" xref="S5.T1.8.8.8.2.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.8.8.8.2.m1.1.1.1.cmml" xref="S5.T1.8.8.8.2.m1.1.1">subscript</csymbol><apply id="S5.T1.8.8.8.2.m1.1.1.2.cmml" xref="S5.T1.8.8.8.2.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.8.8.8.2.m1.1.1.2.1.cmml" xref="S5.T1.8.8.8.2.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.8.8.8.2.m1.1.1.2.2.cmml" type="float" xref="S5.T1.8.8.8.2.m1.1.1.2.2">48.22</cn></apply><apply id="S5.T1.8.8.8.2.m1.1.1.3.cmml" xref="S5.T1.8.8.8.2.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.8.8.8.2.m1.1.1.3.1.cmml" xref="S5.T1.8.8.8.2.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.8.8.8.2.m1.1.1.3.2.cmml" xref="S5.T1.8.8.8.2.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.8.8.8.2.m1.1.1.3.2.1.cmml" xref="S5.T1.8.8.8.2.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.8.8.8.2.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.8.8.8.2.m1.1.1.3.2.2">1.36</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.8.8.8.2.m1.1c">48.22\%_{\pm 1.36\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.8.8.8.2.m1.1d">48.22 % start_POSTSUBSCRIPT ± 1.36 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td" id="S5.T1.12.12.12.8"></td> <td class="ltx_td ltx_align_center" id="S5.T1.9.9.9.3"><math alttext="49.94\%_{\pm 1.24\%}" class="ltx_Math" display="inline" id="S5.T1.9.9.9.3.m1.1"><semantics id="S5.T1.9.9.9.3.m1.1a"><msub id="S5.T1.9.9.9.3.m1.1.1" xref="S5.T1.9.9.9.3.m1.1.1.cmml"><mrow id="S5.T1.9.9.9.3.m1.1.1.2" xref="S5.T1.9.9.9.3.m1.1.1.2.cmml"><mn id="S5.T1.9.9.9.3.m1.1.1.2.2" mathsize="90%" xref="S5.T1.9.9.9.3.m1.1.1.2.2.cmml">49.94</mn><mo id="S5.T1.9.9.9.3.m1.1.1.2.1" mathsize="90%" xref="S5.T1.9.9.9.3.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.9.9.9.3.m1.1.1.3" xref="S5.T1.9.9.9.3.m1.1.1.3.cmml"><mo id="S5.T1.9.9.9.3.m1.1.1.3a" mathsize="90%" xref="S5.T1.9.9.9.3.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.9.9.9.3.m1.1.1.3.2" xref="S5.T1.9.9.9.3.m1.1.1.3.2.cmml"><mn id="S5.T1.9.9.9.3.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.9.9.9.3.m1.1.1.3.2.2.cmml">1.24</mn><mo id="S5.T1.9.9.9.3.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.9.9.9.3.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.9.9.9.3.m1.1b"><apply id="S5.T1.9.9.9.3.m1.1.1.cmml" xref="S5.T1.9.9.9.3.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.9.9.9.3.m1.1.1.1.cmml" xref="S5.T1.9.9.9.3.m1.1.1">subscript</csymbol><apply id="S5.T1.9.9.9.3.m1.1.1.2.cmml" xref="S5.T1.9.9.9.3.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.9.9.9.3.m1.1.1.2.1.cmml" xref="S5.T1.9.9.9.3.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.9.9.9.3.m1.1.1.2.2.cmml" type="float" xref="S5.T1.9.9.9.3.m1.1.1.2.2">49.94</cn></apply><apply id="S5.T1.9.9.9.3.m1.1.1.3.cmml" xref="S5.T1.9.9.9.3.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.9.9.9.3.m1.1.1.3.1.cmml" xref="S5.T1.9.9.9.3.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.9.9.9.3.m1.1.1.3.2.cmml" xref="S5.T1.9.9.9.3.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.9.9.9.3.m1.1.1.3.2.1.cmml" xref="S5.T1.9.9.9.3.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.9.9.9.3.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.9.9.9.3.m1.1.1.3.2.2">1.24</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.9.9.9.3.m1.1c">49.94\%_{\pm 1.24\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.9.9.9.3.m1.1d">49.94 % start_POSTSUBSCRIPT ± 1.24 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T1.10.10.10.4"><math alttext="48.84\%_{\pm 1.74\%}" class="ltx_Math" display="inline" id="S5.T1.10.10.10.4.m1.1"><semantics id="S5.T1.10.10.10.4.m1.1a"><msub id="S5.T1.10.10.10.4.m1.1.1" xref="S5.T1.10.10.10.4.m1.1.1.cmml"><mrow id="S5.T1.10.10.10.4.m1.1.1.2" xref="S5.T1.10.10.10.4.m1.1.1.2.cmml"><mn id="S5.T1.10.10.10.4.m1.1.1.2.2" mathsize="90%" xref="S5.T1.10.10.10.4.m1.1.1.2.2.cmml">48.84</mn><mo id="S5.T1.10.10.10.4.m1.1.1.2.1" mathsize="90%" xref="S5.T1.10.10.10.4.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.10.10.10.4.m1.1.1.3" xref="S5.T1.10.10.10.4.m1.1.1.3.cmml"><mo id="S5.T1.10.10.10.4.m1.1.1.3a" mathsize="90%" xref="S5.T1.10.10.10.4.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.10.10.10.4.m1.1.1.3.2" xref="S5.T1.10.10.10.4.m1.1.1.3.2.cmml"><mn id="S5.T1.10.10.10.4.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.10.10.10.4.m1.1.1.3.2.2.cmml">1.74</mn><mo id="S5.T1.10.10.10.4.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.10.10.10.4.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.10.10.10.4.m1.1b"><apply id="S5.T1.10.10.10.4.m1.1.1.cmml" xref="S5.T1.10.10.10.4.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.10.10.10.4.m1.1.1.1.cmml" xref="S5.T1.10.10.10.4.m1.1.1">subscript</csymbol><apply id="S5.T1.10.10.10.4.m1.1.1.2.cmml" xref="S5.T1.10.10.10.4.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.10.10.10.4.m1.1.1.2.1.cmml" xref="S5.T1.10.10.10.4.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.10.10.10.4.m1.1.1.2.2.cmml" type="float" xref="S5.T1.10.10.10.4.m1.1.1.2.2">48.84</cn></apply><apply id="S5.T1.10.10.10.4.m1.1.1.3.cmml" xref="S5.T1.10.10.10.4.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.10.10.10.4.m1.1.1.3.1.cmml" xref="S5.T1.10.10.10.4.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.10.10.10.4.m1.1.1.3.2.cmml" xref="S5.T1.10.10.10.4.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.10.10.10.4.m1.1.1.3.2.1.cmml" xref="S5.T1.10.10.10.4.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.10.10.10.4.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.10.10.10.4.m1.1.1.3.2.2">1.74</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.10.10.10.4.m1.1c">48.84\%_{\pm 1.74\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.10.10.10.4.m1.1d">48.84 % start_POSTSUBSCRIPT ± 1.74 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td" id="S5.T1.12.12.12.9"></td> <td class="ltx_td ltx_align_center" id="S5.T1.11.11.11.5"><math alttext="51.13\%_{\pm 1.55\%}" class="ltx_Math" display="inline" id="S5.T1.11.11.11.5.m1.1"><semantics id="S5.T1.11.11.11.5.m1.1a"><msub id="S5.T1.11.11.11.5.m1.1.1" xref="S5.T1.11.11.11.5.m1.1.1.cmml"><mrow id="S5.T1.11.11.11.5.m1.1.1.2" xref="S5.T1.11.11.11.5.m1.1.1.2.cmml"><mn id="S5.T1.11.11.11.5.m1.1.1.2.2" mathsize="90%" xref="S5.T1.11.11.11.5.m1.1.1.2.2.cmml">51.13</mn><mo id="S5.T1.11.11.11.5.m1.1.1.2.1" mathsize="90%" xref="S5.T1.11.11.11.5.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.11.11.11.5.m1.1.1.3" xref="S5.T1.11.11.11.5.m1.1.1.3.cmml"><mo id="S5.T1.11.11.11.5.m1.1.1.3a" mathsize="90%" xref="S5.T1.11.11.11.5.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.11.11.11.5.m1.1.1.3.2" xref="S5.T1.11.11.11.5.m1.1.1.3.2.cmml"><mn id="S5.T1.11.11.11.5.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.11.11.11.5.m1.1.1.3.2.2.cmml">1.55</mn><mo id="S5.T1.11.11.11.5.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.11.11.11.5.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.11.11.11.5.m1.1b"><apply id="S5.T1.11.11.11.5.m1.1.1.cmml" xref="S5.T1.11.11.11.5.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.11.11.11.5.m1.1.1.1.cmml" xref="S5.T1.11.11.11.5.m1.1.1">subscript</csymbol><apply id="S5.T1.11.11.11.5.m1.1.1.2.cmml" xref="S5.T1.11.11.11.5.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.11.11.11.5.m1.1.1.2.1.cmml" xref="S5.T1.11.11.11.5.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.11.11.11.5.m1.1.1.2.2.cmml" type="float" xref="S5.T1.11.11.11.5.m1.1.1.2.2">51.13</cn></apply><apply id="S5.T1.11.11.11.5.m1.1.1.3.cmml" xref="S5.T1.11.11.11.5.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.11.11.11.5.m1.1.1.3.1.cmml" xref="S5.T1.11.11.11.5.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.11.11.11.5.m1.1.1.3.2.cmml" xref="S5.T1.11.11.11.5.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.11.11.11.5.m1.1.1.3.2.1.cmml" xref="S5.T1.11.11.11.5.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.11.11.11.5.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.11.11.11.5.m1.1.1.3.2.2">1.55</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.11.11.11.5.m1.1c">51.13\%_{\pm 1.55\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.11.11.11.5.m1.1d">51.13 % start_POSTSUBSCRIPT ± 1.55 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T1.12.12.12.6"><math alttext="52.39\%_{\pm 1.62\%}" class="ltx_Math" display="inline" id="S5.T1.12.12.12.6.m1.1"><semantics id="S5.T1.12.12.12.6.m1.1a"><msub id="S5.T1.12.12.12.6.m1.1.1" xref="S5.T1.12.12.12.6.m1.1.1.cmml"><mrow id="S5.T1.12.12.12.6.m1.1.1.2" xref="S5.T1.12.12.12.6.m1.1.1.2.cmml"><mn id="S5.T1.12.12.12.6.m1.1.1.2.2" mathsize="90%" xref="S5.T1.12.12.12.6.m1.1.1.2.2.cmml">52.39</mn><mo id="S5.T1.12.12.12.6.m1.1.1.2.1" mathsize="90%" xref="S5.T1.12.12.12.6.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.12.12.12.6.m1.1.1.3" xref="S5.T1.12.12.12.6.m1.1.1.3.cmml"><mo id="S5.T1.12.12.12.6.m1.1.1.3a" mathsize="90%" xref="S5.T1.12.12.12.6.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.12.12.12.6.m1.1.1.3.2" xref="S5.T1.12.12.12.6.m1.1.1.3.2.cmml"><mn id="S5.T1.12.12.12.6.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.12.12.12.6.m1.1.1.3.2.2.cmml">1.62</mn><mo id="S5.T1.12.12.12.6.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.12.12.12.6.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.12.12.12.6.m1.1b"><apply id="S5.T1.12.12.12.6.m1.1.1.cmml" xref="S5.T1.12.12.12.6.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.12.12.12.6.m1.1.1.1.cmml" xref="S5.T1.12.12.12.6.m1.1.1">subscript</csymbol><apply id="S5.T1.12.12.12.6.m1.1.1.2.cmml" xref="S5.T1.12.12.12.6.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.12.12.12.6.m1.1.1.2.1.cmml" xref="S5.T1.12.12.12.6.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.12.12.12.6.m1.1.1.2.2.cmml" type="float" xref="S5.T1.12.12.12.6.m1.1.1.2.2">52.39</cn></apply><apply id="S5.T1.12.12.12.6.m1.1.1.3.cmml" xref="S5.T1.12.12.12.6.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.12.12.12.6.m1.1.1.3.1.cmml" xref="S5.T1.12.12.12.6.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.12.12.12.6.m1.1.1.3.2.cmml" xref="S5.T1.12.12.12.6.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.12.12.12.6.m1.1.1.3.2.1.cmml" xref="S5.T1.12.12.12.6.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.12.12.12.6.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.12.12.12.6.m1.1.1.3.2.2">1.62</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.12.12.12.6.m1.1c">52.39\%_{\pm 1.62\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.12.12.12.6.m1.1d">52.39 % start_POSTSUBSCRIPT ± 1.62 % end_POSTSUBSCRIPT</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T1.18.18.18"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S5.T1.18.18.18.7"><span class="ltx_text" id="S5.T1.18.18.18.7.1" style="font-size:90%;">KE</span></th> <td class="ltx_td ltx_align_center" id="S5.T1.13.13.13.1"><math alttext="51.50\%_{\pm 0.32\%}" class="ltx_Math" display="inline" id="S5.T1.13.13.13.1.m1.1"><semantics id="S5.T1.13.13.13.1.m1.1a"><msub id="S5.T1.13.13.13.1.m1.1.1" xref="S5.T1.13.13.13.1.m1.1.1.cmml"><mrow id="S5.T1.13.13.13.1.m1.1.1.2" xref="S5.T1.13.13.13.1.m1.1.1.2.cmml"><mn id="S5.T1.13.13.13.1.m1.1.1.2.2" mathsize="90%" xref="S5.T1.13.13.13.1.m1.1.1.2.2.cmml">51.50</mn><mo id="S5.T1.13.13.13.1.m1.1.1.2.1" mathsize="90%" xref="S5.T1.13.13.13.1.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.13.13.13.1.m1.1.1.3" xref="S5.T1.13.13.13.1.m1.1.1.3.cmml"><mo id="S5.T1.13.13.13.1.m1.1.1.3a" mathsize="90%" xref="S5.T1.13.13.13.1.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.13.13.13.1.m1.1.1.3.2" xref="S5.T1.13.13.13.1.m1.1.1.3.2.cmml"><mn id="S5.T1.13.13.13.1.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.13.13.13.1.m1.1.1.3.2.2.cmml">0.32</mn><mo id="S5.T1.13.13.13.1.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.13.13.13.1.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.13.13.13.1.m1.1b"><apply id="S5.T1.13.13.13.1.m1.1.1.cmml" xref="S5.T1.13.13.13.1.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.13.13.13.1.m1.1.1.1.cmml" xref="S5.T1.13.13.13.1.m1.1.1">subscript</csymbol><apply id="S5.T1.13.13.13.1.m1.1.1.2.cmml" xref="S5.T1.13.13.13.1.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.13.13.13.1.m1.1.1.2.1.cmml" xref="S5.T1.13.13.13.1.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.13.13.13.1.m1.1.1.2.2.cmml" type="float" xref="S5.T1.13.13.13.1.m1.1.1.2.2">51.50</cn></apply><apply id="S5.T1.13.13.13.1.m1.1.1.3.cmml" xref="S5.T1.13.13.13.1.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.13.13.13.1.m1.1.1.3.1.cmml" xref="S5.T1.13.13.13.1.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.13.13.13.1.m1.1.1.3.2.cmml" xref="S5.T1.13.13.13.1.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.13.13.13.1.m1.1.1.3.2.1.cmml" xref="S5.T1.13.13.13.1.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.13.13.13.1.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.13.13.13.1.m1.1.1.3.2.2">0.32</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.13.13.13.1.m1.1c">51.50\%_{\pm 0.32\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.13.13.13.1.m1.1d">51.50 % start_POSTSUBSCRIPT ± 0.32 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T1.14.14.14.2"><math alttext="51.85\%_{\pm 0.27\%}" class="ltx_Math" display="inline" id="S5.T1.14.14.14.2.m1.1"><semantics id="S5.T1.14.14.14.2.m1.1a"><msub id="S5.T1.14.14.14.2.m1.1.1" xref="S5.T1.14.14.14.2.m1.1.1.cmml"><mrow id="S5.T1.14.14.14.2.m1.1.1.2" xref="S5.T1.14.14.14.2.m1.1.1.2.cmml"><mn id="S5.T1.14.14.14.2.m1.1.1.2.2" mathsize="90%" xref="S5.T1.14.14.14.2.m1.1.1.2.2.cmml">51.85</mn><mo id="S5.T1.14.14.14.2.m1.1.1.2.1" mathsize="90%" xref="S5.T1.14.14.14.2.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.14.14.14.2.m1.1.1.3" xref="S5.T1.14.14.14.2.m1.1.1.3.cmml"><mo id="S5.T1.14.14.14.2.m1.1.1.3a" mathsize="90%" xref="S5.T1.14.14.14.2.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.14.14.14.2.m1.1.1.3.2" xref="S5.T1.14.14.14.2.m1.1.1.3.2.cmml"><mn id="S5.T1.14.14.14.2.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.14.14.14.2.m1.1.1.3.2.2.cmml">0.27</mn><mo id="S5.T1.14.14.14.2.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.14.14.14.2.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.14.14.14.2.m1.1b"><apply id="S5.T1.14.14.14.2.m1.1.1.cmml" xref="S5.T1.14.14.14.2.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.14.14.14.2.m1.1.1.1.cmml" xref="S5.T1.14.14.14.2.m1.1.1">subscript</csymbol><apply id="S5.T1.14.14.14.2.m1.1.1.2.cmml" xref="S5.T1.14.14.14.2.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.14.14.14.2.m1.1.1.2.1.cmml" xref="S5.T1.14.14.14.2.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.14.14.14.2.m1.1.1.2.2.cmml" type="float" xref="S5.T1.14.14.14.2.m1.1.1.2.2">51.85</cn></apply><apply id="S5.T1.14.14.14.2.m1.1.1.3.cmml" xref="S5.T1.14.14.14.2.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.14.14.14.2.m1.1.1.3.1.cmml" xref="S5.T1.14.14.14.2.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.14.14.14.2.m1.1.1.3.2.cmml" xref="S5.T1.14.14.14.2.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.14.14.14.2.m1.1.1.3.2.1.cmml" xref="S5.T1.14.14.14.2.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.14.14.14.2.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.14.14.14.2.m1.1.1.3.2.2">0.27</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.14.14.14.2.m1.1c">51.85\%_{\pm 0.27\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.14.14.14.2.m1.1d">51.85 % start_POSTSUBSCRIPT ± 0.27 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td" id="S5.T1.18.18.18.8"></td> <td class="ltx_td ltx_align_center" id="S5.T1.15.15.15.3"><math alttext="34.25\%_{\pm 1.63\%}" class="ltx_Math" display="inline" id="S5.T1.15.15.15.3.m1.1"><semantics id="S5.T1.15.15.15.3.m1.1a"><msub id="S5.T1.15.15.15.3.m1.1.1" xref="S5.T1.15.15.15.3.m1.1.1.cmml"><mrow id="S5.T1.15.15.15.3.m1.1.1.2" xref="S5.T1.15.15.15.3.m1.1.1.2.cmml"><mn id="S5.T1.15.15.15.3.m1.1.1.2.2" mathsize="90%" xref="S5.T1.15.15.15.3.m1.1.1.2.2.cmml">34.25</mn><mo id="S5.T1.15.15.15.3.m1.1.1.2.1" mathsize="90%" xref="S5.T1.15.15.15.3.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.15.15.15.3.m1.1.1.3" xref="S5.T1.15.15.15.3.m1.1.1.3.cmml"><mo id="S5.T1.15.15.15.3.m1.1.1.3a" mathsize="90%" xref="S5.T1.15.15.15.3.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.15.15.15.3.m1.1.1.3.2" xref="S5.T1.15.15.15.3.m1.1.1.3.2.cmml"><mn id="S5.T1.15.15.15.3.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.15.15.15.3.m1.1.1.3.2.2.cmml">1.63</mn><mo id="S5.T1.15.15.15.3.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.15.15.15.3.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.15.15.15.3.m1.1b"><apply id="S5.T1.15.15.15.3.m1.1.1.cmml" xref="S5.T1.15.15.15.3.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.15.15.15.3.m1.1.1.1.cmml" xref="S5.T1.15.15.15.3.m1.1.1">subscript</csymbol><apply id="S5.T1.15.15.15.3.m1.1.1.2.cmml" xref="S5.T1.15.15.15.3.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.15.15.15.3.m1.1.1.2.1.cmml" xref="S5.T1.15.15.15.3.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.15.15.15.3.m1.1.1.2.2.cmml" type="float" xref="S5.T1.15.15.15.3.m1.1.1.2.2">34.25</cn></apply><apply id="S5.T1.15.15.15.3.m1.1.1.3.cmml" xref="S5.T1.15.15.15.3.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.15.15.15.3.m1.1.1.3.1.cmml" xref="S5.T1.15.15.15.3.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.15.15.15.3.m1.1.1.3.2.cmml" xref="S5.T1.15.15.15.3.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.15.15.15.3.m1.1.1.3.2.1.cmml" xref="S5.T1.15.15.15.3.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.15.15.15.3.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.15.15.15.3.m1.1.1.3.2.2">1.63</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.15.15.15.3.m1.1c">34.25\%_{\pm 1.63\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.15.15.15.3.m1.1d">34.25 % start_POSTSUBSCRIPT ± 1.63 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T1.16.16.16.4"><math alttext="30.45\%_{\pm 1.43\%}" class="ltx_Math" display="inline" id="S5.T1.16.16.16.4.m1.1"><semantics id="S5.T1.16.16.16.4.m1.1a"><msub id="S5.T1.16.16.16.4.m1.1.1" xref="S5.T1.16.16.16.4.m1.1.1.cmml"><mrow id="S5.T1.16.16.16.4.m1.1.1.2" xref="S5.T1.16.16.16.4.m1.1.1.2.cmml"><mn id="S5.T1.16.16.16.4.m1.1.1.2.2" mathsize="90%" xref="S5.T1.16.16.16.4.m1.1.1.2.2.cmml">30.45</mn><mo id="S5.T1.16.16.16.4.m1.1.1.2.1" mathsize="90%" xref="S5.T1.16.16.16.4.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.16.16.16.4.m1.1.1.3" xref="S5.T1.16.16.16.4.m1.1.1.3.cmml"><mo id="S5.T1.16.16.16.4.m1.1.1.3a" mathsize="90%" xref="S5.T1.16.16.16.4.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.16.16.16.4.m1.1.1.3.2" xref="S5.T1.16.16.16.4.m1.1.1.3.2.cmml"><mn id="S5.T1.16.16.16.4.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.16.16.16.4.m1.1.1.3.2.2.cmml">1.43</mn><mo id="S5.T1.16.16.16.4.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.16.16.16.4.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.16.16.16.4.m1.1b"><apply id="S5.T1.16.16.16.4.m1.1.1.cmml" xref="S5.T1.16.16.16.4.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.16.16.16.4.m1.1.1.1.cmml" xref="S5.T1.16.16.16.4.m1.1.1">subscript</csymbol><apply id="S5.T1.16.16.16.4.m1.1.1.2.cmml" xref="S5.T1.16.16.16.4.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.16.16.16.4.m1.1.1.2.1.cmml" xref="S5.T1.16.16.16.4.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.16.16.16.4.m1.1.1.2.2.cmml" type="float" xref="S5.T1.16.16.16.4.m1.1.1.2.2">30.45</cn></apply><apply id="S5.T1.16.16.16.4.m1.1.1.3.cmml" xref="S5.T1.16.16.16.4.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.16.16.16.4.m1.1.1.3.1.cmml" xref="S5.T1.16.16.16.4.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.16.16.16.4.m1.1.1.3.2.cmml" xref="S5.T1.16.16.16.4.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.16.16.16.4.m1.1.1.3.2.1.cmml" xref="S5.T1.16.16.16.4.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.16.16.16.4.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.16.16.16.4.m1.1.1.3.2.2">1.43</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.16.16.16.4.m1.1c">30.45\%_{\pm 1.43\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.16.16.16.4.m1.1d">30.45 % start_POSTSUBSCRIPT ± 1.43 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td" id="S5.T1.18.18.18.9"></td> <td class="ltx_td ltx_align_center" id="S5.T1.17.17.17.5"><math alttext="40.56\%_{\pm 1.21\%}" class="ltx_Math" display="inline" id="S5.T1.17.17.17.5.m1.1"><semantics id="S5.T1.17.17.17.5.m1.1a"><msub id="S5.T1.17.17.17.5.m1.1.1" xref="S5.T1.17.17.17.5.m1.1.1.cmml"><mrow id="S5.T1.17.17.17.5.m1.1.1.2" xref="S5.T1.17.17.17.5.m1.1.1.2.cmml"><mn id="S5.T1.17.17.17.5.m1.1.1.2.2" mathsize="90%" xref="S5.T1.17.17.17.5.m1.1.1.2.2.cmml">40.56</mn><mo id="S5.T1.17.17.17.5.m1.1.1.2.1" mathsize="90%" xref="S5.T1.17.17.17.5.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.17.17.17.5.m1.1.1.3" xref="S5.T1.17.17.17.5.m1.1.1.3.cmml"><mo id="S5.T1.17.17.17.5.m1.1.1.3a" mathsize="90%" xref="S5.T1.17.17.17.5.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.17.17.17.5.m1.1.1.3.2" xref="S5.T1.17.17.17.5.m1.1.1.3.2.cmml"><mn id="S5.T1.17.17.17.5.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.17.17.17.5.m1.1.1.3.2.2.cmml">1.21</mn><mo id="S5.T1.17.17.17.5.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.17.17.17.5.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.17.17.17.5.m1.1b"><apply id="S5.T1.17.17.17.5.m1.1.1.cmml" xref="S5.T1.17.17.17.5.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.17.17.17.5.m1.1.1.1.cmml" xref="S5.T1.17.17.17.5.m1.1.1">subscript</csymbol><apply id="S5.T1.17.17.17.5.m1.1.1.2.cmml" xref="S5.T1.17.17.17.5.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.17.17.17.5.m1.1.1.2.1.cmml" xref="S5.T1.17.17.17.5.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.17.17.17.5.m1.1.1.2.2.cmml" type="float" xref="S5.T1.17.17.17.5.m1.1.1.2.2">40.56</cn></apply><apply id="S5.T1.17.17.17.5.m1.1.1.3.cmml" xref="S5.T1.17.17.17.5.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.17.17.17.5.m1.1.1.3.1.cmml" xref="S5.T1.17.17.17.5.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.17.17.17.5.m1.1.1.3.2.cmml" xref="S5.T1.17.17.17.5.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.17.17.17.5.m1.1.1.3.2.1.cmml" xref="S5.T1.17.17.17.5.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.17.17.17.5.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.17.17.17.5.m1.1.1.3.2.2">1.21</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.17.17.17.5.m1.1c">40.56\%_{\pm 1.21\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.17.17.17.5.m1.1d">40.56 % start_POSTSUBSCRIPT ± 1.21 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T1.18.18.18.6"><math alttext="41.00\%_{\pm 0.57\%}" class="ltx_Math" display="inline" id="S5.T1.18.18.18.6.m1.1"><semantics id="S5.T1.18.18.18.6.m1.1a"><msub id="S5.T1.18.18.18.6.m1.1.1" xref="S5.T1.18.18.18.6.m1.1.1.cmml"><mrow id="S5.T1.18.18.18.6.m1.1.1.2" xref="S5.T1.18.18.18.6.m1.1.1.2.cmml"><mn id="S5.T1.18.18.18.6.m1.1.1.2.2" mathsize="90%" xref="S5.T1.18.18.18.6.m1.1.1.2.2.cmml">41.00</mn><mo id="S5.T1.18.18.18.6.m1.1.1.2.1" mathsize="90%" xref="S5.T1.18.18.18.6.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.18.18.18.6.m1.1.1.3" xref="S5.T1.18.18.18.6.m1.1.1.3.cmml"><mo id="S5.T1.18.18.18.6.m1.1.1.3a" mathsize="90%" xref="S5.T1.18.18.18.6.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.18.18.18.6.m1.1.1.3.2" xref="S5.T1.18.18.18.6.m1.1.1.3.2.cmml"><mn id="S5.T1.18.18.18.6.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.18.18.18.6.m1.1.1.3.2.2.cmml">0.57</mn><mo id="S5.T1.18.18.18.6.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.18.18.18.6.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.18.18.18.6.m1.1b"><apply id="S5.T1.18.18.18.6.m1.1.1.cmml" xref="S5.T1.18.18.18.6.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.18.18.18.6.m1.1.1.1.cmml" xref="S5.T1.18.18.18.6.m1.1.1">subscript</csymbol><apply id="S5.T1.18.18.18.6.m1.1.1.2.cmml" xref="S5.T1.18.18.18.6.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.18.18.18.6.m1.1.1.2.1.cmml" xref="S5.T1.18.18.18.6.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.18.18.18.6.m1.1.1.2.2.cmml" type="float" xref="S5.T1.18.18.18.6.m1.1.1.2.2">41.00</cn></apply><apply id="S5.T1.18.18.18.6.m1.1.1.3.cmml" xref="S5.T1.18.18.18.6.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.18.18.18.6.m1.1.1.3.1.cmml" xref="S5.T1.18.18.18.6.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.18.18.18.6.m1.1.1.3.2.cmml" xref="S5.T1.18.18.18.6.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.18.18.18.6.m1.1.1.3.2.1.cmml" xref="S5.T1.18.18.18.6.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.18.18.18.6.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.18.18.18.6.m1.1.1.3.2.2">0.57</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.18.18.18.6.m1.1c">41.00\%_{\pm 0.57\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.18.18.18.6.m1.1d">41.00 % start_POSTSUBSCRIPT ± 0.57 % end_POSTSUBSCRIPT</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T1.24.24.24"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S5.T1.24.24.24.7"><span class="ltx_text" id="S5.T1.24.24.24.7.1" style="font-size:90%;">KN</span></th> <td class="ltx_td ltx_align_center" id="S5.T1.19.19.19.1"><math alttext="50.66\%_{\pm 0.52\%}" class="ltx_Math" display="inline" id="S5.T1.19.19.19.1.m1.1"><semantics id="S5.T1.19.19.19.1.m1.1a"><msub id="S5.T1.19.19.19.1.m1.1.1" xref="S5.T1.19.19.19.1.m1.1.1.cmml"><mrow id="S5.T1.19.19.19.1.m1.1.1.2" xref="S5.T1.19.19.19.1.m1.1.1.2.cmml"><mn id="S5.T1.19.19.19.1.m1.1.1.2.2" mathsize="90%" xref="S5.T1.19.19.19.1.m1.1.1.2.2.cmml">50.66</mn><mo id="S5.T1.19.19.19.1.m1.1.1.2.1" mathsize="90%" xref="S5.T1.19.19.19.1.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.19.19.19.1.m1.1.1.3" xref="S5.T1.19.19.19.1.m1.1.1.3.cmml"><mo id="S5.T1.19.19.19.1.m1.1.1.3a" mathsize="90%" xref="S5.T1.19.19.19.1.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.19.19.19.1.m1.1.1.3.2" xref="S5.T1.19.19.19.1.m1.1.1.3.2.cmml"><mn id="S5.T1.19.19.19.1.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.19.19.19.1.m1.1.1.3.2.2.cmml">0.52</mn><mo id="S5.T1.19.19.19.1.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.19.19.19.1.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.19.19.19.1.m1.1b"><apply id="S5.T1.19.19.19.1.m1.1.1.cmml" xref="S5.T1.19.19.19.1.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.19.19.19.1.m1.1.1.1.cmml" xref="S5.T1.19.19.19.1.m1.1.1">subscript</csymbol><apply id="S5.T1.19.19.19.1.m1.1.1.2.cmml" xref="S5.T1.19.19.19.1.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.19.19.19.1.m1.1.1.2.1.cmml" xref="S5.T1.19.19.19.1.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.19.19.19.1.m1.1.1.2.2.cmml" type="float" xref="S5.T1.19.19.19.1.m1.1.1.2.2">50.66</cn></apply><apply id="S5.T1.19.19.19.1.m1.1.1.3.cmml" xref="S5.T1.19.19.19.1.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.19.19.19.1.m1.1.1.3.1.cmml" xref="S5.T1.19.19.19.1.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.19.19.19.1.m1.1.1.3.2.cmml" xref="S5.T1.19.19.19.1.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.19.19.19.1.m1.1.1.3.2.1.cmml" xref="S5.T1.19.19.19.1.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.19.19.19.1.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.19.19.19.1.m1.1.1.3.2.2">0.52</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.19.19.19.1.m1.1c">50.66\%_{\pm 0.52\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.19.19.19.1.m1.1d">50.66 % start_POSTSUBSCRIPT ± 0.52 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T1.20.20.20.2"><math alttext="51.06\%_{\pm 0.11\%}" class="ltx_Math" display="inline" id="S5.T1.20.20.20.2.m1.1"><semantics id="S5.T1.20.20.20.2.m1.1a"><msub id="S5.T1.20.20.20.2.m1.1.1" xref="S5.T1.20.20.20.2.m1.1.1.cmml"><mrow id="S5.T1.20.20.20.2.m1.1.1.2" xref="S5.T1.20.20.20.2.m1.1.1.2.cmml"><mn id="S5.T1.20.20.20.2.m1.1.1.2.2" mathsize="90%" xref="S5.T1.20.20.20.2.m1.1.1.2.2.cmml">51.06</mn><mo id="S5.T1.20.20.20.2.m1.1.1.2.1" mathsize="90%" xref="S5.T1.20.20.20.2.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.20.20.20.2.m1.1.1.3" xref="S5.T1.20.20.20.2.m1.1.1.3.cmml"><mo id="S5.T1.20.20.20.2.m1.1.1.3a" mathsize="90%" xref="S5.T1.20.20.20.2.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.20.20.20.2.m1.1.1.3.2" xref="S5.T1.20.20.20.2.m1.1.1.3.2.cmml"><mn id="S5.T1.20.20.20.2.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.20.20.20.2.m1.1.1.3.2.2.cmml">0.11</mn><mo id="S5.T1.20.20.20.2.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.20.20.20.2.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.20.20.20.2.m1.1b"><apply id="S5.T1.20.20.20.2.m1.1.1.cmml" xref="S5.T1.20.20.20.2.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.20.20.20.2.m1.1.1.1.cmml" xref="S5.T1.20.20.20.2.m1.1.1">subscript</csymbol><apply id="S5.T1.20.20.20.2.m1.1.1.2.cmml" xref="S5.T1.20.20.20.2.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.20.20.20.2.m1.1.1.2.1.cmml" xref="S5.T1.20.20.20.2.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.20.20.20.2.m1.1.1.2.2.cmml" type="float" xref="S5.T1.20.20.20.2.m1.1.1.2.2">51.06</cn></apply><apply id="S5.T1.20.20.20.2.m1.1.1.3.cmml" xref="S5.T1.20.20.20.2.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.20.20.20.2.m1.1.1.3.1.cmml" xref="S5.T1.20.20.20.2.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.20.20.20.2.m1.1.1.3.2.cmml" xref="S5.T1.20.20.20.2.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.20.20.20.2.m1.1.1.3.2.1.cmml" xref="S5.T1.20.20.20.2.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.20.20.20.2.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.20.20.20.2.m1.1.1.3.2.2">0.11</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.20.20.20.2.m1.1c">51.06\%_{\pm 0.11\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.20.20.20.2.m1.1d">51.06 % start_POSTSUBSCRIPT ± 0.11 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td" id="S5.T1.24.24.24.8"></td> <td class="ltx_td ltx_align_center" id="S5.T1.21.21.21.3"><math alttext="49.41\%_{\pm 0.47\%}" class="ltx_Math" display="inline" id="S5.T1.21.21.21.3.m1.1"><semantics id="S5.T1.21.21.21.3.m1.1a"><msub id="S5.T1.21.21.21.3.m1.1.1" xref="S5.T1.21.21.21.3.m1.1.1.cmml"><mrow id="S5.T1.21.21.21.3.m1.1.1.2" xref="S5.T1.21.21.21.3.m1.1.1.2.cmml"><mn id="S5.T1.21.21.21.3.m1.1.1.2.2" mathsize="90%" xref="S5.T1.21.21.21.3.m1.1.1.2.2.cmml">49.41</mn><mo id="S5.T1.21.21.21.3.m1.1.1.2.1" mathsize="90%" xref="S5.T1.21.21.21.3.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.21.21.21.3.m1.1.1.3" xref="S5.T1.21.21.21.3.m1.1.1.3.cmml"><mo id="S5.T1.21.21.21.3.m1.1.1.3a" mathsize="90%" xref="S5.T1.21.21.21.3.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.21.21.21.3.m1.1.1.3.2" xref="S5.T1.21.21.21.3.m1.1.1.3.2.cmml"><mn id="S5.T1.21.21.21.3.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.21.21.21.3.m1.1.1.3.2.2.cmml">0.47</mn><mo id="S5.T1.21.21.21.3.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.21.21.21.3.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.21.21.21.3.m1.1b"><apply id="S5.T1.21.21.21.3.m1.1.1.cmml" xref="S5.T1.21.21.21.3.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.21.21.21.3.m1.1.1.1.cmml" xref="S5.T1.21.21.21.3.m1.1.1">subscript</csymbol><apply id="S5.T1.21.21.21.3.m1.1.1.2.cmml" xref="S5.T1.21.21.21.3.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.21.21.21.3.m1.1.1.2.1.cmml" xref="S5.T1.21.21.21.3.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.21.21.21.3.m1.1.1.2.2.cmml" type="float" xref="S5.T1.21.21.21.3.m1.1.1.2.2">49.41</cn></apply><apply id="S5.T1.21.21.21.3.m1.1.1.3.cmml" xref="S5.T1.21.21.21.3.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.21.21.21.3.m1.1.1.3.1.cmml" xref="S5.T1.21.21.21.3.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.21.21.21.3.m1.1.1.3.2.cmml" xref="S5.T1.21.21.21.3.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.21.21.21.3.m1.1.1.3.2.1.cmml" xref="S5.T1.21.21.21.3.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.21.21.21.3.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.21.21.21.3.m1.1.1.3.2.2">0.47</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.21.21.21.3.m1.1c">49.41\%_{\pm 0.47\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.21.21.21.3.m1.1d">49.41 % start_POSTSUBSCRIPT ± 0.47 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T1.22.22.22.4"><math alttext="51.20\%_{\pm 1.38\%}" class="ltx_Math" display="inline" id="S5.T1.22.22.22.4.m1.1"><semantics id="S5.T1.22.22.22.4.m1.1a"><msub id="S5.T1.22.22.22.4.m1.1.1" xref="S5.T1.22.22.22.4.m1.1.1.cmml"><mrow id="S5.T1.22.22.22.4.m1.1.1.2" xref="S5.T1.22.22.22.4.m1.1.1.2.cmml"><mn id="S5.T1.22.22.22.4.m1.1.1.2.2" mathsize="90%" xref="S5.T1.22.22.22.4.m1.1.1.2.2.cmml">51.20</mn><mo id="S5.T1.22.22.22.4.m1.1.1.2.1" mathsize="90%" xref="S5.T1.22.22.22.4.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.22.22.22.4.m1.1.1.3" xref="S5.T1.22.22.22.4.m1.1.1.3.cmml"><mo id="S5.T1.22.22.22.4.m1.1.1.3a" mathsize="90%" xref="S5.T1.22.22.22.4.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.22.22.22.4.m1.1.1.3.2" xref="S5.T1.22.22.22.4.m1.1.1.3.2.cmml"><mn id="S5.T1.22.22.22.4.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.22.22.22.4.m1.1.1.3.2.2.cmml">1.38</mn><mo id="S5.T1.22.22.22.4.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.22.22.22.4.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.22.22.22.4.m1.1b"><apply id="S5.T1.22.22.22.4.m1.1.1.cmml" xref="S5.T1.22.22.22.4.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.22.22.22.4.m1.1.1.1.cmml" xref="S5.T1.22.22.22.4.m1.1.1">subscript</csymbol><apply id="S5.T1.22.22.22.4.m1.1.1.2.cmml" xref="S5.T1.22.22.22.4.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.22.22.22.4.m1.1.1.2.1.cmml" xref="S5.T1.22.22.22.4.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.22.22.22.4.m1.1.1.2.2.cmml" type="float" xref="S5.T1.22.22.22.4.m1.1.1.2.2">51.20</cn></apply><apply id="S5.T1.22.22.22.4.m1.1.1.3.cmml" xref="S5.T1.22.22.22.4.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.22.22.22.4.m1.1.1.3.1.cmml" xref="S5.T1.22.22.22.4.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.22.22.22.4.m1.1.1.3.2.cmml" xref="S5.T1.22.22.22.4.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.22.22.22.4.m1.1.1.3.2.1.cmml" xref="S5.T1.22.22.22.4.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.22.22.22.4.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.22.22.22.4.m1.1.1.3.2.2">1.38</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.22.22.22.4.m1.1c">51.20\%_{\pm 1.38\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.22.22.22.4.m1.1d">51.20 % start_POSTSUBSCRIPT ± 1.38 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td" id="S5.T1.24.24.24.9"></td> <td class="ltx_td ltx_align_center" id="S5.T1.23.23.23.5"><math alttext="50.52\%_{\pm 1.05\%}" class="ltx_Math" display="inline" id="S5.T1.23.23.23.5.m1.1"><semantics id="S5.T1.23.23.23.5.m1.1a"><msub id="S5.T1.23.23.23.5.m1.1.1" xref="S5.T1.23.23.23.5.m1.1.1.cmml"><mrow id="S5.T1.23.23.23.5.m1.1.1.2" xref="S5.T1.23.23.23.5.m1.1.1.2.cmml"><mn id="S5.T1.23.23.23.5.m1.1.1.2.2" mathsize="90%" xref="S5.T1.23.23.23.5.m1.1.1.2.2.cmml">50.52</mn><mo id="S5.T1.23.23.23.5.m1.1.1.2.1" mathsize="90%" xref="S5.T1.23.23.23.5.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.23.23.23.5.m1.1.1.3" xref="S5.T1.23.23.23.5.m1.1.1.3.cmml"><mo id="S5.T1.23.23.23.5.m1.1.1.3a" mathsize="90%" xref="S5.T1.23.23.23.5.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.23.23.23.5.m1.1.1.3.2" xref="S5.T1.23.23.23.5.m1.1.1.3.2.cmml"><mn id="S5.T1.23.23.23.5.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.23.23.23.5.m1.1.1.3.2.2.cmml">1.05</mn><mo id="S5.T1.23.23.23.5.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.23.23.23.5.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.23.23.23.5.m1.1b"><apply id="S5.T1.23.23.23.5.m1.1.1.cmml" xref="S5.T1.23.23.23.5.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.23.23.23.5.m1.1.1.1.cmml" xref="S5.T1.23.23.23.5.m1.1.1">subscript</csymbol><apply id="S5.T1.23.23.23.5.m1.1.1.2.cmml" xref="S5.T1.23.23.23.5.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.23.23.23.5.m1.1.1.2.1.cmml" xref="S5.T1.23.23.23.5.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.23.23.23.5.m1.1.1.2.2.cmml" type="float" xref="S5.T1.23.23.23.5.m1.1.1.2.2">50.52</cn></apply><apply id="S5.T1.23.23.23.5.m1.1.1.3.cmml" xref="S5.T1.23.23.23.5.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.23.23.23.5.m1.1.1.3.1.cmml" xref="S5.T1.23.23.23.5.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.23.23.23.5.m1.1.1.3.2.cmml" xref="S5.T1.23.23.23.5.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.23.23.23.5.m1.1.1.3.2.1.cmml" xref="S5.T1.23.23.23.5.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.23.23.23.5.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.23.23.23.5.m1.1.1.3.2.2">1.05</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.23.23.23.5.m1.1c">50.52\%_{\pm 1.05\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.23.23.23.5.m1.1d">50.52 % start_POSTSUBSCRIPT ± 1.05 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T1.24.24.24.6"><math alttext="50.67\%_{\pm 1.10\%}" class="ltx_Math" display="inline" id="S5.T1.24.24.24.6.m1.1"><semantics id="S5.T1.24.24.24.6.m1.1a"><msub id="S5.T1.24.24.24.6.m1.1.1" xref="S5.T1.24.24.24.6.m1.1.1.cmml"><mrow id="S5.T1.24.24.24.6.m1.1.1.2" xref="S5.T1.24.24.24.6.m1.1.1.2.cmml"><mn id="S5.T1.24.24.24.6.m1.1.1.2.2" mathsize="90%" xref="S5.T1.24.24.24.6.m1.1.1.2.2.cmml">50.67</mn><mo id="S5.T1.24.24.24.6.m1.1.1.2.1" mathsize="90%" xref="S5.T1.24.24.24.6.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.24.24.24.6.m1.1.1.3" xref="S5.T1.24.24.24.6.m1.1.1.3.cmml"><mo id="S5.T1.24.24.24.6.m1.1.1.3a" mathsize="90%" xref="S5.T1.24.24.24.6.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.24.24.24.6.m1.1.1.3.2" xref="S5.T1.24.24.24.6.m1.1.1.3.2.cmml"><mn id="S5.T1.24.24.24.6.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.24.24.24.6.m1.1.1.3.2.2.cmml">1.10</mn><mo id="S5.T1.24.24.24.6.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.24.24.24.6.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.24.24.24.6.m1.1b"><apply id="S5.T1.24.24.24.6.m1.1.1.cmml" xref="S5.T1.24.24.24.6.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.24.24.24.6.m1.1.1.1.cmml" xref="S5.T1.24.24.24.6.m1.1.1">subscript</csymbol><apply id="S5.T1.24.24.24.6.m1.1.1.2.cmml" xref="S5.T1.24.24.24.6.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.24.24.24.6.m1.1.1.2.1.cmml" xref="S5.T1.24.24.24.6.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.24.24.24.6.m1.1.1.2.2.cmml" type="float" xref="S5.T1.24.24.24.6.m1.1.1.2.2">50.67</cn></apply><apply id="S5.T1.24.24.24.6.m1.1.1.3.cmml" xref="S5.T1.24.24.24.6.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.24.24.24.6.m1.1.1.3.1.cmml" xref="S5.T1.24.24.24.6.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.24.24.24.6.m1.1.1.3.2.cmml" xref="S5.T1.24.24.24.6.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.24.24.24.6.m1.1.1.3.2.1.cmml" xref="S5.T1.24.24.24.6.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.24.24.24.6.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.24.24.24.6.m1.1.1.3.2.2">1.10</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.24.24.24.6.m1.1c">50.67\%_{\pm 1.10\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.24.24.24.6.m1.1d">50.67 % start_POSTSUBSCRIPT ± 1.10 % end_POSTSUBSCRIPT</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T1.30.30.30"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S5.T1.30.30.30.7"><span class="ltx_text" id="S5.T1.30.30.30.7.1" style="font-size:90%;">MEND</span></th> <td class="ltx_td ltx_align_center" id="S5.T1.25.25.25.1"><math alttext="50.41\%_{\pm 0.18\%}" class="ltx_Math" display="inline" id="S5.T1.25.25.25.1.m1.1"><semantics id="S5.T1.25.25.25.1.m1.1a"><msub id="S5.T1.25.25.25.1.m1.1.1" xref="S5.T1.25.25.25.1.m1.1.1.cmml"><mrow id="S5.T1.25.25.25.1.m1.1.1.2" xref="S5.T1.25.25.25.1.m1.1.1.2.cmml"><mn id="S5.T1.25.25.25.1.m1.1.1.2.2" mathsize="90%" xref="S5.T1.25.25.25.1.m1.1.1.2.2.cmml">50.41</mn><mo id="S5.T1.25.25.25.1.m1.1.1.2.1" mathsize="90%" xref="S5.T1.25.25.25.1.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.25.25.25.1.m1.1.1.3" xref="S5.T1.25.25.25.1.m1.1.1.3.cmml"><mo id="S5.T1.25.25.25.1.m1.1.1.3a" mathsize="90%" xref="S5.T1.25.25.25.1.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.25.25.25.1.m1.1.1.3.2" xref="S5.T1.25.25.25.1.m1.1.1.3.2.cmml"><mn id="S5.T1.25.25.25.1.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.25.25.25.1.m1.1.1.3.2.2.cmml">0.18</mn><mo id="S5.T1.25.25.25.1.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.25.25.25.1.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.25.25.25.1.m1.1b"><apply id="S5.T1.25.25.25.1.m1.1.1.cmml" xref="S5.T1.25.25.25.1.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.25.25.25.1.m1.1.1.1.cmml" xref="S5.T1.25.25.25.1.m1.1.1">subscript</csymbol><apply id="S5.T1.25.25.25.1.m1.1.1.2.cmml" xref="S5.T1.25.25.25.1.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.25.25.25.1.m1.1.1.2.1.cmml" xref="S5.T1.25.25.25.1.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.25.25.25.1.m1.1.1.2.2.cmml" type="float" xref="S5.T1.25.25.25.1.m1.1.1.2.2">50.41</cn></apply><apply id="S5.T1.25.25.25.1.m1.1.1.3.cmml" xref="S5.T1.25.25.25.1.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.25.25.25.1.m1.1.1.3.1.cmml" xref="S5.T1.25.25.25.1.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.25.25.25.1.m1.1.1.3.2.cmml" xref="S5.T1.25.25.25.1.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.25.25.25.1.m1.1.1.3.2.1.cmml" xref="S5.T1.25.25.25.1.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.25.25.25.1.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.25.25.25.1.m1.1.1.3.2.2">0.18</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.25.25.25.1.m1.1c">50.41\%_{\pm 0.18\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.25.25.25.1.m1.1d">50.41 % start_POSTSUBSCRIPT ± 0.18 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T1.26.26.26.2"><math alttext="50.20\%_{\pm 0.02\%}" class="ltx_Math" display="inline" id="S5.T1.26.26.26.2.m1.1"><semantics id="S5.T1.26.26.26.2.m1.1a"><msub id="S5.T1.26.26.26.2.m1.1.1" xref="S5.T1.26.26.26.2.m1.1.1.cmml"><mrow id="S5.T1.26.26.26.2.m1.1.1.2" xref="S5.T1.26.26.26.2.m1.1.1.2.cmml"><mn id="S5.T1.26.26.26.2.m1.1.1.2.2" mathsize="90%" xref="S5.T1.26.26.26.2.m1.1.1.2.2.cmml">50.20</mn><mo id="S5.T1.26.26.26.2.m1.1.1.2.1" mathsize="90%" xref="S5.T1.26.26.26.2.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.26.26.26.2.m1.1.1.3" xref="S5.T1.26.26.26.2.m1.1.1.3.cmml"><mo id="S5.T1.26.26.26.2.m1.1.1.3a" mathsize="90%" xref="S5.T1.26.26.26.2.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.26.26.26.2.m1.1.1.3.2" xref="S5.T1.26.26.26.2.m1.1.1.3.2.cmml"><mn id="S5.T1.26.26.26.2.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.26.26.26.2.m1.1.1.3.2.2.cmml">0.02</mn><mo id="S5.T1.26.26.26.2.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.26.26.26.2.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.26.26.26.2.m1.1b"><apply id="S5.T1.26.26.26.2.m1.1.1.cmml" xref="S5.T1.26.26.26.2.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.26.26.26.2.m1.1.1.1.cmml" xref="S5.T1.26.26.26.2.m1.1.1">subscript</csymbol><apply id="S5.T1.26.26.26.2.m1.1.1.2.cmml" xref="S5.T1.26.26.26.2.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.26.26.26.2.m1.1.1.2.1.cmml" xref="S5.T1.26.26.26.2.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.26.26.26.2.m1.1.1.2.2.cmml" type="float" xref="S5.T1.26.26.26.2.m1.1.1.2.2">50.20</cn></apply><apply id="S5.T1.26.26.26.2.m1.1.1.3.cmml" xref="S5.T1.26.26.26.2.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.26.26.26.2.m1.1.1.3.1.cmml" xref="S5.T1.26.26.26.2.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.26.26.26.2.m1.1.1.3.2.cmml" xref="S5.T1.26.26.26.2.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.26.26.26.2.m1.1.1.3.2.1.cmml" xref="S5.T1.26.26.26.2.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.26.26.26.2.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.26.26.26.2.m1.1.1.3.2.2">0.02</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.26.26.26.2.m1.1c">50.20\%_{\pm 0.02\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.26.26.26.2.m1.1d">50.20 % start_POSTSUBSCRIPT ± 0.02 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td" id="S5.T1.30.30.30.8"></td> <td class="ltx_td ltx_align_center" id="S5.T1.27.27.27.3"><math alttext="49.35\%_{\pm 0.47\%}" class="ltx_Math" display="inline" id="S5.T1.27.27.27.3.m1.1"><semantics id="S5.T1.27.27.27.3.m1.1a"><msub id="S5.T1.27.27.27.3.m1.1.1" xref="S5.T1.27.27.27.3.m1.1.1.cmml"><mrow id="S5.T1.27.27.27.3.m1.1.1.2" xref="S5.T1.27.27.27.3.m1.1.1.2.cmml"><mn id="S5.T1.27.27.27.3.m1.1.1.2.2" mathsize="90%" xref="S5.T1.27.27.27.3.m1.1.1.2.2.cmml">49.35</mn><mo id="S5.T1.27.27.27.3.m1.1.1.2.1" mathsize="90%" xref="S5.T1.27.27.27.3.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.27.27.27.3.m1.1.1.3" xref="S5.T1.27.27.27.3.m1.1.1.3.cmml"><mo id="S5.T1.27.27.27.3.m1.1.1.3a" mathsize="90%" xref="S5.T1.27.27.27.3.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.27.27.27.3.m1.1.1.3.2" xref="S5.T1.27.27.27.3.m1.1.1.3.2.cmml"><mn id="S5.T1.27.27.27.3.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.27.27.27.3.m1.1.1.3.2.2.cmml">0.47</mn><mo id="S5.T1.27.27.27.3.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.27.27.27.3.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.27.27.27.3.m1.1b"><apply id="S5.T1.27.27.27.3.m1.1.1.cmml" xref="S5.T1.27.27.27.3.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.27.27.27.3.m1.1.1.1.cmml" xref="S5.T1.27.27.27.3.m1.1.1">subscript</csymbol><apply id="S5.T1.27.27.27.3.m1.1.1.2.cmml" xref="S5.T1.27.27.27.3.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.27.27.27.3.m1.1.1.2.1.cmml" xref="S5.T1.27.27.27.3.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.27.27.27.3.m1.1.1.2.2.cmml" type="float" xref="S5.T1.27.27.27.3.m1.1.1.2.2">49.35</cn></apply><apply id="S5.T1.27.27.27.3.m1.1.1.3.cmml" xref="S5.T1.27.27.27.3.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.27.27.27.3.m1.1.1.3.1.cmml" xref="S5.T1.27.27.27.3.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.27.27.27.3.m1.1.1.3.2.cmml" xref="S5.T1.27.27.27.3.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.27.27.27.3.m1.1.1.3.2.1.cmml" xref="S5.T1.27.27.27.3.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.27.27.27.3.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.27.27.27.3.m1.1.1.3.2.2">0.47</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.27.27.27.3.m1.1c">49.35\%_{\pm 0.47\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.27.27.27.3.m1.1d">49.35 % start_POSTSUBSCRIPT ± 0.47 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T1.28.28.28.4"><math alttext="50.88\%_{\pm 0.30\%}" class="ltx_Math" display="inline" id="S5.T1.28.28.28.4.m1.1"><semantics id="S5.T1.28.28.28.4.m1.1a"><msub id="S5.T1.28.28.28.4.m1.1.1" xref="S5.T1.28.28.28.4.m1.1.1.cmml"><mrow id="S5.T1.28.28.28.4.m1.1.1.2" xref="S5.T1.28.28.28.4.m1.1.1.2.cmml"><mn id="S5.T1.28.28.28.4.m1.1.1.2.2" mathsize="90%" xref="S5.T1.28.28.28.4.m1.1.1.2.2.cmml">50.88</mn><mo id="S5.T1.28.28.28.4.m1.1.1.2.1" mathsize="90%" xref="S5.T1.28.28.28.4.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.28.28.28.4.m1.1.1.3" xref="S5.T1.28.28.28.4.m1.1.1.3.cmml"><mo id="S5.T1.28.28.28.4.m1.1.1.3a" mathsize="90%" xref="S5.T1.28.28.28.4.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.28.28.28.4.m1.1.1.3.2" xref="S5.T1.28.28.28.4.m1.1.1.3.2.cmml"><mn id="S5.T1.28.28.28.4.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.28.28.28.4.m1.1.1.3.2.2.cmml">0.30</mn><mo id="S5.T1.28.28.28.4.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.28.28.28.4.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.28.28.28.4.m1.1b"><apply id="S5.T1.28.28.28.4.m1.1.1.cmml" xref="S5.T1.28.28.28.4.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.28.28.28.4.m1.1.1.1.cmml" xref="S5.T1.28.28.28.4.m1.1.1">subscript</csymbol><apply id="S5.T1.28.28.28.4.m1.1.1.2.cmml" xref="S5.T1.28.28.28.4.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.28.28.28.4.m1.1.1.2.1.cmml" xref="S5.T1.28.28.28.4.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.28.28.28.4.m1.1.1.2.2.cmml" type="float" xref="S5.T1.28.28.28.4.m1.1.1.2.2">50.88</cn></apply><apply id="S5.T1.28.28.28.4.m1.1.1.3.cmml" xref="S5.T1.28.28.28.4.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.28.28.28.4.m1.1.1.3.1.cmml" xref="S5.T1.28.28.28.4.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.28.28.28.4.m1.1.1.3.2.cmml" xref="S5.T1.28.28.28.4.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.28.28.28.4.m1.1.1.3.2.1.cmml" xref="S5.T1.28.28.28.4.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.28.28.28.4.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.28.28.28.4.m1.1.1.3.2.2">0.30</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.28.28.28.4.m1.1c">50.88\%_{\pm 0.30\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.28.28.28.4.m1.1d">50.88 % start_POSTSUBSCRIPT ± 0.30 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td" id="S5.T1.30.30.30.9"></td> <td class="ltx_td ltx_align_center" id="S5.T1.29.29.29.5"><math alttext="50.29\%_{\pm 0.71\%}" class="ltx_Math" display="inline" id="S5.T1.29.29.29.5.m1.1"><semantics id="S5.T1.29.29.29.5.m1.1a"><msub id="S5.T1.29.29.29.5.m1.1.1" xref="S5.T1.29.29.29.5.m1.1.1.cmml"><mrow id="S5.T1.29.29.29.5.m1.1.1.2" xref="S5.T1.29.29.29.5.m1.1.1.2.cmml"><mn id="S5.T1.29.29.29.5.m1.1.1.2.2" mathsize="90%" xref="S5.T1.29.29.29.5.m1.1.1.2.2.cmml">50.29</mn><mo id="S5.T1.29.29.29.5.m1.1.1.2.1" mathsize="90%" xref="S5.T1.29.29.29.5.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.29.29.29.5.m1.1.1.3" xref="S5.T1.29.29.29.5.m1.1.1.3.cmml"><mo id="S5.T1.29.29.29.5.m1.1.1.3a" mathsize="90%" xref="S5.T1.29.29.29.5.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.29.29.29.5.m1.1.1.3.2" xref="S5.T1.29.29.29.5.m1.1.1.3.2.cmml"><mn id="S5.T1.29.29.29.5.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.29.29.29.5.m1.1.1.3.2.2.cmml">0.71</mn><mo id="S5.T1.29.29.29.5.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.29.29.29.5.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.29.29.29.5.m1.1b"><apply id="S5.T1.29.29.29.5.m1.1.1.cmml" xref="S5.T1.29.29.29.5.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.29.29.29.5.m1.1.1.1.cmml" xref="S5.T1.29.29.29.5.m1.1.1">subscript</csymbol><apply id="S5.T1.29.29.29.5.m1.1.1.2.cmml" xref="S5.T1.29.29.29.5.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.29.29.29.5.m1.1.1.2.1.cmml" xref="S5.T1.29.29.29.5.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.29.29.29.5.m1.1.1.2.2.cmml" type="float" xref="S5.T1.29.29.29.5.m1.1.1.2.2">50.29</cn></apply><apply id="S5.T1.29.29.29.5.m1.1.1.3.cmml" xref="S5.T1.29.29.29.5.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.29.29.29.5.m1.1.1.3.1.cmml" xref="S5.T1.29.29.29.5.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.29.29.29.5.m1.1.1.3.2.cmml" xref="S5.T1.29.29.29.5.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.29.29.29.5.m1.1.1.3.2.1.cmml" xref="S5.T1.29.29.29.5.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.29.29.29.5.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.29.29.29.5.m1.1.1.3.2.2">0.71</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.29.29.29.5.m1.1c">50.29\%_{\pm 0.71\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.29.29.29.5.m1.1d">50.29 % start_POSTSUBSCRIPT ± 0.71 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T1.30.30.30.6"><math alttext="54.65\%_{\pm 1.12\%}" class="ltx_Math" display="inline" id="S5.T1.30.30.30.6.m1.1"><semantics id="S5.T1.30.30.30.6.m1.1a"><msub id="S5.T1.30.30.30.6.m1.1.1" xref="S5.T1.30.30.30.6.m1.1.1.cmml"><mrow id="S5.T1.30.30.30.6.m1.1.1.2" xref="S5.T1.30.30.30.6.m1.1.1.2.cmml"><mn id="S5.T1.30.30.30.6.m1.1.1.2.2" mathsize="90%" xref="S5.T1.30.30.30.6.m1.1.1.2.2.cmml">54.65</mn><mo id="S5.T1.30.30.30.6.m1.1.1.2.1" mathsize="90%" xref="S5.T1.30.30.30.6.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.30.30.30.6.m1.1.1.3" xref="S5.T1.30.30.30.6.m1.1.1.3.cmml"><mo id="S5.T1.30.30.30.6.m1.1.1.3a" mathsize="90%" xref="S5.T1.30.30.30.6.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.30.30.30.6.m1.1.1.3.2" xref="S5.T1.30.30.30.6.m1.1.1.3.2.cmml"><mn id="S5.T1.30.30.30.6.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.30.30.30.6.m1.1.1.3.2.2.cmml">1.12</mn><mo id="S5.T1.30.30.30.6.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.30.30.30.6.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.30.30.30.6.m1.1b"><apply id="S5.T1.30.30.30.6.m1.1.1.cmml" xref="S5.T1.30.30.30.6.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.30.30.30.6.m1.1.1.1.cmml" xref="S5.T1.30.30.30.6.m1.1.1">subscript</csymbol><apply id="S5.T1.30.30.30.6.m1.1.1.2.cmml" xref="S5.T1.30.30.30.6.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.30.30.30.6.m1.1.1.2.1.cmml" xref="S5.T1.30.30.30.6.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.30.30.30.6.m1.1.1.2.2.cmml" type="float" xref="S5.T1.30.30.30.6.m1.1.1.2.2">54.65</cn></apply><apply id="S5.T1.30.30.30.6.m1.1.1.3.cmml" xref="S5.T1.30.30.30.6.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.30.30.30.6.m1.1.1.3.1.cmml" xref="S5.T1.30.30.30.6.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.30.30.30.6.m1.1.1.3.2.cmml" xref="S5.T1.30.30.30.6.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.30.30.30.6.m1.1.1.3.2.1.cmml" xref="S5.T1.30.30.30.6.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.30.30.30.6.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.30.30.30.6.m1.1.1.3.2.2">1.12</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.30.30.30.6.m1.1c">54.65\%_{\pm 1.12\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.30.30.30.6.m1.1d">54.65 % start_POSTSUBSCRIPT ± 1.12 % end_POSTSUBSCRIPT</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T1.36.36.39.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S5.T1.36.36.39.1.1"><span class="ltx_text" id="S5.T1.36.36.39.1.1.1" style="font-size:90%;">no edit</span></th> <td class="ltx_td ltx_align_center" id="S5.T1.36.36.39.1.2"><span class="ltx_text" id="S5.T1.36.36.39.1.2.1" style="font-size:90%;">35.87%</span></td> <td class="ltx_td ltx_align_center" id="S5.T1.36.36.39.1.3"><span class="ltx_text" id="S5.T1.36.36.39.1.3.1" style="font-size:90%;">29.74%</span></td> <td class="ltx_td" id="S5.T1.36.36.39.1.4"></td> <td class="ltx_td ltx_align_center" id="S5.T1.36.36.39.1.5"><span class="ltx_text" id="S5.T1.36.36.39.1.5.1" style="font-size:90%;">30.58%</span></td> <td class="ltx_td ltx_align_center" id="S5.T1.36.36.39.1.6"><span class="ltx_text" id="S5.T1.36.36.39.1.6.1" style="font-size:90%;">28.21%</span></td> <td class="ltx_td" id="S5.T1.36.36.39.1.7"></td> <td class="ltx_td ltx_align_center" id="S5.T1.36.36.39.1.8"><span class="ltx_text" id="S5.T1.36.36.39.1.8.1" style="font-size:90%;">33.52%</span></td> <td class="ltx_td ltx_align_center" id="S5.T1.36.36.39.1.9"><span class="ltx_text" id="S5.T1.36.36.39.1.9.1" style="font-size:90%;">52.16%</span></td> </tr> <tr class="ltx_tr" id="S5.T1.36.36.36"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb ltx_border_t" id="S5.T1.36.36.36.7"><span class="ltx_text ltx_font_italic" id="S5.T1.36.36.36.7.1" style="font-size:90%;">KGT</span></th> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T1.31.31.31.1"><math alttext="91.77\%_{\pm 1.37\%}" class="ltx_Math" display="inline" id="S5.T1.31.31.31.1.m1.1"><semantics id="S5.T1.31.31.31.1.m1.1a"><msub id="S5.T1.31.31.31.1.m1.1.1" xref="S5.T1.31.31.31.1.m1.1.1.cmml"><mrow id="S5.T1.31.31.31.1.m1.1.1.2" xref="S5.T1.31.31.31.1.m1.1.1.2.cmml"><mn id="S5.T1.31.31.31.1.m1.1.1.2.2" mathsize="90%" xref="S5.T1.31.31.31.1.m1.1.1.2.2.cmml">91.77</mn><mo id="S5.T1.31.31.31.1.m1.1.1.2.1" mathsize="90%" xref="S5.T1.31.31.31.1.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.31.31.31.1.m1.1.1.3" xref="S5.T1.31.31.31.1.m1.1.1.3.cmml"><mo id="S5.T1.31.31.31.1.m1.1.1.3a" mathsize="90%" xref="S5.T1.31.31.31.1.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.31.31.31.1.m1.1.1.3.2" xref="S5.T1.31.31.31.1.m1.1.1.3.2.cmml"><mn id="S5.T1.31.31.31.1.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.31.31.31.1.m1.1.1.3.2.2.cmml">1.37</mn><mo id="S5.T1.31.31.31.1.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.31.31.31.1.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.31.31.31.1.m1.1b"><apply id="S5.T1.31.31.31.1.m1.1.1.cmml" xref="S5.T1.31.31.31.1.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.31.31.31.1.m1.1.1.1.cmml" xref="S5.T1.31.31.31.1.m1.1.1">subscript</csymbol><apply id="S5.T1.31.31.31.1.m1.1.1.2.cmml" xref="S5.T1.31.31.31.1.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.31.31.31.1.m1.1.1.2.1.cmml" xref="S5.T1.31.31.31.1.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.31.31.31.1.m1.1.1.2.2.cmml" type="float" xref="S5.T1.31.31.31.1.m1.1.1.2.2">91.77</cn></apply><apply id="S5.T1.31.31.31.1.m1.1.1.3.cmml" xref="S5.T1.31.31.31.1.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.31.31.31.1.m1.1.1.3.1.cmml" xref="S5.T1.31.31.31.1.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.31.31.31.1.m1.1.1.3.2.cmml" xref="S5.T1.31.31.31.1.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.31.31.31.1.m1.1.1.3.2.1.cmml" xref="S5.T1.31.31.31.1.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.31.31.31.1.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.31.31.31.1.m1.1.1.3.2.2">1.37</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.31.31.31.1.m1.1c">91.77\%_{\pm 1.37\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.31.31.31.1.m1.1d">91.77 % start_POSTSUBSCRIPT ± 1.37 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T1.32.32.32.2"><math alttext="91.75\%_{\pm 1.84\%}" class="ltx_Math" display="inline" id="S5.T1.32.32.32.2.m1.1"><semantics id="S5.T1.32.32.32.2.m1.1a"><msub id="S5.T1.32.32.32.2.m1.1.1" xref="S5.T1.32.32.32.2.m1.1.1.cmml"><mrow id="S5.T1.32.32.32.2.m1.1.1.2" xref="S5.T1.32.32.32.2.m1.1.1.2.cmml"><mn id="S5.T1.32.32.32.2.m1.1.1.2.2" mathsize="90%" xref="S5.T1.32.32.32.2.m1.1.1.2.2.cmml">91.75</mn><mo id="S5.T1.32.32.32.2.m1.1.1.2.1" mathsize="90%" xref="S5.T1.32.32.32.2.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.32.32.32.2.m1.1.1.3" xref="S5.T1.32.32.32.2.m1.1.1.3.cmml"><mo id="S5.T1.32.32.32.2.m1.1.1.3a" mathsize="90%" xref="S5.T1.32.32.32.2.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.32.32.32.2.m1.1.1.3.2" xref="S5.T1.32.32.32.2.m1.1.1.3.2.cmml"><mn id="S5.T1.32.32.32.2.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.32.32.32.2.m1.1.1.3.2.2.cmml">1.84</mn><mo id="S5.T1.32.32.32.2.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.32.32.32.2.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.32.32.32.2.m1.1b"><apply id="S5.T1.32.32.32.2.m1.1.1.cmml" xref="S5.T1.32.32.32.2.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.32.32.32.2.m1.1.1.1.cmml" xref="S5.T1.32.32.32.2.m1.1.1">subscript</csymbol><apply id="S5.T1.32.32.32.2.m1.1.1.2.cmml" xref="S5.T1.32.32.32.2.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.32.32.32.2.m1.1.1.2.1.cmml" xref="S5.T1.32.32.32.2.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.32.32.32.2.m1.1.1.2.2.cmml" type="float" xref="S5.T1.32.32.32.2.m1.1.1.2.2">91.75</cn></apply><apply id="S5.T1.32.32.32.2.m1.1.1.3.cmml" xref="S5.T1.32.32.32.2.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.32.32.32.2.m1.1.1.3.1.cmml" xref="S5.T1.32.32.32.2.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.32.32.32.2.m1.1.1.3.2.cmml" xref="S5.T1.32.32.32.2.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.32.32.32.2.m1.1.1.3.2.1.cmml" xref="S5.T1.32.32.32.2.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.32.32.32.2.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.32.32.32.2.m1.1.1.3.2.2">1.84</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.32.32.32.2.m1.1c">91.75\%_{\pm 1.84\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.32.32.32.2.m1.1d">91.75 % start_POSTSUBSCRIPT ± 1.84 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_border_bb ltx_border_t" id="S5.T1.36.36.36.8"></td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T1.33.33.33.3"><math alttext="91.1\%_{\pm 1.43\%}" class="ltx_Math" display="inline" id="S5.T1.33.33.33.3.m1.1"><semantics id="S5.T1.33.33.33.3.m1.1a"><msub id="S5.T1.33.33.33.3.m1.1.1" xref="S5.T1.33.33.33.3.m1.1.1.cmml"><mrow id="S5.T1.33.33.33.3.m1.1.1.2" xref="S5.T1.33.33.33.3.m1.1.1.2.cmml"><mn id="S5.T1.33.33.33.3.m1.1.1.2.2" mathsize="90%" xref="S5.T1.33.33.33.3.m1.1.1.2.2.cmml">91.1</mn><mo id="S5.T1.33.33.33.3.m1.1.1.2.1" mathsize="90%" xref="S5.T1.33.33.33.3.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.33.33.33.3.m1.1.1.3" xref="S5.T1.33.33.33.3.m1.1.1.3.cmml"><mo id="S5.T1.33.33.33.3.m1.1.1.3a" mathsize="90%" xref="S5.T1.33.33.33.3.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.33.33.33.3.m1.1.1.3.2" xref="S5.T1.33.33.33.3.m1.1.1.3.2.cmml"><mn id="S5.T1.33.33.33.3.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.33.33.33.3.m1.1.1.3.2.2.cmml">1.43</mn><mo id="S5.T1.33.33.33.3.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.33.33.33.3.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.33.33.33.3.m1.1b"><apply id="S5.T1.33.33.33.3.m1.1.1.cmml" xref="S5.T1.33.33.33.3.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.33.33.33.3.m1.1.1.1.cmml" xref="S5.T1.33.33.33.3.m1.1.1">subscript</csymbol><apply id="S5.T1.33.33.33.3.m1.1.1.2.cmml" xref="S5.T1.33.33.33.3.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.33.33.33.3.m1.1.1.2.1.cmml" xref="S5.T1.33.33.33.3.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.33.33.33.3.m1.1.1.2.2.cmml" type="float" xref="S5.T1.33.33.33.3.m1.1.1.2.2">91.1</cn></apply><apply id="S5.T1.33.33.33.3.m1.1.1.3.cmml" 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id="S5.T1.34.34.34.4.m1.1.1" xref="S5.T1.34.34.34.4.m1.1.1.cmml"><mrow id="S5.T1.34.34.34.4.m1.1.1.2" xref="S5.T1.34.34.34.4.m1.1.1.2.cmml"><mn id="S5.T1.34.34.34.4.m1.1.1.2.2" mathsize="90%" xref="S5.T1.34.34.34.4.m1.1.1.2.2.cmml">83.86</mn><mo id="S5.T1.34.34.34.4.m1.1.1.2.1" mathsize="90%" xref="S5.T1.34.34.34.4.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.34.34.34.4.m1.1.1.3" xref="S5.T1.34.34.34.4.m1.1.1.3.cmml"><mo id="S5.T1.34.34.34.4.m1.1.1.3a" mathsize="90%" xref="S5.T1.34.34.34.4.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.34.34.34.4.m1.1.1.3.2" xref="S5.T1.34.34.34.4.m1.1.1.3.2.cmml"><mn id="S5.T1.34.34.34.4.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.34.34.34.4.m1.1.1.3.2.2.cmml">1.03</mn><mo id="S5.T1.34.34.34.4.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.34.34.34.4.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.34.34.34.4.m1.1b"><apply id="S5.T1.34.34.34.4.m1.1.1.cmml" xref="S5.T1.34.34.34.4.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.34.34.34.4.m1.1.1.1.cmml" xref="S5.T1.34.34.34.4.m1.1.1">subscript</csymbol><apply id="S5.T1.34.34.34.4.m1.1.1.2.cmml" xref="S5.T1.34.34.34.4.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.34.34.34.4.m1.1.1.2.1.cmml" xref="S5.T1.34.34.34.4.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.34.34.34.4.m1.1.1.2.2.cmml" type="float" xref="S5.T1.34.34.34.4.m1.1.1.2.2">83.86</cn></apply><apply id="S5.T1.34.34.34.4.m1.1.1.3.cmml" xref="S5.T1.34.34.34.4.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.34.34.34.4.m1.1.1.3.1.cmml" xref="S5.T1.34.34.34.4.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.34.34.34.4.m1.1.1.3.2.cmml" xref="S5.T1.34.34.34.4.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.34.34.34.4.m1.1.1.3.2.1.cmml" xref="S5.T1.34.34.34.4.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.34.34.34.4.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.34.34.34.4.m1.1.1.3.2.2">1.03</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.34.34.34.4.m1.1c">83.86\%_{\pm 1.03\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.34.34.34.4.m1.1d">83.86 % start_POSTSUBSCRIPT ± 1.03 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_border_bb ltx_border_t" id="S5.T1.36.36.36.9"></td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T1.35.35.35.5"><math alttext="94.58\%_{\pm 0.96\%}" class="ltx_Math" display="inline" id="S5.T1.35.35.35.5.m1.1"><semantics id="S5.T1.35.35.35.5.m1.1a"><msub id="S5.T1.35.35.35.5.m1.1.1" xref="S5.T1.35.35.35.5.m1.1.1.cmml"><mrow id="S5.T1.35.35.35.5.m1.1.1.2" xref="S5.T1.35.35.35.5.m1.1.1.2.cmml"><mn id="S5.T1.35.35.35.5.m1.1.1.2.2" mathsize="90%" xref="S5.T1.35.35.35.5.m1.1.1.2.2.cmml">94.58</mn><mo id="S5.T1.35.35.35.5.m1.1.1.2.1" mathsize="90%" xref="S5.T1.35.35.35.5.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T1.35.35.35.5.m1.1.1.3" xref="S5.T1.35.35.35.5.m1.1.1.3.cmml"><mo id="S5.T1.35.35.35.5.m1.1.1.3a" mathsize="90%" xref="S5.T1.35.35.35.5.m1.1.1.3.cmml">±</mo><mrow id="S5.T1.35.35.35.5.m1.1.1.3.2" xref="S5.T1.35.35.35.5.m1.1.1.3.2.cmml"><mn id="S5.T1.35.35.35.5.m1.1.1.3.2.2" mathsize="90%" xref="S5.T1.35.35.35.5.m1.1.1.3.2.2.cmml">0.96</mn><mo id="S5.T1.35.35.35.5.m1.1.1.3.2.1" mathsize="90%" xref="S5.T1.35.35.35.5.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T1.35.35.35.5.m1.1b"><apply id="S5.T1.35.35.35.5.m1.1.1.cmml" xref="S5.T1.35.35.35.5.m1.1.1"><csymbol cd="ambiguous" id="S5.T1.35.35.35.5.m1.1.1.1.cmml" xref="S5.T1.35.35.35.5.m1.1.1">subscript</csymbol><apply id="S5.T1.35.35.35.5.m1.1.1.2.cmml" xref="S5.T1.35.35.35.5.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.35.35.35.5.m1.1.1.2.1.cmml" xref="S5.T1.35.35.35.5.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.35.35.35.5.m1.1.1.2.2.cmml" type="float" xref="S5.T1.35.35.35.5.m1.1.1.2.2">94.58</cn></apply><apply id="S5.T1.35.35.35.5.m1.1.1.3.cmml" xref="S5.T1.35.35.35.5.m1.1.1.3"><csymbol cd="latexml" 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xref="S5.T1.36.36.36.6.m1.1.1">subscript</csymbol><apply id="S5.T1.36.36.36.6.m1.1.1.2.cmml" xref="S5.T1.36.36.36.6.m1.1.1.2"><csymbol cd="latexml" id="S5.T1.36.36.36.6.m1.1.1.2.1.cmml" xref="S5.T1.36.36.36.6.m1.1.1.2.1">percent</csymbol><cn id="S5.T1.36.36.36.6.m1.1.1.2.2.cmml" type="float" xref="S5.T1.36.36.36.6.m1.1.1.2.2">86.89</cn></apply><apply id="S5.T1.36.36.36.6.m1.1.1.3.cmml" xref="S5.T1.36.36.36.6.m1.1.1.3"><csymbol cd="latexml" id="S5.T1.36.36.36.6.m1.1.1.3.1.cmml" xref="S5.T1.36.36.36.6.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T1.36.36.36.6.m1.1.1.3.2.cmml" xref="S5.T1.36.36.36.6.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T1.36.36.36.6.m1.1.1.3.2.1.cmml" xref="S5.T1.36.36.36.6.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T1.36.36.36.6.m1.1.1.3.2.2.cmml" type="float" xref="S5.T1.36.36.36.6.m1.1.1.3.2.2">1.37</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T1.36.36.36.6.m1.1c">86.89\%_{\pm 1.37\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T1.36.36.36.6.m1.1d">86.89 % start_POSTSUBSCRIPT ± 1.37 % end_POSTSUBSCRIPT</annotation></semantics></math></td> </tr> </tbody> </table> </span></div> </figure> <div class="ltx_para" id="S5.SS2.p2"> <p class="ltx_p" id="S5.SS2.p2.1">KGT also shows significant improvement compared with the baselines on <span class="ltx_text ltx_font_italic" id="S5.SS2.p2.1.1">CounterFactExtension</span>, and the results are shown in <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S5.T2" title="In 5.2 Personalization Performance ‣ 5 Experiments ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_tag">Table</span> <span class="ltx_text ltx_ref_tag">2</span></a>. Specifically, KGT improves efficacy by more than 31%, 34%, 51%, 42%, 48%, and 54% on efficacy when adopting Llama3-8B compared with FT, KE, KN, MEND, and no-edit, respectively. For paraphrase score on Llama3-8B, KGT outperforms by more than 27%, 35%, 41%, 36%, 44%, and 42% compared with FT, KE, KN, MEND, and no-edit, respectively.</p> </div> <figure class="ltx_table" id="S5.T2"> <figcaption class="ltx_caption ltx_centering" style="font-size:90%;"><span class="ltx_tag ltx_tag_table">Table 2: </span>Results on <span class="ltx_text ltx_font_italic" id="S5.T2.41.1">CounterFactExtend</span> dataset when the user only provides the answers to the queries as feedback.</figcaption> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S5.T2.36" style="width:433.6pt;height:165.6pt;vertical-align:-1.0pt;"><span class="ltx_transformed_inner" style="transform:translate(-3.7pt,1.4pt) scale(0.983118514620529,0.983118514620529) ;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S5.T2.36.36"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S5.T2.36.36.37.1"> <th class="ltx_td ltx_th ltx_th_row ltx_border_tt" id="S5.T2.36.36.37.1.1"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="2" id="S5.T2.36.36.37.1.2"><span class="ltx_text ltx_font_bold" id="S5.T2.36.36.37.1.2.1" style="font-size:90%;">GPT2</span></th> <th class="ltx_td ltx_th ltx_th_column ltx_border_tt" id="S5.T2.36.36.37.1.3"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="2" id="S5.T2.36.36.37.1.4"><span class="ltx_text ltx_font_bold" id="S5.T2.36.36.37.1.4.1" style="font-size:90%;">Llama2-7B</span></th> <th class="ltx_td ltx_th ltx_th_column ltx_border_tt" id="S5.T2.36.36.37.1.5"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="2" id="S5.T2.36.36.37.1.6"><span class="ltx_text ltx_font_bold" id="S5.T2.36.36.37.1.6.1" style="font-size:90%;">Llama3-8B</span></th> </tr> <tr class="ltx_tr" id="S5.T2.36.36.38.2"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row" id="S5.T2.36.36.38.2.1"><span class="ltx_text ltx_font_bold" id="S5.T2.36.36.38.2.1.1" style="font-size:90%;">Method</span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.36.36.38.2.2"> <table class="ltx_tabular ltx_align_middle" id="S5.T2.36.36.38.2.2.1"> <tr class="ltx_tr" id="S5.T2.36.36.38.2.2.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T2.36.36.38.2.2.1.1.1"><span class="ltx_text" id="S5.T2.36.36.38.2.2.1.1.1.1" style="font-size:90%;">Efficacy</span></td> </tr> </table> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.36.36.38.2.3"> <table class="ltx_tabular ltx_align_middle" id="S5.T2.36.36.38.2.3.1"> <tr class="ltx_tr" id="S5.T2.36.36.38.2.3.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T2.36.36.38.2.3.1.1.1"><span class="ltx_text" id="S5.T2.36.36.38.2.3.1.1.1.1" style="font-size:90%;">Paraphrase</span></td> </tr> </table> </th> <th class="ltx_td ltx_th ltx_th_column" id="S5.T2.36.36.38.2.4"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.36.36.38.2.5"> <table class="ltx_tabular ltx_align_middle" id="S5.T2.36.36.38.2.5.1"> <tr class="ltx_tr" id="S5.T2.36.36.38.2.5.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T2.36.36.38.2.5.1.1.1"><span class="ltx_text" id="S5.T2.36.36.38.2.5.1.1.1.1" style="font-size:90%;">Efficacy</span></td> </tr> </table> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.36.36.38.2.6"> <table class="ltx_tabular ltx_align_middle" id="S5.T2.36.36.38.2.6.1"> <tr class="ltx_tr" id="S5.T2.36.36.38.2.6.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T2.36.36.38.2.6.1.1.1"><span class="ltx_text" id="S5.T2.36.36.38.2.6.1.1.1.1" style="font-size:90%;">Paraphrase</span></td> </tr> </table> </th> <th class="ltx_td ltx_th ltx_th_column" id="S5.T2.36.36.38.2.7"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.36.36.38.2.8"> <table class="ltx_tabular ltx_align_middle" id="S5.T2.36.36.38.2.8.1"> <tr class="ltx_tr" id="S5.T2.36.36.38.2.8.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T2.36.36.38.2.8.1.1.1"><span class="ltx_text" id="S5.T2.36.36.38.2.8.1.1.1.1" style="font-size:90%;">Efficacy</span></td> </tr> </table> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.36.36.38.2.9"> <table class="ltx_tabular ltx_align_middle" id="S5.T2.36.36.38.2.9.1"> <tr class="ltx_tr" id="S5.T2.36.36.38.2.9.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T2.36.36.38.2.9.1.1.1"><span class="ltx_text" id="S5.T2.36.36.38.2.9.1.1.1.1" style="font-size:90%;">Paraphrase</span></td> </tr> </table> </th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S5.T2.6.6.6"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_tt" id="S5.T2.6.6.6.7"><span class="ltx_text" id="S5.T2.6.6.6.7.1" style="font-size:90%;">FT</span></th> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T2.1.1.1.1"><math alttext="58.67\%_{\pm 0.11\%}" class="ltx_Math" display="inline" id="S5.T2.1.1.1.1.m1.1"><semantics id="S5.T2.1.1.1.1.m1.1a"><msub id="S5.T2.1.1.1.1.m1.1.1" xref="S5.T2.1.1.1.1.m1.1.1.cmml"><mrow id="S5.T2.1.1.1.1.m1.1.1.2" xref="S5.T2.1.1.1.1.m1.1.1.2.cmml"><mn id="S5.T2.1.1.1.1.m1.1.1.2.2" mathsize="90%" xref="S5.T2.1.1.1.1.m1.1.1.2.2.cmml">58.67</mn><mo id="S5.T2.1.1.1.1.m1.1.1.2.1" mathsize="90%" xref="S5.T2.1.1.1.1.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.1.1.1.1.m1.1.1.3" xref="S5.T2.1.1.1.1.m1.1.1.3.cmml"><mo id="S5.T2.1.1.1.1.m1.1.1.3a" mathsize="90%" xref="S5.T2.1.1.1.1.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.1.1.1.1.m1.1.1.3.2" xref="S5.T2.1.1.1.1.m1.1.1.3.2.cmml"><mn id="S5.T2.1.1.1.1.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.1.1.1.1.m1.1.1.3.2.2.cmml">0.11</mn><mo id="S5.T2.1.1.1.1.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.1.1.1.1.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.1.1.1.1.m1.1b"><apply id="S5.T2.1.1.1.1.m1.1.1.cmml" xref="S5.T2.1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.1.1.1.1.m1.1.1.1.cmml" xref="S5.T2.1.1.1.1.m1.1.1">subscript</csymbol><apply id="S5.T2.1.1.1.1.m1.1.1.2.cmml" xref="S5.T2.1.1.1.1.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.1.1.1.1.m1.1.1.2.1.cmml" xref="S5.T2.1.1.1.1.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.1.1.1.1.m1.1.1.2.2.cmml" type="float" xref="S5.T2.1.1.1.1.m1.1.1.2.2">58.67</cn></apply><apply id="S5.T2.1.1.1.1.m1.1.1.3.cmml" xref="S5.T2.1.1.1.1.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.1.1.1.1.m1.1.1.3.1.cmml" xref="S5.T2.1.1.1.1.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.1.1.1.1.m1.1.1.3.2.cmml" xref="S5.T2.1.1.1.1.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.1.1.1.1.m1.1.1.3.2.1.cmml" xref="S5.T2.1.1.1.1.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.1.1.1.1.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.1.1.1.1.m1.1.1.3.2.2">0.11</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.1.1.1.1.m1.1c">58.67\%_{\pm 0.11\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.1.1.1.1.m1.1d">58.67 % start_POSTSUBSCRIPT ± 0.11 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T2.2.2.2.2"><math alttext="53.71\%_{\pm 0.06\%}" class="ltx_Math" display="inline" id="S5.T2.2.2.2.2.m1.1"><semantics id="S5.T2.2.2.2.2.m1.1a"><msub id="S5.T2.2.2.2.2.m1.1.1" xref="S5.T2.2.2.2.2.m1.1.1.cmml"><mrow id="S5.T2.2.2.2.2.m1.1.1.2" xref="S5.T2.2.2.2.2.m1.1.1.2.cmml"><mn id="S5.T2.2.2.2.2.m1.1.1.2.2" mathsize="90%" xref="S5.T2.2.2.2.2.m1.1.1.2.2.cmml">53.71</mn><mo id="S5.T2.2.2.2.2.m1.1.1.2.1" mathsize="90%" xref="S5.T2.2.2.2.2.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.2.2.2.2.m1.1.1.3" xref="S5.T2.2.2.2.2.m1.1.1.3.cmml"><mo id="S5.T2.2.2.2.2.m1.1.1.3a" mathsize="90%" xref="S5.T2.2.2.2.2.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.2.2.2.2.m1.1.1.3.2" xref="S5.T2.2.2.2.2.m1.1.1.3.2.cmml"><mn id="S5.T2.2.2.2.2.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.2.2.2.2.m1.1.1.3.2.2.cmml">0.06</mn><mo id="S5.T2.2.2.2.2.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.2.2.2.2.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.2.2.2.2.m1.1b"><apply id="S5.T2.2.2.2.2.m1.1.1.cmml" xref="S5.T2.2.2.2.2.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.2.2.2.2.m1.1.1.1.cmml" xref="S5.T2.2.2.2.2.m1.1.1">subscript</csymbol><apply id="S5.T2.2.2.2.2.m1.1.1.2.cmml" xref="S5.T2.2.2.2.2.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.2.2.2.2.m1.1.1.2.1.cmml" xref="S5.T2.2.2.2.2.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.2.2.2.2.m1.1.1.2.2.cmml" type="float" xref="S5.T2.2.2.2.2.m1.1.1.2.2">53.71</cn></apply><apply id="S5.T2.2.2.2.2.m1.1.1.3.cmml" xref="S5.T2.2.2.2.2.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.2.2.2.2.m1.1.1.3.1.cmml" xref="S5.T2.2.2.2.2.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.2.2.2.2.m1.1.1.3.2.cmml" xref="S5.T2.2.2.2.2.m1.1.1.3.2"><csymbol 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id="S5.T2.3.3.3.3.m1.1.1.2.1" mathsize="90%" xref="S5.T2.3.3.3.3.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.3.3.3.3.m1.1.1.3" xref="S5.T2.3.3.3.3.m1.1.1.3.cmml"><mo id="S5.T2.3.3.3.3.m1.1.1.3a" mathsize="90%" xref="S5.T2.3.3.3.3.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.3.3.3.3.m1.1.1.3.2" xref="S5.T2.3.3.3.3.m1.1.1.3.2.cmml"><mn id="S5.T2.3.3.3.3.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.3.3.3.3.m1.1.1.3.2.2.cmml">0.15</mn><mo id="S5.T2.3.3.3.3.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.3.3.3.3.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.3.3.3.3.m1.1b"><apply id="S5.T2.3.3.3.3.m1.1.1.cmml" xref="S5.T2.3.3.3.3.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.3.3.3.3.m1.1.1.1.cmml" xref="S5.T2.3.3.3.3.m1.1.1">subscript</csymbol><apply id="S5.T2.3.3.3.3.m1.1.1.2.cmml" xref="S5.T2.3.3.3.3.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.3.3.3.3.m1.1.1.2.1.cmml" xref="S5.T2.3.3.3.3.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.3.3.3.3.m1.1.1.2.2.cmml" type="float" xref="S5.T2.3.3.3.3.m1.1.1.2.2">59.70</cn></apply><apply id="S5.T2.3.3.3.3.m1.1.1.3.cmml" xref="S5.T2.3.3.3.3.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.3.3.3.3.m1.1.1.3.1.cmml" xref="S5.T2.3.3.3.3.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.3.3.3.3.m1.1.1.3.2.cmml" xref="S5.T2.3.3.3.3.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.3.3.3.3.m1.1.1.3.2.1.cmml" xref="S5.T2.3.3.3.3.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.3.3.3.3.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.3.3.3.3.m1.1.1.3.2.2">0.15</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.3.3.3.3.m1.1c">59.70\%_{\pm 0.15\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.3.3.3.3.m1.1d">59.70 % start_POSTSUBSCRIPT ± 0.15 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T2.4.4.4.4"><math alttext="63.09\%_{\pm 0.09\%}" class="ltx_Math" display="inline" id="S5.T2.4.4.4.4.m1.1"><semantics id="S5.T2.4.4.4.4.m1.1a"><msub id="S5.T2.4.4.4.4.m1.1.1" xref="S5.T2.4.4.4.4.m1.1.1.cmml"><mrow id="S5.T2.4.4.4.4.m1.1.1.2" xref="S5.T2.4.4.4.4.m1.1.1.2.cmml"><mn id="S5.T2.4.4.4.4.m1.1.1.2.2" mathsize="90%" xref="S5.T2.4.4.4.4.m1.1.1.2.2.cmml">63.09</mn><mo id="S5.T2.4.4.4.4.m1.1.1.2.1" mathsize="90%" xref="S5.T2.4.4.4.4.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.4.4.4.4.m1.1.1.3" xref="S5.T2.4.4.4.4.m1.1.1.3.cmml"><mo id="S5.T2.4.4.4.4.m1.1.1.3a" mathsize="90%" xref="S5.T2.4.4.4.4.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.4.4.4.4.m1.1.1.3.2" xref="S5.T2.4.4.4.4.m1.1.1.3.2.cmml"><mn id="S5.T2.4.4.4.4.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.4.4.4.4.m1.1.1.3.2.2.cmml">0.09</mn><mo id="S5.T2.4.4.4.4.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.4.4.4.4.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.4.4.4.4.m1.1b"><apply id="S5.T2.4.4.4.4.m1.1.1.cmml" xref="S5.T2.4.4.4.4.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.4.4.4.4.m1.1.1.1.cmml" xref="S5.T2.4.4.4.4.m1.1.1">subscript</csymbol><apply id="S5.T2.4.4.4.4.m1.1.1.2.cmml" xref="S5.T2.4.4.4.4.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.4.4.4.4.m1.1.1.2.1.cmml" xref="S5.T2.4.4.4.4.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.4.4.4.4.m1.1.1.2.2.cmml" type="float" xref="S5.T2.4.4.4.4.m1.1.1.2.2">63.09</cn></apply><apply id="S5.T2.4.4.4.4.m1.1.1.3.cmml" xref="S5.T2.4.4.4.4.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.4.4.4.4.m1.1.1.3.1.cmml" xref="S5.T2.4.4.4.4.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.4.4.4.4.m1.1.1.3.2.cmml" xref="S5.T2.4.4.4.4.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.4.4.4.4.m1.1.1.3.2.1.cmml" xref="S5.T2.4.4.4.4.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.4.4.4.4.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.4.4.4.4.m1.1.1.3.2.2">0.09</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.4.4.4.4.m1.1c">63.09\%_{\pm 0.09\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.4.4.4.4.m1.1d">63.09 % start_POSTSUBSCRIPT ± 0.09 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_border_tt" id="S5.T2.6.6.6.9"></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T2.5.5.5.5"><math alttext="62.29\%_{\pm 0.25\%}" class="ltx_Math" display="inline" id="S5.T2.5.5.5.5.m1.1"><semantics id="S5.T2.5.5.5.5.m1.1a"><msub id="S5.T2.5.5.5.5.m1.1.1" xref="S5.T2.5.5.5.5.m1.1.1.cmml"><mrow id="S5.T2.5.5.5.5.m1.1.1.2" xref="S5.T2.5.5.5.5.m1.1.1.2.cmml"><mn id="S5.T2.5.5.5.5.m1.1.1.2.2" mathsize="90%" xref="S5.T2.5.5.5.5.m1.1.1.2.2.cmml">62.29</mn><mo id="S5.T2.5.5.5.5.m1.1.1.2.1" mathsize="90%" xref="S5.T2.5.5.5.5.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.5.5.5.5.m1.1.1.3" xref="S5.T2.5.5.5.5.m1.1.1.3.cmml"><mo id="S5.T2.5.5.5.5.m1.1.1.3a" mathsize="90%" xref="S5.T2.5.5.5.5.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.5.5.5.5.m1.1.1.3.2" xref="S5.T2.5.5.5.5.m1.1.1.3.2.cmml"><mn id="S5.T2.5.5.5.5.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.5.5.5.5.m1.1.1.3.2.2.cmml">0.25</mn><mo id="S5.T2.5.5.5.5.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.5.5.5.5.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.5.5.5.5.m1.1b"><apply id="S5.T2.5.5.5.5.m1.1.1.cmml" xref="S5.T2.5.5.5.5.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.5.5.5.5.m1.1.1.1.cmml" xref="S5.T2.5.5.5.5.m1.1.1">subscript</csymbol><apply id="S5.T2.5.5.5.5.m1.1.1.2.cmml" xref="S5.T2.5.5.5.5.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.5.5.5.5.m1.1.1.2.1.cmml" xref="S5.T2.5.5.5.5.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.5.5.5.5.m1.1.1.2.2.cmml" type="float" xref="S5.T2.5.5.5.5.m1.1.1.2.2">62.29</cn></apply><apply id="S5.T2.5.5.5.5.m1.1.1.3.cmml" xref="S5.T2.5.5.5.5.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.5.5.5.5.m1.1.1.3.1.cmml" xref="S5.T2.5.5.5.5.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.5.5.5.5.m1.1.1.3.2.cmml" xref="S5.T2.5.5.5.5.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.5.5.5.5.m1.1.1.3.2.1.cmml" xref="S5.T2.5.5.5.5.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.5.5.5.5.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.5.5.5.5.m1.1.1.3.2.2">0.25</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.5.5.5.5.m1.1c">62.29\%_{\pm 0.25\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.5.5.5.5.m1.1d">62.29 % start_POSTSUBSCRIPT ± 0.25 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T2.6.6.6.6"><math alttext="61.97\%_{\pm 0.10\%}" class="ltx_Math" display="inline" id="S5.T2.6.6.6.6.m1.1"><semantics id="S5.T2.6.6.6.6.m1.1a"><msub id="S5.T2.6.6.6.6.m1.1.1" xref="S5.T2.6.6.6.6.m1.1.1.cmml"><mrow id="S5.T2.6.6.6.6.m1.1.1.2" xref="S5.T2.6.6.6.6.m1.1.1.2.cmml"><mn id="S5.T2.6.6.6.6.m1.1.1.2.2" mathsize="90%" xref="S5.T2.6.6.6.6.m1.1.1.2.2.cmml">61.97</mn><mo id="S5.T2.6.6.6.6.m1.1.1.2.1" mathsize="90%" xref="S5.T2.6.6.6.6.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.6.6.6.6.m1.1.1.3" xref="S5.T2.6.6.6.6.m1.1.1.3.cmml"><mo id="S5.T2.6.6.6.6.m1.1.1.3a" mathsize="90%" xref="S5.T2.6.6.6.6.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.6.6.6.6.m1.1.1.3.2" xref="S5.T2.6.6.6.6.m1.1.1.3.2.cmml"><mn id="S5.T2.6.6.6.6.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.6.6.6.6.m1.1.1.3.2.2.cmml">0.10</mn><mo id="S5.T2.6.6.6.6.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.6.6.6.6.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.6.6.6.6.m1.1b"><apply id="S5.T2.6.6.6.6.m1.1.1.cmml" xref="S5.T2.6.6.6.6.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.6.6.6.6.m1.1.1.1.cmml" xref="S5.T2.6.6.6.6.m1.1.1">subscript</csymbol><apply id="S5.T2.6.6.6.6.m1.1.1.2.cmml" xref="S5.T2.6.6.6.6.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.6.6.6.6.m1.1.1.2.1.cmml" xref="S5.T2.6.6.6.6.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.6.6.6.6.m1.1.1.2.2.cmml" type="float" xref="S5.T2.6.6.6.6.m1.1.1.2.2">61.97</cn></apply><apply id="S5.T2.6.6.6.6.m1.1.1.3.cmml" xref="S5.T2.6.6.6.6.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.6.6.6.6.m1.1.1.3.1.cmml" xref="S5.T2.6.6.6.6.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.6.6.6.6.m1.1.1.3.2.cmml" xref="S5.T2.6.6.6.6.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.6.6.6.6.m1.1.1.3.2.1.cmml" xref="S5.T2.6.6.6.6.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.6.6.6.6.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.6.6.6.6.m1.1.1.3.2.2">0.10</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.6.6.6.6.m1.1c">61.97\%_{\pm 0.10\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.6.6.6.6.m1.1d">61.97 % start_POSTSUBSCRIPT ± 0.10 % end_POSTSUBSCRIPT</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T2.12.12.12"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S5.T2.12.12.12.7"><span class="ltx_text" id="S5.T2.12.12.12.7.1" style="font-size:90%;">ROME</span></th> <td class="ltx_td ltx_align_center" id="S5.T2.7.7.7.1"><math alttext="57.44\%_{\pm 1.75\%}" class="ltx_Math" display="inline" id="S5.T2.7.7.7.1.m1.1"><semantics id="S5.T2.7.7.7.1.m1.1a"><msub id="S5.T2.7.7.7.1.m1.1.1" xref="S5.T2.7.7.7.1.m1.1.1.cmml"><mrow id="S5.T2.7.7.7.1.m1.1.1.2" xref="S5.T2.7.7.7.1.m1.1.1.2.cmml"><mn id="S5.T2.7.7.7.1.m1.1.1.2.2" mathsize="90%" xref="S5.T2.7.7.7.1.m1.1.1.2.2.cmml">57.44</mn><mo id="S5.T2.7.7.7.1.m1.1.1.2.1" mathsize="90%" xref="S5.T2.7.7.7.1.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.7.7.7.1.m1.1.1.3" xref="S5.T2.7.7.7.1.m1.1.1.3.cmml"><mo id="S5.T2.7.7.7.1.m1.1.1.3a" mathsize="90%" xref="S5.T2.7.7.7.1.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.7.7.7.1.m1.1.1.3.2" xref="S5.T2.7.7.7.1.m1.1.1.3.2.cmml"><mn id="S5.T2.7.7.7.1.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.7.7.7.1.m1.1.1.3.2.2.cmml">1.75</mn><mo id="S5.T2.7.7.7.1.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.7.7.7.1.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.7.7.7.1.m1.1b"><apply id="S5.T2.7.7.7.1.m1.1.1.cmml" xref="S5.T2.7.7.7.1.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.7.7.7.1.m1.1.1.1.cmml" xref="S5.T2.7.7.7.1.m1.1.1">subscript</csymbol><apply id="S5.T2.7.7.7.1.m1.1.1.2.cmml" xref="S5.T2.7.7.7.1.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.7.7.7.1.m1.1.1.2.1.cmml" xref="S5.T2.7.7.7.1.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.7.7.7.1.m1.1.1.2.2.cmml" type="float" xref="S5.T2.7.7.7.1.m1.1.1.2.2">57.44</cn></apply><apply id="S5.T2.7.7.7.1.m1.1.1.3.cmml" xref="S5.T2.7.7.7.1.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.7.7.7.1.m1.1.1.3.1.cmml" xref="S5.T2.7.7.7.1.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.7.7.7.1.m1.1.1.3.2.cmml" xref="S5.T2.7.7.7.1.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.7.7.7.1.m1.1.1.3.2.1.cmml" xref="S5.T2.7.7.7.1.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.7.7.7.1.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.7.7.7.1.m1.1.1.3.2.2">1.75</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.7.7.7.1.m1.1c">57.44\%_{\pm 1.75\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.7.7.7.1.m1.1d">57.44 % start_POSTSUBSCRIPT ± 1.75 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T2.8.8.8.2"><math alttext="58.45\%_{\pm 1.00\%}" class="ltx_Math" display="inline" id="S5.T2.8.8.8.2.m1.1"><semantics id="S5.T2.8.8.8.2.m1.1a"><msub id="S5.T2.8.8.8.2.m1.1.1" xref="S5.T2.8.8.8.2.m1.1.1.cmml"><mrow id="S5.T2.8.8.8.2.m1.1.1.2" xref="S5.T2.8.8.8.2.m1.1.1.2.cmml"><mn id="S5.T2.8.8.8.2.m1.1.1.2.2" mathsize="90%" xref="S5.T2.8.8.8.2.m1.1.1.2.2.cmml">58.45</mn><mo id="S5.T2.8.8.8.2.m1.1.1.2.1" mathsize="90%" xref="S5.T2.8.8.8.2.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.8.8.8.2.m1.1.1.3" xref="S5.T2.8.8.8.2.m1.1.1.3.cmml"><mo id="S5.T2.8.8.8.2.m1.1.1.3a" mathsize="90%" xref="S5.T2.8.8.8.2.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.8.8.8.2.m1.1.1.3.2" xref="S5.T2.8.8.8.2.m1.1.1.3.2.cmml"><mn id="S5.T2.8.8.8.2.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.8.8.8.2.m1.1.1.3.2.2.cmml">1.00</mn><mo id="S5.T2.8.8.8.2.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.8.8.8.2.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.8.8.8.2.m1.1b"><apply id="S5.T2.8.8.8.2.m1.1.1.cmml" xref="S5.T2.8.8.8.2.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.8.8.8.2.m1.1.1.1.cmml" xref="S5.T2.8.8.8.2.m1.1.1">subscript</csymbol><apply id="S5.T2.8.8.8.2.m1.1.1.2.cmml" xref="S5.T2.8.8.8.2.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.8.8.8.2.m1.1.1.2.1.cmml" xref="S5.T2.8.8.8.2.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.8.8.8.2.m1.1.1.2.2.cmml" type="float" xref="S5.T2.8.8.8.2.m1.1.1.2.2">58.45</cn></apply><apply id="S5.T2.8.8.8.2.m1.1.1.3.cmml" xref="S5.T2.8.8.8.2.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.8.8.8.2.m1.1.1.3.1.cmml" xref="S5.T2.8.8.8.2.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.8.8.8.2.m1.1.1.3.2.cmml" xref="S5.T2.8.8.8.2.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.8.8.8.2.m1.1.1.3.2.1.cmml" xref="S5.T2.8.8.8.2.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.8.8.8.2.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.8.8.8.2.m1.1.1.3.2.2">1.00</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.8.8.8.2.m1.1c">58.45\%_{\pm 1.00\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.8.8.8.2.m1.1d">58.45 % start_POSTSUBSCRIPT ± 1.00 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td" id="S5.T2.12.12.12.8"></td> <td class="ltx_td ltx_align_center" id="S5.T2.9.9.9.3"><math alttext="47.33\%_{\pm 1.60\%}" class="ltx_Math" display="inline" id="S5.T2.9.9.9.3.m1.1"><semantics id="S5.T2.9.9.9.3.m1.1a"><msub id="S5.T2.9.9.9.3.m1.1.1" xref="S5.T2.9.9.9.3.m1.1.1.cmml"><mrow id="S5.T2.9.9.9.3.m1.1.1.2" xref="S5.T2.9.9.9.3.m1.1.1.2.cmml"><mn id="S5.T2.9.9.9.3.m1.1.1.2.2" mathsize="90%" xref="S5.T2.9.9.9.3.m1.1.1.2.2.cmml">47.33</mn><mo id="S5.T2.9.9.9.3.m1.1.1.2.1" mathsize="90%" xref="S5.T2.9.9.9.3.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.9.9.9.3.m1.1.1.3" xref="S5.T2.9.9.9.3.m1.1.1.3.cmml"><mo id="S5.T2.9.9.9.3.m1.1.1.3a" mathsize="90%" xref="S5.T2.9.9.9.3.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.9.9.9.3.m1.1.1.3.2" xref="S5.T2.9.9.9.3.m1.1.1.3.2.cmml"><mn id="S5.T2.9.9.9.3.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.9.9.9.3.m1.1.1.3.2.2.cmml">1.60</mn><mo id="S5.T2.9.9.9.3.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.9.9.9.3.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.9.9.9.3.m1.1b"><apply id="S5.T2.9.9.9.3.m1.1.1.cmml" xref="S5.T2.9.9.9.3.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.9.9.9.3.m1.1.1.1.cmml" xref="S5.T2.9.9.9.3.m1.1.1">subscript</csymbol><apply id="S5.T2.9.9.9.3.m1.1.1.2.cmml" xref="S5.T2.9.9.9.3.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.9.9.9.3.m1.1.1.2.1.cmml" xref="S5.T2.9.9.9.3.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.9.9.9.3.m1.1.1.2.2.cmml" type="float" xref="S5.T2.9.9.9.3.m1.1.1.2.2">47.33</cn></apply><apply id="S5.T2.9.9.9.3.m1.1.1.3.cmml" xref="S5.T2.9.9.9.3.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.9.9.9.3.m1.1.1.3.1.cmml" xref="S5.T2.9.9.9.3.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.9.9.9.3.m1.1.1.3.2.cmml" xref="S5.T2.9.9.9.3.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.9.9.9.3.m1.1.1.3.2.1.cmml" xref="S5.T2.9.9.9.3.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.9.9.9.3.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.9.9.9.3.m1.1.1.3.2.2">1.60</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.9.9.9.3.m1.1c">47.33\%_{\pm 1.60\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.9.9.9.3.m1.1d">47.33 % start_POSTSUBSCRIPT ± 1.60 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T2.10.10.10.4"><math alttext="48.36\%_{\pm 0.71\%}" class="ltx_Math" display="inline" id="S5.T2.10.10.10.4.m1.1"><semantics id="S5.T2.10.10.10.4.m1.1a"><msub id="S5.T2.10.10.10.4.m1.1.1" xref="S5.T2.10.10.10.4.m1.1.1.cmml"><mrow id="S5.T2.10.10.10.4.m1.1.1.2" xref="S5.T2.10.10.10.4.m1.1.1.2.cmml"><mn id="S5.T2.10.10.10.4.m1.1.1.2.2" mathsize="90%" xref="S5.T2.10.10.10.4.m1.1.1.2.2.cmml">48.36</mn><mo id="S5.T2.10.10.10.4.m1.1.1.2.1" mathsize="90%" xref="S5.T2.10.10.10.4.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.10.10.10.4.m1.1.1.3" xref="S5.T2.10.10.10.4.m1.1.1.3.cmml"><mo id="S5.T2.10.10.10.4.m1.1.1.3a" mathsize="90%" xref="S5.T2.10.10.10.4.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.10.10.10.4.m1.1.1.3.2" xref="S5.T2.10.10.10.4.m1.1.1.3.2.cmml"><mn id="S5.T2.10.10.10.4.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.10.10.10.4.m1.1.1.3.2.2.cmml">0.71</mn><mo id="S5.T2.10.10.10.4.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.10.10.10.4.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.10.10.10.4.m1.1b"><apply id="S5.T2.10.10.10.4.m1.1.1.cmml" xref="S5.T2.10.10.10.4.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.10.10.10.4.m1.1.1.1.cmml" xref="S5.T2.10.10.10.4.m1.1.1">subscript</csymbol><apply id="S5.T2.10.10.10.4.m1.1.1.2.cmml" xref="S5.T2.10.10.10.4.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.10.10.10.4.m1.1.1.2.1.cmml" xref="S5.T2.10.10.10.4.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.10.10.10.4.m1.1.1.2.2.cmml" type="float" xref="S5.T2.10.10.10.4.m1.1.1.2.2">48.36</cn></apply><apply id="S5.T2.10.10.10.4.m1.1.1.3.cmml" xref="S5.T2.10.10.10.4.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.10.10.10.4.m1.1.1.3.1.cmml" xref="S5.T2.10.10.10.4.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.10.10.10.4.m1.1.1.3.2.cmml" xref="S5.T2.10.10.10.4.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.10.10.10.4.m1.1.1.3.2.1.cmml" xref="S5.T2.10.10.10.4.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.10.10.10.4.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.10.10.10.4.m1.1.1.3.2.2">0.71</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.10.10.10.4.m1.1c">48.36\%_{\pm 0.71\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.10.10.10.4.m1.1d">48.36 % start_POSTSUBSCRIPT ± 0.71 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td" id="S5.T2.12.12.12.9"></td> <td class="ltx_td ltx_align_center" id="S5.T2.11.11.11.5"><math alttext="59.28\%_{\pm 1.79\%}" class="ltx_Math" display="inline" id="S5.T2.11.11.11.5.m1.1"><semantics id="S5.T2.11.11.11.5.m1.1a"><msub id="S5.T2.11.11.11.5.m1.1.1" xref="S5.T2.11.11.11.5.m1.1.1.cmml"><mrow id="S5.T2.11.11.11.5.m1.1.1.2" xref="S5.T2.11.11.11.5.m1.1.1.2.cmml"><mn id="S5.T2.11.11.11.5.m1.1.1.2.2" mathsize="90%" xref="S5.T2.11.11.11.5.m1.1.1.2.2.cmml">59.28</mn><mo id="S5.T2.11.11.11.5.m1.1.1.2.1" mathsize="90%" xref="S5.T2.11.11.11.5.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.11.11.11.5.m1.1.1.3" xref="S5.T2.11.11.11.5.m1.1.1.3.cmml"><mo id="S5.T2.11.11.11.5.m1.1.1.3a" mathsize="90%" xref="S5.T2.11.11.11.5.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.11.11.11.5.m1.1.1.3.2" xref="S5.T2.11.11.11.5.m1.1.1.3.2.cmml"><mn id="S5.T2.11.11.11.5.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.11.11.11.5.m1.1.1.3.2.2.cmml">1.79</mn><mo id="S5.T2.11.11.11.5.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.11.11.11.5.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.11.11.11.5.m1.1b"><apply id="S5.T2.11.11.11.5.m1.1.1.cmml" xref="S5.T2.11.11.11.5.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.11.11.11.5.m1.1.1.1.cmml" xref="S5.T2.11.11.11.5.m1.1.1">subscript</csymbol><apply id="S5.T2.11.11.11.5.m1.1.1.2.cmml" xref="S5.T2.11.11.11.5.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.11.11.11.5.m1.1.1.2.1.cmml" xref="S5.T2.11.11.11.5.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.11.11.11.5.m1.1.1.2.2.cmml" type="float" xref="S5.T2.11.11.11.5.m1.1.1.2.2">59.28</cn></apply><apply id="S5.T2.11.11.11.5.m1.1.1.3.cmml" xref="S5.T2.11.11.11.5.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.11.11.11.5.m1.1.1.3.1.cmml" xref="S5.T2.11.11.11.5.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.11.11.11.5.m1.1.1.3.2.cmml" xref="S5.T2.11.11.11.5.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.11.11.11.5.m1.1.1.3.2.1.cmml" xref="S5.T2.11.11.11.5.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.11.11.11.5.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.11.11.11.5.m1.1.1.3.2.2">1.79</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.11.11.11.5.m1.1c">59.28\%_{\pm 1.79\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.11.11.11.5.m1.1d">59.28 % start_POSTSUBSCRIPT ± 1.79 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T2.12.12.12.6"><math alttext="53.77\%_{\pm 1.37\%}" class="ltx_Math" display="inline" id="S5.T2.12.12.12.6.m1.1"><semantics id="S5.T2.12.12.12.6.m1.1a"><msub id="S5.T2.12.12.12.6.m1.1.1" xref="S5.T2.12.12.12.6.m1.1.1.cmml"><mrow id="S5.T2.12.12.12.6.m1.1.1.2" xref="S5.T2.12.12.12.6.m1.1.1.2.cmml"><mn id="S5.T2.12.12.12.6.m1.1.1.2.2" mathsize="90%" xref="S5.T2.12.12.12.6.m1.1.1.2.2.cmml">53.77</mn><mo id="S5.T2.12.12.12.6.m1.1.1.2.1" mathsize="90%" xref="S5.T2.12.12.12.6.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.12.12.12.6.m1.1.1.3" xref="S5.T2.12.12.12.6.m1.1.1.3.cmml"><mo id="S5.T2.12.12.12.6.m1.1.1.3a" mathsize="90%" xref="S5.T2.12.12.12.6.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.12.12.12.6.m1.1.1.3.2" xref="S5.T2.12.12.12.6.m1.1.1.3.2.cmml"><mn id="S5.T2.12.12.12.6.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.12.12.12.6.m1.1.1.3.2.2.cmml">1.37</mn><mo id="S5.T2.12.12.12.6.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.12.12.12.6.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.12.12.12.6.m1.1b"><apply id="S5.T2.12.12.12.6.m1.1.1.cmml" xref="S5.T2.12.12.12.6.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.12.12.12.6.m1.1.1.1.cmml" xref="S5.T2.12.12.12.6.m1.1.1">subscript</csymbol><apply id="S5.T2.12.12.12.6.m1.1.1.2.cmml" xref="S5.T2.12.12.12.6.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.12.12.12.6.m1.1.1.2.1.cmml" xref="S5.T2.12.12.12.6.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.12.12.12.6.m1.1.1.2.2.cmml" type="float" xref="S5.T2.12.12.12.6.m1.1.1.2.2">53.77</cn></apply><apply id="S5.T2.12.12.12.6.m1.1.1.3.cmml" xref="S5.T2.12.12.12.6.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.12.12.12.6.m1.1.1.3.1.cmml" xref="S5.T2.12.12.12.6.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.12.12.12.6.m1.1.1.3.2.cmml" xref="S5.T2.12.12.12.6.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.12.12.12.6.m1.1.1.3.2.1.cmml" xref="S5.T2.12.12.12.6.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.12.12.12.6.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.12.12.12.6.m1.1.1.3.2.2">1.37</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.12.12.12.6.m1.1c">53.77\%_{\pm 1.37\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.12.12.12.6.m1.1d">53.77 % start_POSTSUBSCRIPT ± 1.37 % end_POSTSUBSCRIPT</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T2.18.18.18"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S5.T2.18.18.18.7"><span class="ltx_text" id="S5.T2.18.18.18.7.1" style="font-size:90%;">KE</span></th> <td class="ltx_td ltx_align_center" id="S5.T2.13.13.13.1"><math alttext="52.49\%_{\pm 0.28\%}" class="ltx_Math" display="inline" id="S5.T2.13.13.13.1.m1.1"><semantics id="S5.T2.13.13.13.1.m1.1a"><msub id="S5.T2.13.13.13.1.m1.1.1" xref="S5.T2.13.13.13.1.m1.1.1.cmml"><mrow id="S5.T2.13.13.13.1.m1.1.1.2" xref="S5.T2.13.13.13.1.m1.1.1.2.cmml"><mn id="S5.T2.13.13.13.1.m1.1.1.2.2" mathsize="90%" xref="S5.T2.13.13.13.1.m1.1.1.2.2.cmml">52.49</mn><mo id="S5.T2.13.13.13.1.m1.1.1.2.1" mathsize="90%" xref="S5.T2.13.13.13.1.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.13.13.13.1.m1.1.1.3" xref="S5.T2.13.13.13.1.m1.1.1.3.cmml"><mo id="S5.T2.13.13.13.1.m1.1.1.3a" mathsize="90%" xref="S5.T2.13.13.13.1.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.13.13.13.1.m1.1.1.3.2" xref="S5.T2.13.13.13.1.m1.1.1.3.2.cmml"><mn id="S5.T2.13.13.13.1.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.13.13.13.1.m1.1.1.3.2.2.cmml">0.28</mn><mo id="S5.T2.13.13.13.1.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.13.13.13.1.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.13.13.13.1.m1.1b"><apply id="S5.T2.13.13.13.1.m1.1.1.cmml" xref="S5.T2.13.13.13.1.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.13.13.13.1.m1.1.1.1.cmml" xref="S5.T2.13.13.13.1.m1.1.1">subscript</csymbol><apply id="S5.T2.13.13.13.1.m1.1.1.2.cmml" xref="S5.T2.13.13.13.1.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.13.13.13.1.m1.1.1.2.1.cmml" xref="S5.T2.13.13.13.1.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.13.13.13.1.m1.1.1.2.2.cmml" type="float" xref="S5.T2.13.13.13.1.m1.1.1.2.2">52.49</cn></apply><apply id="S5.T2.13.13.13.1.m1.1.1.3.cmml" xref="S5.T2.13.13.13.1.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.13.13.13.1.m1.1.1.3.1.cmml" xref="S5.T2.13.13.13.1.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.13.13.13.1.m1.1.1.3.2.cmml" xref="S5.T2.13.13.13.1.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.13.13.13.1.m1.1.1.3.2.1.cmml" xref="S5.T2.13.13.13.1.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.13.13.13.1.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.13.13.13.1.m1.1.1.3.2.2">0.28</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.13.13.13.1.m1.1c">52.49\%_{\pm 0.28\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.13.13.13.1.m1.1d">52.49 % start_POSTSUBSCRIPT ± 0.28 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T2.14.14.14.2"><math alttext="52.55\%_{\pm 0.44\%}" class="ltx_Math" display="inline" id="S5.T2.14.14.14.2.m1.1"><semantics id="S5.T2.14.14.14.2.m1.1a"><msub id="S5.T2.14.14.14.2.m1.1.1" xref="S5.T2.14.14.14.2.m1.1.1.cmml"><mrow id="S5.T2.14.14.14.2.m1.1.1.2" xref="S5.T2.14.14.14.2.m1.1.1.2.cmml"><mn id="S5.T2.14.14.14.2.m1.1.1.2.2" mathsize="90%" xref="S5.T2.14.14.14.2.m1.1.1.2.2.cmml">52.55</mn><mo id="S5.T2.14.14.14.2.m1.1.1.2.1" mathsize="90%" xref="S5.T2.14.14.14.2.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.14.14.14.2.m1.1.1.3" xref="S5.T2.14.14.14.2.m1.1.1.3.cmml"><mo id="S5.T2.14.14.14.2.m1.1.1.3a" mathsize="90%" xref="S5.T2.14.14.14.2.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.14.14.14.2.m1.1.1.3.2" xref="S5.T2.14.14.14.2.m1.1.1.3.2.cmml"><mn id="S5.T2.14.14.14.2.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.14.14.14.2.m1.1.1.3.2.2.cmml">0.44</mn><mo id="S5.T2.14.14.14.2.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.14.14.14.2.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.14.14.14.2.m1.1b"><apply id="S5.T2.14.14.14.2.m1.1.1.cmml" xref="S5.T2.14.14.14.2.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.14.14.14.2.m1.1.1.1.cmml" xref="S5.T2.14.14.14.2.m1.1.1">subscript</csymbol><apply id="S5.T2.14.14.14.2.m1.1.1.2.cmml" xref="S5.T2.14.14.14.2.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.14.14.14.2.m1.1.1.2.1.cmml" xref="S5.T2.14.14.14.2.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.14.14.14.2.m1.1.1.2.2.cmml" type="float" xref="S5.T2.14.14.14.2.m1.1.1.2.2">52.55</cn></apply><apply id="S5.T2.14.14.14.2.m1.1.1.3.cmml" xref="S5.T2.14.14.14.2.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.14.14.14.2.m1.1.1.3.1.cmml" xref="S5.T2.14.14.14.2.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.14.14.14.2.m1.1.1.3.2.cmml" xref="S5.T2.14.14.14.2.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.14.14.14.2.m1.1.1.3.2.1.cmml" xref="S5.T2.14.14.14.2.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.14.14.14.2.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.14.14.14.2.m1.1.1.3.2.2">0.44</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.14.14.14.2.m1.1c">52.55\%_{\pm 0.44\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.14.14.14.2.m1.1d">52.55 % start_POSTSUBSCRIPT ± 0.44 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td" id="S5.T2.18.18.18.8"></td> <td class="ltx_td ltx_align_center" id="S5.T2.15.15.15.3"><math alttext="33.83\%_{\pm 1.91\%}" class="ltx_Math" display="inline" id="S5.T2.15.15.15.3.m1.1"><semantics id="S5.T2.15.15.15.3.m1.1a"><msub id="S5.T2.15.15.15.3.m1.1.1" xref="S5.T2.15.15.15.3.m1.1.1.cmml"><mrow id="S5.T2.15.15.15.3.m1.1.1.2" xref="S5.T2.15.15.15.3.m1.1.1.2.cmml"><mn id="S5.T2.15.15.15.3.m1.1.1.2.2" mathsize="90%" xref="S5.T2.15.15.15.3.m1.1.1.2.2.cmml">33.83</mn><mo id="S5.T2.15.15.15.3.m1.1.1.2.1" mathsize="90%" xref="S5.T2.15.15.15.3.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.15.15.15.3.m1.1.1.3" xref="S5.T2.15.15.15.3.m1.1.1.3.cmml"><mo id="S5.T2.15.15.15.3.m1.1.1.3a" mathsize="90%" xref="S5.T2.15.15.15.3.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.15.15.15.3.m1.1.1.3.2" xref="S5.T2.15.15.15.3.m1.1.1.3.2.cmml"><mn id="S5.T2.15.15.15.3.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.15.15.15.3.m1.1.1.3.2.2.cmml">1.91</mn><mo id="S5.T2.15.15.15.3.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.15.15.15.3.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.15.15.15.3.m1.1b"><apply id="S5.T2.15.15.15.3.m1.1.1.cmml" xref="S5.T2.15.15.15.3.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.15.15.15.3.m1.1.1.1.cmml" xref="S5.T2.15.15.15.3.m1.1.1">subscript</csymbol><apply id="S5.T2.15.15.15.3.m1.1.1.2.cmml" xref="S5.T2.15.15.15.3.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.15.15.15.3.m1.1.1.2.1.cmml" xref="S5.T2.15.15.15.3.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.15.15.15.3.m1.1.1.2.2.cmml" type="float" xref="S5.T2.15.15.15.3.m1.1.1.2.2">33.83</cn></apply><apply id="S5.T2.15.15.15.3.m1.1.1.3.cmml" xref="S5.T2.15.15.15.3.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.15.15.15.3.m1.1.1.3.1.cmml" xref="S5.T2.15.15.15.3.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.15.15.15.3.m1.1.1.3.2.cmml" xref="S5.T2.15.15.15.3.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.15.15.15.3.m1.1.1.3.2.1.cmml" xref="S5.T2.15.15.15.3.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.15.15.15.3.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.15.15.15.3.m1.1.1.3.2.2">1.91</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.15.15.15.3.m1.1c">33.83\%_{\pm 1.91\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.15.15.15.3.m1.1d">33.83 % start_POSTSUBSCRIPT ± 1.91 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T2.16.16.16.4"><math alttext="44.49\%_{\pm 0.92\%}" class="ltx_Math" display="inline" id="S5.T2.16.16.16.4.m1.1"><semantics id="S5.T2.16.16.16.4.m1.1a"><msub id="S5.T2.16.16.16.4.m1.1.1" xref="S5.T2.16.16.16.4.m1.1.1.cmml"><mrow id="S5.T2.16.16.16.4.m1.1.1.2" xref="S5.T2.16.16.16.4.m1.1.1.2.cmml"><mn id="S5.T2.16.16.16.4.m1.1.1.2.2" mathsize="90%" xref="S5.T2.16.16.16.4.m1.1.1.2.2.cmml">44.49</mn><mo id="S5.T2.16.16.16.4.m1.1.1.2.1" mathsize="90%" xref="S5.T2.16.16.16.4.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.16.16.16.4.m1.1.1.3" xref="S5.T2.16.16.16.4.m1.1.1.3.cmml"><mo id="S5.T2.16.16.16.4.m1.1.1.3a" mathsize="90%" xref="S5.T2.16.16.16.4.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.16.16.16.4.m1.1.1.3.2" xref="S5.T2.16.16.16.4.m1.1.1.3.2.cmml"><mn id="S5.T2.16.16.16.4.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.16.16.16.4.m1.1.1.3.2.2.cmml">0.92</mn><mo id="S5.T2.16.16.16.4.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.16.16.16.4.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.16.16.16.4.m1.1b"><apply id="S5.T2.16.16.16.4.m1.1.1.cmml" xref="S5.T2.16.16.16.4.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.16.16.16.4.m1.1.1.1.cmml" xref="S5.T2.16.16.16.4.m1.1.1">subscript</csymbol><apply id="S5.T2.16.16.16.4.m1.1.1.2.cmml" xref="S5.T2.16.16.16.4.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.16.16.16.4.m1.1.1.2.1.cmml" xref="S5.T2.16.16.16.4.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.16.16.16.4.m1.1.1.2.2.cmml" type="float" xref="S5.T2.16.16.16.4.m1.1.1.2.2">44.49</cn></apply><apply id="S5.T2.16.16.16.4.m1.1.1.3.cmml" xref="S5.T2.16.16.16.4.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.16.16.16.4.m1.1.1.3.1.cmml" xref="S5.T2.16.16.16.4.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.16.16.16.4.m1.1.1.3.2.cmml" xref="S5.T2.16.16.16.4.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.16.16.16.4.m1.1.1.3.2.1.cmml" xref="S5.T2.16.16.16.4.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.16.16.16.4.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.16.16.16.4.m1.1.1.3.2.2">0.92</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.16.16.16.4.m1.1c">44.49\%_{\pm 0.92\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.16.16.16.4.m1.1d">44.49 % start_POSTSUBSCRIPT ± 0.92 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td" id="S5.T2.18.18.18.9"></td> <td class="ltx_td ltx_align_center" id="S5.T2.17.17.17.5"><math alttext="41.92\%_{\pm 1.16\%}" class="ltx_Math" display="inline" id="S5.T2.17.17.17.5.m1.1"><semantics id="S5.T2.17.17.17.5.m1.1a"><msub id="S5.T2.17.17.17.5.m1.1.1" xref="S5.T2.17.17.17.5.m1.1.1.cmml"><mrow id="S5.T2.17.17.17.5.m1.1.1.2" xref="S5.T2.17.17.17.5.m1.1.1.2.cmml"><mn id="S5.T2.17.17.17.5.m1.1.1.2.2" mathsize="90%" xref="S5.T2.17.17.17.5.m1.1.1.2.2.cmml">41.92</mn><mo id="S5.T2.17.17.17.5.m1.1.1.2.1" mathsize="90%" xref="S5.T2.17.17.17.5.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.17.17.17.5.m1.1.1.3" xref="S5.T2.17.17.17.5.m1.1.1.3.cmml"><mo id="S5.T2.17.17.17.5.m1.1.1.3a" mathsize="90%" xref="S5.T2.17.17.17.5.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.17.17.17.5.m1.1.1.3.2" xref="S5.T2.17.17.17.5.m1.1.1.3.2.cmml"><mn id="S5.T2.17.17.17.5.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.17.17.17.5.m1.1.1.3.2.2.cmml">1.16</mn><mo id="S5.T2.17.17.17.5.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.17.17.17.5.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.17.17.17.5.m1.1b"><apply id="S5.T2.17.17.17.5.m1.1.1.cmml" xref="S5.T2.17.17.17.5.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.17.17.17.5.m1.1.1.1.cmml" xref="S5.T2.17.17.17.5.m1.1.1">subscript</csymbol><apply id="S5.T2.17.17.17.5.m1.1.1.2.cmml" xref="S5.T2.17.17.17.5.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.17.17.17.5.m1.1.1.2.1.cmml" xref="S5.T2.17.17.17.5.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.17.17.17.5.m1.1.1.2.2.cmml" type="float" xref="S5.T2.17.17.17.5.m1.1.1.2.2">41.92</cn></apply><apply id="S5.T2.17.17.17.5.m1.1.1.3.cmml" xref="S5.T2.17.17.17.5.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.17.17.17.5.m1.1.1.3.1.cmml" xref="S5.T2.17.17.17.5.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.17.17.17.5.m1.1.1.3.2.cmml" xref="S5.T2.17.17.17.5.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.17.17.17.5.m1.1.1.3.2.1.cmml" xref="S5.T2.17.17.17.5.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.17.17.17.5.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.17.17.17.5.m1.1.1.3.2.2">1.16</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.17.17.17.5.m1.1c">41.92\%_{\pm 1.16\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.17.17.17.5.m1.1d">41.92 % start_POSTSUBSCRIPT ± 1.16 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T2.18.18.18.6"><math alttext="47.35\%_{\pm 0.47\%}" class="ltx_Math" display="inline" id="S5.T2.18.18.18.6.m1.1"><semantics id="S5.T2.18.18.18.6.m1.1a"><msub id="S5.T2.18.18.18.6.m1.1.1" xref="S5.T2.18.18.18.6.m1.1.1.cmml"><mrow id="S5.T2.18.18.18.6.m1.1.1.2" xref="S5.T2.18.18.18.6.m1.1.1.2.cmml"><mn id="S5.T2.18.18.18.6.m1.1.1.2.2" mathsize="90%" xref="S5.T2.18.18.18.6.m1.1.1.2.2.cmml">47.35</mn><mo id="S5.T2.18.18.18.6.m1.1.1.2.1" mathsize="90%" xref="S5.T2.18.18.18.6.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.18.18.18.6.m1.1.1.3" xref="S5.T2.18.18.18.6.m1.1.1.3.cmml"><mo id="S5.T2.18.18.18.6.m1.1.1.3a" mathsize="90%" xref="S5.T2.18.18.18.6.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.18.18.18.6.m1.1.1.3.2" xref="S5.T2.18.18.18.6.m1.1.1.3.2.cmml"><mn id="S5.T2.18.18.18.6.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.18.18.18.6.m1.1.1.3.2.2.cmml">0.47</mn><mo id="S5.T2.18.18.18.6.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.18.18.18.6.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.18.18.18.6.m1.1b"><apply id="S5.T2.18.18.18.6.m1.1.1.cmml" xref="S5.T2.18.18.18.6.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.18.18.18.6.m1.1.1.1.cmml" xref="S5.T2.18.18.18.6.m1.1.1">subscript</csymbol><apply id="S5.T2.18.18.18.6.m1.1.1.2.cmml" xref="S5.T2.18.18.18.6.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.18.18.18.6.m1.1.1.2.1.cmml" xref="S5.T2.18.18.18.6.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.18.18.18.6.m1.1.1.2.2.cmml" type="float" xref="S5.T2.18.18.18.6.m1.1.1.2.2">47.35</cn></apply><apply id="S5.T2.18.18.18.6.m1.1.1.3.cmml" xref="S5.T2.18.18.18.6.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.18.18.18.6.m1.1.1.3.1.cmml" xref="S5.T2.18.18.18.6.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.18.18.18.6.m1.1.1.3.2.cmml" xref="S5.T2.18.18.18.6.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.18.18.18.6.m1.1.1.3.2.1.cmml" xref="S5.T2.18.18.18.6.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.18.18.18.6.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.18.18.18.6.m1.1.1.3.2.2">0.47</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.18.18.18.6.m1.1c">47.35\%_{\pm 0.47\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.18.18.18.6.m1.1d">47.35 % start_POSTSUBSCRIPT ± 0.47 % end_POSTSUBSCRIPT</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T2.24.24.24"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S5.T2.24.24.24.7"><span class="ltx_text" id="S5.T2.24.24.24.7.1" style="font-size:90%;">KN</span></th> <td class="ltx_td ltx_align_center" id="S5.T2.19.19.19.1"><math alttext="47.40\%_{\pm 0.37\%}" class="ltx_Math" display="inline" id="S5.T2.19.19.19.1.m1.1"><semantics id="S5.T2.19.19.19.1.m1.1a"><msub id="S5.T2.19.19.19.1.m1.1.1" xref="S5.T2.19.19.19.1.m1.1.1.cmml"><mrow id="S5.T2.19.19.19.1.m1.1.1.2" xref="S5.T2.19.19.19.1.m1.1.1.2.cmml"><mn id="S5.T2.19.19.19.1.m1.1.1.2.2" mathsize="90%" xref="S5.T2.19.19.19.1.m1.1.1.2.2.cmml">47.40</mn><mo id="S5.T2.19.19.19.1.m1.1.1.2.1" mathsize="90%" xref="S5.T2.19.19.19.1.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.19.19.19.1.m1.1.1.3" xref="S5.T2.19.19.19.1.m1.1.1.3.cmml"><mo id="S5.T2.19.19.19.1.m1.1.1.3a" mathsize="90%" xref="S5.T2.19.19.19.1.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.19.19.19.1.m1.1.1.3.2" xref="S5.T2.19.19.19.1.m1.1.1.3.2.cmml"><mn id="S5.T2.19.19.19.1.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.19.19.19.1.m1.1.1.3.2.2.cmml">0.37</mn><mo id="S5.T2.19.19.19.1.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.19.19.19.1.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.19.19.19.1.m1.1b"><apply id="S5.T2.19.19.19.1.m1.1.1.cmml" xref="S5.T2.19.19.19.1.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.19.19.19.1.m1.1.1.1.cmml" xref="S5.T2.19.19.19.1.m1.1.1">subscript</csymbol><apply id="S5.T2.19.19.19.1.m1.1.1.2.cmml" xref="S5.T2.19.19.19.1.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.19.19.19.1.m1.1.1.2.1.cmml" xref="S5.T2.19.19.19.1.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.19.19.19.1.m1.1.1.2.2.cmml" type="float" xref="S5.T2.19.19.19.1.m1.1.1.2.2">47.40</cn></apply><apply id="S5.T2.19.19.19.1.m1.1.1.3.cmml" xref="S5.T2.19.19.19.1.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.19.19.19.1.m1.1.1.3.1.cmml" xref="S5.T2.19.19.19.1.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.19.19.19.1.m1.1.1.3.2.cmml" xref="S5.T2.19.19.19.1.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.19.19.19.1.m1.1.1.3.2.1.cmml" xref="S5.T2.19.19.19.1.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.19.19.19.1.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.19.19.19.1.m1.1.1.3.2.2">0.37</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.19.19.19.1.m1.1c">47.40\%_{\pm 0.37\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.19.19.19.1.m1.1d">47.40 % start_POSTSUBSCRIPT ± 0.37 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T2.20.20.20.2"><math alttext="47.22\%_{\pm 0.04\%}" class="ltx_Math" display="inline" id="S5.T2.20.20.20.2.m1.1"><semantics id="S5.T2.20.20.20.2.m1.1a"><msub id="S5.T2.20.20.20.2.m1.1.1" xref="S5.T2.20.20.20.2.m1.1.1.cmml"><mrow id="S5.T2.20.20.20.2.m1.1.1.2" xref="S5.T2.20.20.20.2.m1.1.1.2.cmml"><mn id="S5.T2.20.20.20.2.m1.1.1.2.2" mathsize="90%" xref="S5.T2.20.20.20.2.m1.1.1.2.2.cmml">47.22</mn><mo id="S5.T2.20.20.20.2.m1.1.1.2.1" mathsize="90%" xref="S5.T2.20.20.20.2.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.20.20.20.2.m1.1.1.3" xref="S5.T2.20.20.20.2.m1.1.1.3.cmml"><mo id="S5.T2.20.20.20.2.m1.1.1.3a" mathsize="90%" xref="S5.T2.20.20.20.2.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.20.20.20.2.m1.1.1.3.2" xref="S5.T2.20.20.20.2.m1.1.1.3.2.cmml"><mn id="S5.T2.20.20.20.2.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.20.20.20.2.m1.1.1.3.2.2.cmml">0.04</mn><mo id="S5.T2.20.20.20.2.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.20.20.20.2.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.20.20.20.2.m1.1b"><apply id="S5.T2.20.20.20.2.m1.1.1.cmml" xref="S5.T2.20.20.20.2.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.20.20.20.2.m1.1.1.1.cmml" xref="S5.T2.20.20.20.2.m1.1.1">subscript</csymbol><apply id="S5.T2.20.20.20.2.m1.1.1.2.cmml" xref="S5.T2.20.20.20.2.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.20.20.20.2.m1.1.1.2.1.cmml" xref="S5.T2.20.20.20.2.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.20.20.20.2.m1.1.1.2.2.cmml" type="float" xref="S5.T2.20.20.20.2.m1.1.1.2.2">47.22</cn></apply><apply id="S5.T2.20.20.20.2.m1.1.1.3.cmml" xref="S5.T2.20.20.20.2.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.20.20.20.2.m1.1.1.3.1.cmml" xref="S5.T2.20.20.20.2.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.20.20.20.2.m1.1.1.3.2.cmml" xref="S5.T2.20.20.20.2.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.20.20.20.2.m1.1.1.3.2.1.cmml" xref="S5.T2.20.20.20.2.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.20.20.20.2.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.20.20.20.2.m1.1.1.3.2.2">0.04</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.20.20.20.2.m1.1c">47.22\%_{\pm 0.04\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.20.20.20.2.m1.1d">47.22 % start_POSTSUBSCRIPT ± 0.04 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td" id="S5.T2.24.24.24.8"></td> <td class="ltx_td ltx_align_center" id="S5.T2.21.21.21.3"><math alttext="49.74\%_{\pm 0.05\%}" class="ltx_Math" display="inline" id="S5.T2.21.21.21.3.m1.1"><semantics id="S5.T2.21.21.21.3.m1.1a"><msub id="S5.T2.21.21.21.3.m1.1.1" xref="S5.T2.21.21.21.3.m1.1.1.cmml"><mrow id="S5.T2.21.21.21.3.m1.1.1.2" xref="S5.T2.21.21.21.3.m1.1.1.2.cmml"><mn id="S5.T2.21.21.21.3.m1.1.1.2.2" mathsize="90%" xref="S5.T2.21.21.21.3.m1.1.1.2.2.cmml">49.74</mn><mo id="S5.T2.21.21.21.3.m1.1.1.2.1" mathsize="90%" xref="S5.T2.21.21.21.3.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.21.21.21.3.m1.1.1.3" xref="S5.T2.21.21.21.3.m1.1.1.3.cmml"><mo id="S5.T2.21.21.21.3.m1.1.1.3a" mathsize="90%" xref="S5.T2.21.21.21.3.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.21.21.21.3.m1.1.1.3.2" xref="S5.T2.21.21.21.3.m1.1.1.3.2.cmml"><mn id="S5.T2.21.21.21.3.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.21.21.21.3.m1.1.1.3.2.2.cmml">0.05</mn><mo id="S5.T2.21.21.21.3.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.21.21.21.3.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.21.21.21.3.m1.1b"><apply id="S5.T2.21.21.21.3.m1.1.1.cmml" xref="S5.T2.21.21.21.3.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.21.21.21.3.m1.1.1.1.cmml" xref="S5.T2.21.21.21.3.m1.1.1">subscript</csymbol><apply id="S5.T2.21.21.21.3.m1.1.1.2.cmml" xref="S5.T2.21.21.21.3.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.21.21.21.3.m1.1.1.2.1.cmml" xref="S5.T2.21.21.21.3.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.21.21.21.3.m1.1.1.2.2.cmml" type="float" xref="S5.T2.21.21.21.3.m1.1.1.2.2">49.74</cn></apply><apply id="S5.T2.21.21.21.3.m1.1.1.3.cmml" xref="S5.T2.21.21.21.3.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.21.21.21.3.m1.1.1.3.1.cmml" xref="S5.T2.21.21.21.3.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.21.21.21.3.m1.1.1.3.2.cmml" xref="S5.T2.21.21.21.3.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.21.21.21.3.m1.1.1.3.2.1.cmml" xref="S5.T2.21.21.21.3.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.21.21.21.3.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.21.21.21.3.m1.1.1.3.2.2">0.05</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.21.21.21.3.m1.1c">49.74\%_{\pm 0.05\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.21.21.21.3.m1.1d">49.74 % start_POSTSUBSCRIPT ± 0.05 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T2.22.22.22.4"><math alttext="49.67\%_{\pm 1.36\%}" class="ltx_Math" display="inline" id="S5.T2.22.22.22.4.m1.1"><semantics id="S5.T2.22.22.22.4.m1.1a"><msub id="S5.T2.22.22.22.4.m1.1.1" xref="S5.T2.22.22.22.4.m1.1.1.cmml"><mrow id="S5.T2.22.22.22.4.m1.1.1.2" xref="S5.T2.22.22.22.4.m1.1.1.2.cmml"><mn id="S5.T2.22.22.22.4.m1.1.1.2.2" mathsize="90%" xref="S5.T2.22.22.22.4.m1.1.1.2.2.cmml">49.67</mn><mo id="S5.T2.22.22.22.4.m1.1.1.2.1" mathsize="90%" xref="S5.T2.22.22.22.4.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.22.22.22.4.m1.1.1.3" xref="S5.T2.22.22.22.4.m1.1.1.3.cmml"><mo id="S5.T2.22.22.22.4.m1.1.1.3a" mathsize="90%" xref="S5.T2.22.22.22.4.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.22.22.22.4.m1.1.1.3.2" xref="S5.T2.22.22.22.4.m1.1.1.3.2.cmml"><mn id="S5.T2.22.22.22.4.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.22.22.22.4.m1.1.1.3.2.2.cmml">1.36</mn><mo id="S5.T2.22.22.22.4.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.22.22.22.4.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.22.22.22.4.m1.1b"><apply id="S5.T2.22.22.22.4.m1.1.1.cmml" xref="S5.T2.22.22.22.4.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.22.22.22.4.m1.1.1.1.cmml" xref="S5.T2.22.22.22.4.m1.1.1">subscript</csymbol><apply id="S5.T2.22.22.22.4.m1.1.1.2.cmml" xref="S5.T2.22.22.22.4.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.22.22.22.4.m1.1.1.2.1.cmml" xref="S5.T2.22.22.22.4.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.22.22.22.4.m1.1.1.2.2.cmml" type="float" xref="S5.T2.22.22.22.4.m1.1.1.2.2">49.67</cn></apply><apply id="S5.T2.22.22.22.4.m1.1.1.3.cmml" xref="S5.T2.22.22.22.4.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.22.22.22.4.m1.1.1.3.1.cmml" xref="S5.T2.22.22.22.4.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.22.22.22.4.m1.1.1.3.2.cmml" xref="S5.T2.22.22.22.4.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.22.22.22.4.m1.1.1.3.2.1.cmml" xref="S5.T2.22.22.22.4.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.22.22.22.4.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.22.22.22.4.m1.1.1.3.2.2">1.36</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.22.22.22.4.m1.1c">49.67\%_{\pm 1.36\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.22.22.22.4.m1.1d">49.67 % start_POSTSUBSCRIPT ± 1.36 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td" id="S5.T2.24.24.24.9"></td> <td class="ltx_td ltx_align_center" id="S5.T2.23.23.23.5"><math alttext="51.51\%_{\pm 0.58\%}" class="ltx_Math" display="inline" id="S5.T2.23.23.23.5.m1.1"><semantics id="S5.T2.23.23.23.5.m1.1a"><msub id="S5.T2.23.23.23.5.m1.1.1" xref="S5.T2.23.23.23.5.m1.1.1.cmml"><mrow id="S5.T2.23.23.23.5.m1.1.1.2" xref="S5.T2.23.23.23.5.m1.1.1.2.cmml"><mn id="S5.T2.23.23.23.5.m1.1.1.2.2" mathsize="90%" xref="S5.T2.23.23.23.5.m1.1.1.2.2.cmml">51.51</mn><mo id="S5.T2.23.23.23.5.m1.1.1.2.1" mathsize="90%" xref="S5.T2.23.23.23.5.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.23.23.23.5.m1.1.1.3" xref="S5.T2.23.23.23.5.m1.1.1.3.cmml"><mo id="S5.T2.23.23.23.5.m1.1.1.3a" mathsize="90%" xref="S5.T2.23.23.23.5.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.23.23.23.5.m1.1.1.3.2" xref="S5.T2.23.23.23.5.m1.1.1.3.2.cmml"><mn id="S5.T2.23.23.23.5.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.23.23.23.5.m1.1.1.3.2.2.cmml">0.58</mn><mo id="S5.T2.23.23.23.5.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.23.23.23.5.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.23.23.23.5.m1.1b"><apply id="S5.T2.23.23.23.5.m1.1.1.cmml" xref="S5.T2.23.23.23.5.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.23.23.23.5.m1.1.1.1.cmml" xref="S5.T2.23.23.23.5.m1.1.1">subscript</csymbol><apply id="S5.T2.23.23.23.5.m1.1.1.2.cmml" xref="S5.T2.23.23.23.5.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.23.23.23.5.m1.1.1.2.1.cmml" xref="S5.T2.23.23.23.5.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.23.23.23.5.m1.1.1.2.2.cmml" type="float" xref="S5.T2.23.23.23.5.m1.1.1.2.2">51.51</cn></apply><apply id="S5.T2.23.23.23.5.m1.1.1.3.cmml" xref="S5.T2.23.23.23.5.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.23.23.23.5.m1.1.1.3.1.cmml" xref="S5.T2.23.23.23.5.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.23.23.23.5.m1.1.1.3.2.cmml" xref="S5.T2.23.23.23.5.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.23.23.23.5.m1.1.1.3.2.1.cmml" xref="S5.T2.23.23.23.5.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.23.23.23.5.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.23.23.23.5.m1.1.1.3.2.2">0.58</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.23.23.23.5.m1.1c">51.51\%_{\pm 0.58\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.23.23.23.5.m1.1d">51.51 % start_POSTSUBSCRIPT ± 0.58 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T2.24.24.24.6"><math alttext="52.62\%_{\pm 1.91\%}" class="ltx_Math" display="inline" id="S5.T2.24.24.24.6.m1.1"><semantics id="S5.T2.24.24.24.6.m1.1a"><msub id="S5.T2.24.24.24.6.m1.1.1" xref="S5.T2.24.24.24.6.m1.1.1.cmml"><mrow id="S5.T2.24.24.24.6.m1.1.1.2" xref="S5.T2.24.24.24.6.m1.1.1.2.cmml"><mn id="S5.T2.24.24.24.6.m1.1.1.2.2" mathsize="90%" xref="S5.T2.24.24.24.6.m1.1.1.2.2.cmml">52.62</mn><mo id="S5.T2.24.24.24.6.m1.1.1.2.1" mathsize="90%" xref="S5.T2.24.24.24.6.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.24.24.24.6.m1.1.1.3" xref="S5.T2.24.24.24.6.m1.1.1.3.cmml"><mo id="S5.T2.24.24.24.6.m1.1.1.3a" mathsize="90%" xref="S5.T2.24.24.24.6.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.24.24.24.6.m1.1.1.3.2" xref="S5.T2.24.24.24.6.m1.1.1.3.2.cmml"><mn id="S5.T2.24.24.24.6.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.24.24.24.6.m1.1.1.3.2.2.cmml">1.91</mn><mo id="S5.T2.24.24.24.6.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.24.24.24.6.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.24.24.24.6.m1.1b"><apply id="S5.T2.24.24.24.6.m1.1.1.cmml" xref="S5.T2.24.24.24.6.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.24.24.24.6.m1.1.1.1.cmml" xref="S5.T2.24.24.24.6.m1.1.1">subscript</csymbol><apply id="S5.T2.24.24.24.6.m1.1.1.2.cmml" xref="S5.T2.24.24.24.6.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.24.24.24.6.m1.1.1.2.1.cmml" xref="S5.T2.24.24.24.6.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.24.24.24.6.m1.1.1.2.2.cmml" type="float" xref="S5.T2.24.24.24.6.m1.1.1.2.2">52.62</cn></apply><apply id="S5.T2.24.24.24.6.m1.1.1.3.cmml" xref="S5.T2.24.24.24.6.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.24.24.24.6.m1.1.1.3.1.cmml" xref="S5.T2.24.24.24.6.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.24.24.24.6.m1.1.1.3.2.cmml" xref="S5.T2.24.24.24.6.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.24.24.24.6.m1.1.1.3.2.1.cmml" xref="S5.T2.24.24.24.6.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.24.24.24.6.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.24.24.24.6.m1.1.1.3.2.2">1.91</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.24.24.24.6.m1.1c">52.62\%_{\pm 1.91\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.24.24.24.6.m1.1d">52.62 % start_POSTSUBSCRIPT ± 1.91 % end_POSTSUBSCRIPT</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T2.30.30.30"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S5.T2.30.30.30.7"><span class="ltx_text" id="S5.T2.30.30.30.7.1" style="font-size:90%;">MEND</span></th> <td class="ltx_td ltx_align_center" id="S5.T2.25.25.25.1"><math alttext="58.30\%_{\pm 0.12\%}" class="ltx_Math" display="inline" id="S5.T2.25.25.25.1.m1.1"><semantics id="S5.T2.25.25.25.1.m1.1a"><msub id="S5.T2.25.25.25.1.m1.1.1" xref="S5.T2.25.25.25.1.m1.1.1.cmml"><mrow id="S5.T2.25.25.25.1.m1.1.1.2" xref="S5.T2.25.25.25.1.m1.1.1.2.cmml"><mn id="S5.T2.25.25.25.1.m1.1.1.2.2" mathsize="90%" xref="S5.T2.25.25.25.1.m1.1.1.2.2.cmml">58.30</mn><mo id="S5.T2.25.25.25.1.m1.1.1.2.1" mathsize="90%" xref="S5.T2.25.25.25.1.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.25.25.25.1.m1.1.1.3" xref="S5.T2.25.25.25.1.m1.1.1.3.cmml"><mo id="S5.T2.25.25.25.1.m1.1.1.3a" mathsize="90%" xref="S5.T2.25.25.25.1.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.25.25.25.1.m1.1.1.3.2" xref="S5.T2.25.25.25.1.m1.1.1.3.2.cmml"><mn id="S5.T2.25.25.25.1.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.25.25.25.1.m1.1.1.3.2.2.cmml">0.12</mn><mo id="S5.T2.25.25.25.1.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.25.25.25.1.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.25.25.25.1.m1.1b"><apply id="S5.T2.25.25.25.1.m1.1.1.cmml" xref="S5.T2.25.25.25.1.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.25.25.25.1.m1.1.1.1.cmml" xref="S5.T2.25.25.25.1.m1.1.1">subscript</csymbol><apply id="S5.T2.25.25.25.1.m1.1.1.2.cmml" xref="S5.T2.25.25.25.1.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.25.25.25.1.m1.1.1.2.1.cmml" xref="S5.T2.25.25.25.1.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.25.25.25.1.m1.1.1.2.2.cmml" type="float" xref="S5.T2.25.25.25.1.m1.1.1.2.2">58.30</cn></apply><apply id="S5.T2.25.25.25.1.m1.1.1.3.cmml" xref="S5.T2.25.25.25.1.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.25.25.25.1.m1.1.1.3.1.cmml" xref="S5.T2.25.25.25.1.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.25.25.25.1.m1.1.1.3.2.cmml" xref="S5.T2.25.25.25.1.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.25.25.25.1.m1.1.1.3.2.1.cmml" xref="S5.T2.25.25.25.1.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.25.25.25.1.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.25.25.25.1.m1.1.1.3.2.2">0.12</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.25.25.25.1.m1.1c">58.30\%_{\pm 0.12\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.25.25.25.1.m1.1d">58.30 % start_POSTSUBSCRIPT ± 0.12 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T2.26.26.26.2"><math alttext="58.51\%_{\pm 0.07\%}" class="ltx_Math" display="inline" id="S5.T2.26.26.26.2.m1.1"><semantics id="S5.T2.26.26.26.2.m1.1a"><msub id="S5.T2.26.26.26.2.m1.1.1" xref="S5.T2.26.26.26.2.m1.1.1.cmml"><mrow id="S5.T2.26.26.26.2.m1.1.1.2" xref="S5.T2.26.26.26.2.m1.1.1.2.cmml"><mn id="S5.T2.26.26.26.2.m1.1.1.2.2" mathsize="90%" xref="S5.T2.26.26.26.2.m1.1.1.2.2.cmml">58.51</mn><mo id="S5.T2.26.26.26.2.m1.1.1.2.1" mathsize="90%" xref="S5.T2.26.26.26.2.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.26.26.26.2.m1.1.1.3" xref="S5.T2.26.26.26.2.m1.1.1.3.cmml"><mo id="S5.T2.26.26.26.2.m1.1.1.3a" mathsize="90%" xref="S5.T2.26.26.26.2.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.26.26.26.2.m1.1.1.3.2" xref="S5.T2.26.26.26.2.m1.1.1.3.2.cmml"><mn id="S5.T2.26.26.26.2.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.26.26.26.2.m1.1.1.3.2.2.cmml">0.07</mn><mo id="S5.T2.26.26.26.2.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.26.26.26.2.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.26.26.26.2.m1.1b"><apply id="S5.T2.26.26.26.2.m1.1.1.cmml" xref="S5.T2.26.26.26.2.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.26.26.26.2.m1.1.1.1.cmml" xref="S5.T2.26.26.26.2.m1.1.1">subscript</csymbol><apply id="S5.T2.26.26.26.2.m1.1.1.2.cmml" xref="S5.T2.26.26.26.2.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.26.26.26.2.m1.1.1.2.1.cmml" xref="S5.T2.26.26.26.2.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.26.26.26.2.m1.1.1.2.2.cmml" type="float" xref="S5.T2.26.26.26.2.m1.1.1.2.2">58.51</cn></apply><apply id="S5.T2.26.26.26.2.m1.1.1.3.cmml" xref="S5.T2.26.26.26.2.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.26.26.26.2.m1.1.1.3.1.cmml" xref="S5.T2.26.26.26.2.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.26.26.26.2.m1.1.1.3.2.cmml" xref="S5.T2.26.26.26.2.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.26.26.26.2.m1.1.1.3.2.1.cmml" xref="S5.T2.26.26.26.2.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.26.26.26.2.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.26.26.26.2.m1.1.1.3.2.2">0.07</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.26.26.26.2.m1.1c">58.51\%_{\pm 0.07\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.26.26.26.2.m1.1d">58.51 % start_POSTSUBSCRIPT ± 0.07 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td" id="S5.T2.30.30.30.8"></td> <td class="ltx_td ltx_align_center" id="S5.T2.27.27.27.3"><math alttext="40.73\%_{\pm 0.05\%}" class="ltx_Math" display="inline" id="S5.T2.27.27.27.3.m1.1"><semantics id="S5.T2.27.27.27.3.m1.1a"><msub id="S5.T2.27.27.27.3.m1.1.1" xref="S5.T2.27.27.27.3.m1.1.1.cmml"><mrow id="S5.T2.27.27.27.3.m1.1.1.2" xref="S5.T2.27.27.27.3.m1.1.1.2.cmml"><mn id="S5.T2.27.27.27.3.m1.1.1.2.2" mathsize="90%" xref="S5.T2.27.27.27.3.m1.1.1.2.2.cmml">40.73</mn><mo id="S5.T2.27.27.27.3.m1.1.1.2.1" mathsize="90%" xref="S5.T2.27.27.27.3.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.27.27.27.3.m1.1.1.3" xref="S5.T2.27.27.27.3.m1.1.1.3.cmml"><mo id="S5.T2.27.27.27.3.m1.1.1.3a" mathsize="90%" xref="S5.T2.27.27.27.3.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.27.27.27.3.m1.1.1.3.2" xref="S5.T2.27.27.27.3.m1.1.1.3.2.cmml"><mn id="S5.T2.27.27.27.3.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.27.27.27.3.m1.1.1.3.2.2.cmml">0.05</mn><mo id="S5.T2.27.27.27.3.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.27.27.27.3.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.27.27.27.3.m1.1b"><apply id="S5.T2.27.27.27.3.m1.1.1.cmml" xref="S5.T2.27.27.27.3.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.27.27.27.3.m1.1.1.1.cmml" xref="S5.T2.27.27.27.3.m1.1.1">subscript</csymbol><apply id="S5.T2.27.27.27.3.m1.1.1.2.cmml" xref="S5.T2.27.27.27.3.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.27.27.27.3.m1.1.1.2.1.cmml" xref="S5.T2.27.27.27.3.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.27.27.27.3.m1.1.1.2.2.cmml" type="float" xref="S5.T2.27.27.27.3.m1.1.1.2.2">40.73</cn></apply><apply id="S5.T2.27.27.27.3.m1.1.1.3.cmml" xref="S5.T2.27.27.27.3.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.27.27.27.3.m1.1.1.3.1.cmml" xref="S5.T2.27.27.27.3.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.27.27.27.3.m1.1.1.3.2.cmml" xref="S5.T2.27.27.27.3.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.27.27.27.3.m1.1.1.3.2.1.cmml" xref="S5.T2.27.27.27.3.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.27.27.27.3.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.27.27.27.3.m1.1.1.3.2.2">0.05</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.27.27.27.3.m1.1c">40.73\%_{\pm 0.05\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.27.27.27.3.m1.1d">40.73 % start_POSTSUBSCRIPT ± 0.05 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T2.28.28.28.4"><math alttext="43.61\%_{\pm 0.04\%}" class="ltx_Math" display="inline" id="S5.T2.28.28.28.4.m1.1"><semantics id="S5.T2.28.28.28.4.m1.1a"><msub id="S5.T2.28.28.28.4.m1.1.1" xref="S5.T2.28.28.28.4.m1.1.1.cmml"><mrow id="S5.T2.28.28.28.4.m1.1.1.2" xref="S5.T2.28.28.28.4.m1.1.1.2.cmml"><mn id="S5.T2.28.28.28.4.m1.1.1.2.2" mathsize="90%" xref="S5.T2.28.28.28.4.m1.1.1.2.2.cmml">43.61</mn><mo id="S5.T2.28.28.28.4.m1.1.1.2.1" mathsize="90%" xref="S5.T2.28.28.28.4.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.28.28.28.4.m1.1.1.3" xref="S5.T2.28.28.28.4.m1.1.1.3.cmml"><mo id="S5.T2.28.28.28.4.m1.1.1.3a" mathsize="90%" xref="S5.T2.28.28.28.4.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.28.28.28.4.m1.1.1.3.2" xref="S5.T2.28.28.28.4.m1.1.1.3.2.cmml"><mn id="S5.T2.28.28.28.4.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.28.28.28.4.m1.1.1.3.2.2.cmml">0.04</mn><mo id="S5.T2.28.28.28.4.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.28.28.28.4.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.28.28.28.4.m1.1b"><apply id="S5.T2.28.28.28.4.m1.1.1.cmml" xref="S5.T2.28.28.28.4.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.28.28.28.4.m1.1.1.1.cmml" xref="S5.T2.28.28.28.4.m1.1.1">subscript</csymbol><apply id="S5.T2.28.28.28.4.m1.1.1.2.cmml" xref="S5.T2.28.28.28.4.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.28.28.28.4.m1.1.1.2.1.cmml" xref="S5.T2.28.28.28.4.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.28.28.28.4.m1.1.1.2.2.cmml" type="float" xref="S5.T2.28.28.28.4.m1.1.1.2.2">43.61</cn></apply><apply id="S5.T2.28.28.28.4.m1.1.1.3.cmml" xref="S5.T2.28.28.28.4.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.28.28.28.4.m1.1.1.3.1.cmml" xref="S5.T2.28.28.28.4.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.28.28.28.4.m1.1.1.3.2.cmml" xref="S5.T2.28.28.28.4.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.28.28.28.4.m1.1.1.3.2.1.cmml" xref="S5.T2.28.28.28.4.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.28.28.28.4.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.28.28.28.4.m1.1.1.3.2.2">0.04</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.28.28.28.4.m1.1c">43.61\%_{\pm 0.04\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.28.28.28.4.m1.1d">43.61 % start_POSTSUBSCRIPT ± 0.04 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td" id="S5.T2.30.30.30.9"></td> <td class="ltx_td ltx_align_center" id="S5.T2.29.29.29.5"><math alttext="45.62\%_{\pm 0.10\%}" class="ltx_Math" display="inline" id="S5.T2.29.29.29.5.m1.1"><semantics id="S5.T2.29.29.29.5.m1.1a"><msub id="S5.T2.29.29.29.5.m1.1.1" xref="S5.T2.29.29.29.5.m1.1.1.cmml"><mrow id="S5.T2.29.29.29.5.m1.1.1.2" xref="S5.T2.29.29.29.5.m1.1.1.2.cmml"><mn id="S5.T2.29.29.29.5.m1.1.1.2.2" mathsize="90%" xref="S5.T2.29.29.29.5.m1.1.1.2.2.cmml">45.62</mn><mo id="S5.T2.29.29.29.5.m1.1.1.2.1" mathsize="90%" xref="S5.T2.29.29.29.5.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.29.29.29.5.m1.1.1.3" xref="S5.T2.29.29.29.5.m1.1.1.3.cmml"><mo id="S5.T2.29.29.29.5.m1.1.1.3a" mathsize="90%" xref="S5.T2.29.29.29.5.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.29.29.29.5.m1.1.1.3.2" xref="S5.T2.29.29.29.5.m1.1.1.3.2.cmml"><mn id="S5.T2.29.29.29.5.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.29.29.29.5.m1.1.1.3.2.2.cmml">0.10</mn><mo id="S5.T2.29.29.29.5.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.29.29.29.5.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.29.29.29.5.m1.1b"><apply id="S5.T2.29.29.29.5.m1.1.1.cmml" xref="S5.T2.29.29.29.5.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.29.29.29.5.m1.1.1.1.cmml" xref="S5.T2.29.29.29.5.m1.1.1">subscript</csymbol><apply id="S5.T2.29.29.29.5.m1.1.1.2.cmml" xref="S5.T2.29.29.29.5.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.29.29.29.5.m1.1.1.2.1.cmml" xref="S5.T2.29.29.29.5.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.29.29.29.5.m1.1.1.2.2.cmml" type="float" xref="S5.T2.29.29.29.5.m1.1.1.2.2">45.62</cn></apply><apply id="S5.T2.29.29.29.5.m1.1.1.3.cmml" xref="S5.T2.29.29.29.5.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.29.29.29.5.m1.1.1.3.1.cmml" xref="S5.T2.29.29.29.5.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.29.29.29.5.m1.1.1.3.2.cmml" xref="S5.T2.29.29.29.5.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.29.29.29.5.m1.1.1.3.2.1.cmml" xref="S5.T2.29.29.29.5.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.29.29.29.5.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.29.29.29.5.m1.1.1.3.2.2">0.10</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.29.29.29.5.m1.1c">45.62\%_{\pm 0.10\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.29.29.29.5.m1.1d">45.62 % start_POSTSUBSCRIPT ± 0.10 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S5.T2.30.30.30.6"><math alttext="44.34\%_{\pm 1.75\%}" class="ltx_Math" display="inline" id="S5.T2.30.30.30.6.m1.1"><semantics id="S5.T2.30.30.30.6.m1.1a"><msub id="S5.T2.30.30.30.6.m1.1.1" xref="S5.T2.30.30.30.6.m1.1.1.cmml"><mrow id="S5.T2.30.30.30.6.m1.1.1.2" xref="S5.T2.30.30.30.6.m1.1.1.2.cmml"><mn id="S5.T2.30.30.30.6.m1.1.1.2.2" mathsize="90%" xref="S5.T2.30.30.30.6.m1.1.1.2.2.cmml">44.34</mn><mo id="S5.T2.30.30.30.6.m1.1.1.2.1" mathsize="90%" xref="S5.T2.30.30.30.6.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.30.30.30.6.m1.1.1.3" xref="S5.T2.30.30.30.6.m1.1.1.3.cmml"><mo id="S5.T2.30.30.30.6.m1.1.1.3a" mathsize="90%" xref="S5.T2.30.30.30.6.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.30.30.30.6.m1.1.1.3.2" xref="S5.T2.30.30.30.6.m1.1.1.3.2.cmml"><mn id="S5.T2.30.30.30.6.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.30.30.30.6.m1.1.1.3.2.2.cmml">1.75</mn><mo id="S5.T2.30.30.30.6.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.30.30.30.6.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.30.30.30.6.m1.1b"><apply id="S5.T2.30.30.30.6.m1.1.1.cmml" xref="S5.T2.30.30.30.6.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.30.30.30.6.m1.1.1.1.cmml" xref="S5.T2.30.30.30.6.m1.1.1">subscript</csymbol><apply id="S5.T2.30.30.30.6.m1.1.1.2.cmml" xref="S5.T2.30.30.30.6.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.30.30.30.6.m1.1.1.2.1.cmml" xref="S5.T2.30.30.30.6.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.30.30.30.6.m1.1.1.2.2.cmml" type="float" xref="S5.T2.30.30.30.6.m1.1.1.2.2">44.34</cn></apply><apply id="S5.T2.30.30.30.6.m1.1.1.3.cmml" xref="S5.T2.30.30.30.6.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.30.30.30.6.m1.1.1.3.1.cmml" xref="S5.T2.30.30.30.6.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.30.30.30.6.m1.1.1.3.2.cmml" xref="S5.T2.30.30.30.6.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.30.30.30.6.m1.1.1.3.2.1.cmml" xref="S5.T2.30.30.30.6.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.30.30.30.6.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.30.30.30.6.m1.1.1.3.2.2">1.75</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.30.30.30.6.m1.1c">44.34\%_{\pm 1.75\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.30.30.30.6.m1.1d">44.34 % start_POSTSUBSCRIPT ± 1.75 % end_POSTSUBSCRIPT</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S5.T2.36.36.39.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S5.T2.36.36.39.1.1"><span class="ltx_text" id="S5.T2.36.36.39.1.1.1" style="font-size:90%;">no edit</span></th> <td class="ltx_td ltx_align_center" id="S5.T2.36.36.39.1.2"><span class="ltx_text" id="S5.T2.36.36.39.1.2.1" style="font-size:90%;">47.14%</span></td> <td class="ltx_td ltx_align_center" id="S5.T2.36.36.39.1.3"><span class="ltx_text" id="S5.T2.36.36.39.1.3.1" style="font-size:90%;">51.74%</span></td> <td class="ltx_td" id="S5.T2.36.36.39.1.4"></td> <td class="ltx_td ltx_align_center" id="S5.T2.36.36.39.1.5"><span class="ltx_text" id="S5.T2.36.36.39.1.5.1" style="font-size:90%;">30.22%</span></td> <td class="ltx_td ltx_align_center" id="S5.T2.36.36.39.1.6"><span class="ltx_text" id="S5.T2.36.36.39.1.6.1" style="font-size:90%;">42.93%</span></td> <td class="ltx_td" id="S5.T2.36.36.39.1.7"></td> <td class="ltx_td ltx_align_center" id="S5.T2.36.36.39.1.8"><span class="ltx_text" id="S5.T2.36.36.39.1.8.1" style="font-size:90%;">39.26%</span></td> <td class="ltx_td ltx_align_center" id="S5.T2.36.36.39.1.9"><span class="ltx_text" id="S5.T2.36.36.39.1.9.1" style="font-size:90%;">47.04%</span></td> </tr> <tr class="ltx_tr" id="S5.T2.36.36.36"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb ltx_border_t" id="S5.T2.36.36.36.7"><span class="ltx_text ltx_font_italic" id="S5.T2.36.36.36.7.1" style="font-size:90%;">KGT</span></th> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T2.31.31.31.1"><math alttext="82.57_{\pm 2.82\%}" class="ltx_Math" display="inline" id="S5.T2.31.31.31.1.m1.1"><semantics id="S5.T2.31.31.31.1.m1.1a"><msub id="S5.T2.31.31.31.1.m1.1.1" xref="S5.T2.31.31.31.1.m1.1.1.cmml"><mn id="S5.T2.31.31.31.1.m1.1.1.2" mathsize="90%" xref="S5.T2.31.31.31.1.m1.1.1.2.cmml">82.57</mn><mrow id="S5.T2.31.31.31.1.m1.1.1.3" xref="S5.T2.31.31.31.1.m1.1.1.3.cmml"><mo id="S5.T2.31.31.31.1.m1.1.1.3a" mathsize="90%" xref="S5.T2.31.31.31.1.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.31.31.31.1.m1.1.1.3.2" xref="S5.T2.31.31.31.1.m1.1.1.3.2.cmml"><mn id="S5.T2.31.31.31.1.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.31.31.31.1.m1.1.1.3.2.2.cmml">2.82</mn><mo id="S5.T2.31.31.31.1.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.31.31.31.1.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.31.31.31.1.m1.1b"><apply id="S5.T2.31.31.31.1.m1.1.1.cmml" xref="S5.T2.31.31.31.1.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.31.31.31.1.m1.1.1.1.cmml" xref="S5.T2.31.31.31.1.m1.1.1">subscript</csymbol><cn id="S5.T2.31.31.31.1.m1.1.1.2.cmml" type="float" xref="S5.T2.31.31.31.1.m1.1.1.2">82.57</cn><apply id="S5.T2.31.31.31.1.m1.1.1.3.cmml" xref="S5.T2.31.31.31.1.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.31.31.31.1.m1.1.1.3.1.cmml" xref="S5.T2.31.31.31.1.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.31.31.31.1.m1.1.1.3.2.cmml" xref="S5.T2.31.31.31.1.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.31.31.31.1.m1.1.1.3.2.1.cmml" xref="S5.T2.31.31.31.1.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.31.31.31.1.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.31.31.31.1.m1.1.1.3.2.2">2.82</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.31.31.31.1.m1.1c">82.57_{\pm 2.82\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.31.31.31.1.m1.1d">82.57 start_POSTSUBSCRIPT ± 2.82 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T2.32.32.32.2"><math alttext="78.35\%_{\pm 3.26\%}" class="ltx_Math" display="inline" id="S5.T2.32.32.32.2.m1.1"><semantics id="S5.T2.32.32.32.2.m1.1a"><msub id="S5.T2.32.32.32.2.m1.1.1" xref="S5.T2.32.32.32.2.m1.1.1.cmml"><mrow id="S5.T2.32.32.32.2.m1.1.1.2" xref="S5.T2.32.32.32.2.m1.1.1.2.cmml"><mn id="S5.T2.32.32.32.2.m1.1.1.2.2" mathsize="90%" xref="S5.T2.32.32.32.2.m1.1.1.2.2.cmml">78.35</mn><mo id="S5.T2.32.32.32.2.m1.1.1.2.1" mathsize="90%" xref="S5.T2.32.32.32.2.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.32.32.32.2.m1.1.1.3" xref="S5.T2.32.32.32.2.m1.1.1.3.cmml"><mo id="S5.T2.32.32.32.2.m1.1.1.3a" mathsize="90%" xref="S5.T2.32.32.32.2.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.32.32.32.2.m1.1.1.3.2" xref="S5.T2.32.32.32.2.m1.1.1.3.2.cmml"><mn id="S5.T2.32.32.32.2.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.32.32.32.2.m1.1.1.3.2.2.cmml">3.26</mn><mo id="S5.T2.32.32.32.2.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.32.32.32.2.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.32.32.32.2.m1.1b"><apply id="S5.T2.32.32.32.2.m1.1.1.cmml" xref="S5.T2.32.32.32.2.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.32.32.32.2.m1.1.1.1.cmml" xref="S5.T2.32.32.32.2.m1.1.1">subscript</csymbol><apply id="S5.T2.32.32.32.2.m1.1.1.2.cmml" xref="S5.T2.32.32.32.2.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.32.32.32.2.m1.1.1.2.1.cmml" xref="S5.T2.32.32.32.2.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.32.32.32.2.m1.1.1.2.2.cmml" type="float" xref="S5.T2.32.32.32.2.m1.1.1.2.2">78.35</cn></apply><apply id="S5.T2.32.32.32.2.m1.1.1.3.cmml" xref="S5.T2.32.32.32.2.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.32.32.32.2.m1.1.1.3.1.cmml" xref="S5.T2.32.32.32.2.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.32.32.32.2.m1.1.1.3.2.cmml" xref="S5.T2.32.32.32.2.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.32.32.32.2.m1.1.1.3.2.1.cmml" xref="S5.T2.32.32.32.2.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.32.32.32.2.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.32.32.32.2.m1.1.1.3.2.2">3.26</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.32.32.32.2.m1.1c">78.35\%_{\pm 3.26\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.32.32.32.2.m1.1d">78.35 % start_POSTSUBSCRIPT ± 3.26 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_border_bb ltx_border_t" id="S5.T2.36.36.36.8"></td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T2.33.33.33.3"><math alttext="90.68_{\pm 0.74\%}" class="ltx_Math" display="inline" id="S5.T2.33.33.33.3.m1.1"><semantics id="S5.T2.33.33.33.3.m1.1a"><msub id="S5.T2.33.33.33.3.m1.1.1" xref="S5.T2.33.33.33.3.m1.1.1.cmml"><mn id="S5.T2.33.33.33.3.m1.1.1.2" mathsize="90%" xref="S5.T2.33.33.33.3.m1.1.1.2.cmml">90.68</mn><mrow id="S5.T2.33.33.33.3.m1.1.1.3" xref="S5.T2.33.33.33.3.m1.1.1.3.cmml"><mo id="S5.T2.33.33.33.3.m1.1.1.3a" mathsize="90%" xref="S5.T2.33.33.33.3.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.33.33.33.3.m1.1.1.3.2" xref="S5.T2.33.33.33.3.m1.1.1.3.2.cmml"><mn id="S5.T2.33.33.33.3.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.33.33.33.3.m1.1.1.3.2.2.cmml">0.74</mn><mo id="S5.T2.33.33.33.3.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.33.33.33.3.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.33.33.33.3.m1.1b"><apply id="S5.T2.33.33.33.3.m1.1.1.cmml" xref="S5.T2.33.33.33.3.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.33.33.33.3.m1.1.1.1.cmml" xref="S5.T2.33.33.33.3.m1.1.1">subscript</csymbol><cn id="S5.T2.33.33.33.3.m1.1.1.2.cmml" type="float" xref="S5.T2.33.33.33.3.m1.1.1.2">90.68</cn><apply id="S5.T2.33.33.33.3.m1.1.1.3.cmml" xref="S5.T2.33.33.33.3.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.33.33.33.3.m1.1.1.3.1.cmml" xref="S5.T2.33.33.33.3.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.33.33.33.3.m1.1.1.3.2.cmml" xref="S5.T2.33.33.33.3.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.33.33.33.3.m1.1.1.3.2.1.cmml" xref="S5.T2.33.33.33.3.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.33.33.33.3.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.33.33.33.3.m1.1.1.3.2.2">0.74</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.33.33.33.3.m1.1c">90.68_{\pm 0.74\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.33.33.33.3.m1.1d">90.68 start_POSTSUBSCRIPT ± 0.74 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T2.34.34.34.4"><math alttext="83.8_{\pm 1.20\%}" class="ltx_Math" display="inline" id="S5.T2.34.34.34.4.m1.1"><semantics id="S5.T2.34.34.34.4.m1.1a"><msub id="S5.T2.34.34.34.4.m1.1.1" xref="S5.T2.34.34.34.4.m1.1.1.cmml"><mn id="S5.T2.34.34.34.4.m1.1.1.2" mathsize="90%" xref="S5.T2.34.34.34.4.m1.1.1.2.cmml">83.8</mn><mrow id="S5.T2.34.34.34.4.m1.1.1.3" xref="S5.T2.34.34.34.4.m1.1.1.3.cmml"><mo id="S5.T2.34.34.34.4.m1.1.1.3a" mathsize="90%" xref="S5.T2.34.34.34.4.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.34.34.34.4.m1.1.1.3.2" xref="S5.T2.34.34.34.4.m1.1.1.3.2.cmml"><mn id="S5.T2.34.34.34.4.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.34.34.34.4.m1.1.1.3.2.2.cmml">1.20</mn><mo id="S5.T2.34.34.34.4.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.34.34.34.4.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.34.34.34.4.m1.1b"><apply id="S5.T2.34.34.34.4.m1.1.1.cmml" xref="S5.T2.34.34.34.4.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.34.34.34.4.m1.1.1.1.cmml" xref="S5.T2.34.34.34.4.m1.1.1">subscript</csymbol><cn id="S5.T2.34.34.34.4.m1.1.1.2.cmml" type="float" xref="S5.T2.34.34.34.4.m1.1.1.2">83.8</cn><apply id="S5.T2.34.34.34.4.m1.1.1.3.cmml" xref="S5.T2.34.34.34.4.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.34.34.34.4.m1.1.1.3.1.cmml" xref="S5.T2.34.34.34.4.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.34.34.34.4.m1.1.1.3.2.cmml" xref="S5.T2.34.34.34.4.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.34.34.34.4.m1.1.1.3.2.1.cmml" xref="S5.T2.34.34.34.4.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.34.34.34.4.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.34.34.34.4.m1.1.1.3.2.2">1.20</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.34.34.34.4.m1.1c">83.8_{\pm 1.20\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.34.34.34.4.m1.1d">83.8 start_POSTSUBSCRIPT ± 1.20 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_border_bb ltx_border_t" id="S5.T2.36.36.36.9"></td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T2.35.35.35.5"><math alttext="93.80\%_{\pm 0.36\%}" class="ltx_Math" display="inline" id="S5.T2.35.35.35.5.m1.1"><semantics id="S5.T2.35.35.35.5.m1.1a"><msub id="S5.T2.35.35.35.5.m1.1.1" xref="S5.T2.35.35.35.5.m1.1.1.cmml"><mrow id="S5.T2.35.35.35.5.m1.1.1.2" xref="S5.T2.35.35.35.5.m1.1.1.2.cmml"><mn id="S5.T2.35.35.35.5.m1.1.1.2.2" mathsize="90%" xref="S5.T2.35.35.35.5.m1.1.1.2.2.cmml">93.80</mn><mo id="S5.T2.35.35.35.5.m1.1.1.2.1" mathsize="90%" xref="S5.T2.35.35.35.5.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.35.35.35.5.m1.1.1.3" xref="S5.T2.35.35.35.5.m1.1.1.3.cmml"><mo id="S5.T2.35.35.35.5.m1.1.1.3a" mathsize="90%" xref="S5.T2.35.35.35.5.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.35.35.35.5.m1.1.1.3.2" xref="S5.T2.35.35.35.5.m1.1.1.3.2.cmml"><mn id="S5.T2.35.35.35.5.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.35.35.35.5.m1.1.1.3.2.2.cmml">0.36</mn><mo id="S5.T2.35.35.35.5.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.35.35.35.5.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.35.35.35.5.m1.1b"><apply id="S5.T2.35.35.35.5.m1.1.1.cmml" xref="S5.T2.35.35.35.5.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.35.35.35.5.m1.1.1.1.cmml" xref="S5.T2.35.35.35.5.m1.1.1">subscript</csymbol><apply id="S5.T2.35.35.35.5.m1.1.1.2.cmml" xref="S5.T2.35.35.35.5.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.35.35.35.5.m1.1.1.2.1.cmml" xref="S5.T2.35.35.35.5.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.35.35.35.5.m1.1.1.2.2.cmml" type="float" xref="S5.T2.35.35.35.5.m1.1.1.2.2">93.80</cn></apply><apply id="S5.T2.35.35.35.5.m1.1.1.3.cmml" xref="S5.T2.35.35.35.5.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.35.35.35.5.m1.1.1.3.1.cmml" xref="S5.T2.35.35.35.5.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.35.35.35.5.m1.1.1.3.2.cmml" xref="S5.T2.35.35.35.5.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.35.35.35.5.m1.1.1.3.2.1.cmml" xref="S5.T2.35.35.35.5.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.35.35.35.5.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.35.35.35.5.m1.1.1.3.2.2">0.36</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.35.35.35.5.m1.1c">93.80\%_{\pm 0.36\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.35.35.35.5.m1.1d">93.80 % start_POSTSUBSCRIPT ± 0.36 % end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T2.36.36.36.6"><math alttext="89.22\%_{\pm 1.17\%}" class="ltx_Math" display="inline" id="S5.T2.36.36.36.6.m1.1"><semantics id="S5.T2.36.36.36.6.m1.1a"><msub id="S5.T2.36.36.36.6.m1.1.1" xref="S5.T2.36.36.36.6.m1.1.1.cmml"><mrow id="S5.T2.36.36.36.6.m1.1.1.2" xref="S5.T2.36.36.36.6.m1.1.1.2.cmml"><mn id="S5.T2.36.36.36.6.m1.1.1.2.2" mathsize="90%" xref="S5.T2.36.36.36.6.m1.1.1.2.2.cmml">89.22</mn><mo id="S5.T2.36.36.36.6.m1.1.1.2.1" mathsize="90%" xref="S5.T2.36.36.36.6.m1.1.1.2.1.cmml">%</mo></mrow><mrow id="S5.T2.36.36.36.6.m1.1.1.3" xref="S5.T2.36.36.36.6.m1.1.1.3.cmml"><mo id="S5.T2.36.36.36.6.m1.1.1.3a" mathsize="90%" xref="S5.T2.36.36.36.6.m1.1.1.3.cmml">±</mo><mrow id="S5.T2.36.36.36.6.m1.1.1.3.2" xref="S5.T2.36.36.36.6.m1.1.1.3.2.cmml"><mn id="S5.T2.36.36.36.6.m1.1.1.3.2.2" mathsize="90%" xref="S5.T2.36.36.36.6.m1.1.1.3.2.2.cmml">1.17</mn><mo id="S5.T2.36.36.36.6.m1.1.1.3.2.1" mathsize="90%" xref="S5.T2.36.36.36.6.m1.1.1.3.2.1.cmml">%</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.T2.36.36.36.6.m1.1b"><apply id="S5.T2.36.36.36.6.m1.1.1.cmml" xref="S5.T2.36.36.36.6.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.36.36.36.6.m1.1.1.1.cmml" xref="S5.T2.36.36.36.6.m1.1.1">subscript</csymbol><apply id="S5.T2.36.36.36.6.m1.1.1.2.cmml" xref="S5.T2.36.36.36.6.m1.1.1.2"><csymbol cd="latexml" id="S5.T2.36.36.36.6.m1.1.1.2.1.cmml" xref="S5.T2.36.36.36.6.m1.1.1.2.1">percent</csymbol><cn id="S5.T2.36.36.36.6.m1.1.1.2.2.cmml" type="float" xref="S5.T2.36.36.36.6.m1.1.1.2.2">89.22</cn></apply><apply id="S5.T2.36.36.36.6.m1.1.1.3.cmml" xref="S5.T2.36.36.36.6.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.36.36.36.6.m1.1.1.3.1.cmml" xref="S5.T2.36.36.36.6.m1.1.1.3">plus-or-minus</csymbol><apply id="S5.T2.36.36.36.6.m1.1.1.3.2.cmml" xref="S5.T2.36.36.36.6.m1.1.1.3.2"><csymbol cd="latexml" id="S5.T2.36.36.36.6.m1.1.1.3.2.1.cmml" xref="S5.T2.36.36.36.6.m1.1.1.3.2.1">percent</csymbol><cn id="S5.T2.36.36.36.6.m1.1.1.3.2.2.cmml" type="float" xref="S5.T2.36.36.36.6.m1.1.1.3.2.2">1.17</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.36.36.36.6.m1.1c">89.22\%_{\pm 1.17\%}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.36.36.36.6.m1.1d">89.22 % start_POSTSUBSCRIPT ± 1.17 % end_POSTSUBSCRIPT</annotation></semantics></math></td> </tr> </tbody> </table> </span></div> </figure> <div class="ltx_para" id="S5.SS2.p3"> <p class="ltx_p" id="S5.SS2.p3.1">The improvement in both efficacy and paraphrase rates demonstrates that KGT outperforms the baseline in personalization performance significantly.</p> </div> </section> <section class="ltx_subsection" id="S5.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">5.3 </span>Efficiency Comparison</h3> <div class="ltx_para" id="S5.SS3.p1"> <p class="ltx_p" id="S5.SS3.p1.1">In addition to the personalization performance, we also evaluate KGT’s efficiency in terms of latency and GPU memory cost. The method must achieve low latency and low GPU memory cost to realize <span class="ltx_text ltx_font_bold" id="S5.SS3.p1.1.1">real-time personalization</span> in practice under the setting with limited computational resources. KGT can improve the time and GPU memory cost efficiency because it only requires inference, which is far more efficient than back-propagation. For latency, we test the average time that our method and the baselines require to complete the personalization for one query-answer pair. The latency results on <span class="ltx_text ltx_font_italic" id="S5.SS3.p1.1.2">CounterFact</span> data are shown in <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S5.T3" title="In 5.3 Efficiency Comparison ‣ 5 Experiments ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_tag">Table</span> <span class="ltx_text ltx_ref_tag">3</span></a>. The results demonstrate that KGT achieves the shortest latency in most cases. Notably, Llama3 requires less time on several baselines than Llama2 and GPT2 because we stop training once the loss converges. The results on the GPU memory cost can also be found in <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S5.T3" title="In 5.3 Efficiency Comparison ‣ 5 Experiments ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_tag">Table</span> <span class="ltx_text ltx_ref_tag">3</span></a>. The memory cost is reduced significantly because only the inference is required for KGT. Specifically, KGT reduces 57%, 56%, 77%, 63%, and 62% GPU memory cost when adopting Llama3-8B compared with FT, KE, KN, and MEND, respectively.</p> </div> <figure class="ltx_table" id="S5.T3"> <figcaption class="ltx_caption ltx_centering" style="font-size:90%;"><span class="ltx_tag ltx_tag_table">Table 3: </span>Results of latency and GPU memory costs on <span class="ltx_text ltx_font_italic" id="S5.T3.5.1">CounterFact</span> dataset when the user only provides the answers to the queries as feedback.</figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S5.T3.6"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S5.T3.6.1.1"> <th class="ltx_td ltx_th ltx_th_row ltx_border_tt" id="S5.T3.6.1.1.1"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="2" id="S5.T3.6.1.1.2"><span class="ltx_text ltx_font_bold" id="S5.T3.6.1.1.2.1" style="font-size:90%;">GPT2</span></th> <th class="ltx_td ltx_th ltx_th_column ltx_border_tt" id="S5.T3.6.1.1.3"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="2" id="S5.T3.6.1.1.4"><span class="ltx_text ltx_font_bold" id="S5.T3.6.1.1.4.1" style="font-size:90%;">Llama2-7B</span></th> <th class="ltx_td ltx_th ltx_th_column ltx_border_tt" id="S5.T3.6.1.1.5"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" colspan="2" id="S5.T3.6.1.1.6"><span class="ltx_text ltx_font_bold" id="S5.T3.6.1.1.6.1" style="font-size:90%;">Llama3-8B</span></th> </tr> <tr class="ltx_tr" id="S5.T3.6.2.2"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row" id="S5.T3.6.2.2.1"><span class="ltx_text ltx_font_bold" id="S5.T3.6.2.2.1.1" style="font-size:90%;">Method</span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T3.6.2.2.2"> <table class="ltx_tabular ltx_align_middle" id="S5.T3.6.2.2.2.1"> <tr class="ltx_tr" id="S5.T3.6.2.2.2.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T3.6.2.2.2.1.1.1"><span class="ltx_text" id="S5.T3.6.2.2.2.1.1.1.1" style="font-size:90%;">Memory</span></td> </tr> </table> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T3.6.2.2.3"> <table class="ltx_tabular ltx_align_middle" id="S5.T3.6.2.2.3.1"> <tr class="ltx_tr" id="S5.T3.6.2.2.3.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T3.6.2.2.3.1.1.1"><span class="ltx_text" id="S5.T3.6.2.2.3.1.1.1.1" style="font-size:90%;">Latency</span></td> </tr> </table> </th> <th class="ltx_td ltx_th ltx_th_column" id="S5.T3.6.2.2.4"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T3.6.2.2.5"> <table class="ltx_tabular ltx_align_middle" id="S5.T3.6.2.2.5.1"> <tr class="ltx_tr" id="S5.T3.6.2.2.5.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T3.6.2.2.5.1.1.1"><span class="ltx_text" id="S5.T3.6.2.2.5.1.1.1.1" style="font-size:90%;">Memory</span></td> </tr> </table> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T3.6.2.2.6"> <table class="ltx_tabular ltx_align_middle" id="S5.T3.6.2.2.6.1"> <tr class="ltx_tr" id="S5.T3.6.2.2.6.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T3.6.2.2.6.1.1.1"><span class="ltx_text" id="S5.T3.6.2.2.6.1.1.1.1" style="font-size:90%;">Latency</span></td> </tr> </table> </th> <th class="ltx_td ltx_th ltx_th_column" id="S5.T3.6.2.2.7"></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T3.6.2.2.8"> <table class="ltx_tabular ltx_align_middle" id="S5.T3.6.2.2.8.1"> <tr class="ltx_tr" id="S5.T3.6.2.2.8.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T3.6.2.2.8.1.1.1"><span class="ltx_text" id="S5.T3.6.2.2.8.1.1.1.1" style="font-size:90%;">Memory</span></td> </tr> </table> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T3.6.2.2.9"> <table class="ltx_tabular ltx_align_middle" id="S5.T3.6.2.2.9.1"> <tr class="ltx_tr" id="S5.T3.6.2.2.9.1.1"> <td class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T3.6.2.2.9.1.1.1"><span class="ltx_text" id="S5.T3.6.2.2.9.1.1.1.1" style="font-size:90%;">Latency</span></td> </tr> </table> </th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S5.T3.6.3.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_tt" id="S5.T3.6.3.1.1"><span class="ltx_text" id="S5.T3.6.3.1.1.1" style="font-size:90%;">FT</span></th> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T3.6.3.1.2"><span class="ltx_text" id="S5.T3.6.3.1.2.1" style="font-size:90%;">8516MB</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T3.6.3.1.3"><span class="ltx_text" id="S5.T3.6.3.1.3.1" style="font-size:90%;">1.80s</span></td> <td class="ltx_td ltx_border_tt" id="S5.T3.6.3.1.4"></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T3.6.3.1.5"><span class="ltx_text" id="S5.T3.6.3.1.5.1" style="font-size:90%;">30990MB</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T3.6.3.1.6"><span class="ltx_text" id="S5.T3.6.3.1.6.1" style="font-size:90%;">0.81s</span></td> <td class="ltx_td ltx_border_tt" id="S5.T3.6.3.1.7"></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T3.6.3.1.8"><span class="ltx_text" id="S5.T3.6.3.1.8.1" style="font-size:90%;">36968MB</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S5.T3.6.3.1.9"><span class="ltx_text" id="S5.T3.6.3.1.9.1" style="font-size:90%;">0.25s</span></td> </tr> <tr class="ltx_tr" id="S5.T3.6.4.2"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S5.T3.6.4.2.1"><span class="ltx_text" id="S5.T3.6.4.2.1.1" style="font-size:90%;">ROME</span></th> <td class="ltx_td ltx_align_center" id="S5.T3.6.4.2.2"><span class="ltx_text" id="S5.T3.6.4.2.2.1" style="font-size:90%;">11948MB</span></td> <td class="ltx_td ltx_align_center" id="S5.T3.6.4.2.3"><span class="ltx_text" id="S5.T3.6.4.2.3.1" style="font-size:90%;">1.39s</span></td> <td class="ltx_td" id="S5.T3.6.4.2.4"></td> <td class="ltx_td ltx_align_center" id="S5.T3.6.4.2.5"><span class="ltx_text" id="S5.T3.6.4.2.5.1" style="font-size:90%;">30452MB</span></td> <td class="ltx_td ltx_align_center" id="S5.T3.6.4.2.6"><span class="ltx_text" id="S5.T3.6.4.2.6.1" style="font-size:90%;">2.33s</span></td> <td class="ltx_td" id="S5.T3.6.4.2.7"></td> <td class="ltx_td ltx_align_center" id="S5.T3.6.4.2.8"><span class="ltx_text" id="S5.T3.6.4.2.8.1" style="font-size:90%;">36660MB</span></td> <td class="ltx_td ltx_align_center" id="S5.T3.6.4.2.9"><span class="ltx_text" id="S5.T3.6.4.2.9.1" style="font-size:90%;">2.05s</span></td> </tr> <tr class="ltx_tr" id="S5.T3.6.5.3"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S5.T3.6.5.3.1"><span class="ltx_text" id="S5.T3.6.5.3.1.1" style="font-size:90%;">KE</span></th> <td class="ltx_td ltx_align_center" id="S5.T3.6.5.3.2"><span class="ltx_text" id="S5.T3.6.5.3.2.1" style="font-size:90%;">31574MB</span></td> <td class="ltx_td ltx_align_center" id="S5.T3.6.5.3.3"><span class="ltx_text" id="S5.T3.6.5.3.3.1" style="font-size:90%;">2.18s</span></td> <td class="ltx_td" id="S5.T3.6.5.3.4"></td> <td class="ltx_td ltx_align_center" id="S5.T3.6.5.3.5"><span class="ltx_text" id="S5.T3.6.5.3.5.1" style="font-size:90%;">33464MB</span></td> <td class="ltx_td ltx_align_center" id="S5.T3.6.5.3.6"><span class="ltx_text" id="S5.T3.6.5.3.6.1" style="font-size:90%;">0.30s</span></td> <td class="ltx_td" id="S5.T3.6.5.3.7"></td> <td class="ltx_td ltx_align_center" id="S5.T3.6.5.3.8"><span class="ltx_text" id="S5.T3.6.5.3.8.1" style="font-size:90%;">69542MB</span></td> <td class="ltx_td ltx_align_center" id="S5.T3.6.5.3.9"><span class="ltx_text" id="S5.T3.6.5.3.9.1" style="font-size:90%;">0.13s</span></td> </tr> <tr class="ltx_tr" id="S5.T3.6.6.4"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S5.T3.6.6.4.1"><span class="ltx_text" id="S5.T3.6.6.4.1.1" style="font-size:90%;">KN</span></th> <td class="ltx_td ltx_align_center" id="S5.T3.6.6.4.2"><span class="ltx_text" id="S5.T3.6.6.4.2.1" style="font-size:90%;">12832MB</span></td> <td class="ltx_td ltx_align_center" id="S5.T3.6.6.4.3"><span class="ltx_text" id="S5.T3.6.6.4.3.1" style="font-size:90%;">3.55s</span></td> <td class="ltx_td" id="S5.T3.6.6.4.4"></td> <td class="ltx_td ltx_align_center" id="S5.T3.6.6.4.5"><span class="ltx_text" id="S5.T3.6.6.4.5.1" style="font-size:90%;">56148MB</span></td> <td class="ltx_td ltx_align_center" id="S5.T3.6.6.4.6"><span class="ltx_text" id="S5.T3.6.6.4.6.1" style="font-size:90%;">0.69s</span></td> <td class="ltx_td" id="S5.T3.6.6.4.7"></td> <td class="ltx_td ltx_align_center" id="S5.T3.6.6.4.8"><span class="ltx_text" id="S5.T3.6.6.4.8.1" style="font-size:90%;">44000MB</span></td> <td class="ltx_td ltx_align_center" id="S5.T3.6.6.4.9"><span class="ltx_text" id="S5.T3.6.6.4.9.1" style="font-size:90%;">0.34s</span></td> </tr> <tr class="ltx_tr" id="S5.T3.6.7.5"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="S5.T3.6.7.5.1"><span class="ltx_text" id="S5.T3.6.7.5.1.1" style="font-size:90%;">MEND</span></th> <td class="ltx_td ltx_align_center" id="S5.T3.6.7.5.2"><span class="ltx_text" id="S5.T3.6.7.5.2.1" style="font-size:90%;">11036MB</span></td> <td class="ltx_td ltx_align_center" id="S5.T3.6.7.5.3"><span class="ltx_text" id="S5.T3.6.7.5.3.1" style="font-size:90%;">0.86s</span></td> <td class="ltx_td" id="S5.T3.6.7.5.4"></td> <td class="ltx_td ltx_align_center" id="S5.T3.6.7.5.5"><span class="ltx_text" id="S5.T3.6.7.5.5.1" style="font-size:90%;">35166MB</span></td> <td class="ltx_td ltx_align_center" id="S5.T3.6.7.5.6"><span class="ltx_text" id="S5.T3.6.7.5.6.1" style="font-size:90%;">1.98s</span></td> <td class="ltx_td" id="S5.T3.6.7.5.7"></td> <td class="ltx_td ltx_align_center" id="S5.T3.6.7.5.8"><span class="ltx_text" id="S5.T3.6.7.5.8.1" style="font-size:90%;">42428MB</span></td> <td class="ltx_td ltx_align_center" id="S5.T3.6.7.5.9"><span class="ltx_text" id="S5.T3.6.7.5.9.1" style="font-size:90%;">1.40s</span></td> </tr> <tr class="ltx_tr" id="S5.T3.6.8.6"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb ltx_border_t" id="S5.T3.6.8.6.1"><span class="ltx_text ltx_font_italic" id="S5.T3.6.8.6.1.1" style="font-size:90%;">KGT</span></th> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T3.6.8.6.2"><span class="ltx_text" id="S5.T3.6.8.6.2.1" style="font-size:90%;">6686MB</span></td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T3.6.8.6.3"><span class="ltx_text" id="S5.T3.6.8.6.3.1" style="font-size:90%;">0.16s</span></td> <td class="ltx_td ltx_border_bb ltx_border_t" id="S5.T3.6.8.6.4"></td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T3.6.8.6.5"><span class="ltx_text" id="S5.T3.6.8.6.5.1" style="font-size:90%;">13516MB</span></td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T3.6.8.6.6"><span class="ltx_text" id="S5.T3.6.8.6.6.1" style="font-size:90%;">0.14s</span></td> <td class="ltx_td ltx_border_bb ltx_border_t" id="S5.T3.6.8.6.7"></td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T3.6.8.6.8"><span class="ltx_text" id="S5.T3.6.8.6.8.1" style="font-size:90%;">15904MB</span></td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S5.T3.6.8.6.9"><span class="ltx_text" id="S5.T3.6.8.6.9.1" style="font-size:90%;">0.15s</span></td> </tr> </tbody> </table> </figure> </section> <section class="ltx_subsection" id="S5.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">5.4 </span>Ablation Study</h3> <section class="ltx_subsubsection" id="S5.SS4.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">5.4.1 </span>Does User Need to Provide Feedback of Relations?</h4> <div class="ltx_para" id="S5.SS4.SSS1.p1"> <p class="ltx_p" id="S5.SS4.SSS1.p1.6">We also conduct experiments under the setting where the user also provides feedback on relations <math alttext="r" class="ltx_Math" display="inline" id="S5.SS4.SSS1.p1.1.m1.1"><semantics id="S5.SS4.SSS1.p1.1.m1.1a"><mi id="S5.SS4.SSS1.p1.1.m1.1.1" xref="S5.SS4.SSS1.p1.1.m1.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S5.SS4.SSS1.p1.1.m1.1b"><ci id="S5.SS4.SSS1.p1.1.m1.1.1.cmml" xref="S5.SS4.SSS1.p1.1.m1.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.SSS1.p1.1.m1.1c">r</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.SSS1.p1.1.m1.1d">italic_r</annotation></semantics></math> between <math alttext="e_{q}" class="ltx_Math" display="inline" id="S5.SS4.SSS1.p1.2.m2.1"><semantics id="S5.SS4.SSS1.p1.2.m2.1a"><msub id="S5.SS4.SSS1.p1.2.m2.1.1" xref="S5.SS4.SSS1.p1.2.m2.1.1.cmml"><mi id="S5.SS4.SSS1.p1.2.m2.1.1.2" xref="S5.SS4.SSS1.p1.2.m2.1.1.2.cmml">e</mi><mi id="S5.SS4.SSS1.p1.2.m2.1.1.3" xref="S5.SS4.SSS1.p1.2.m2.1.1.3.cmml">q</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS4.SSS1.p1.2.m2.1b"><apply id="S5.SS4.SSS1.p1.2.m2.1.1.cmml" xref="S5.SS4.SSS1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.SS4.SSS1.p1.2.m2.1.1.1.cmml" xref="S5.SS4.SSS1.p1.2.m2.1.1">subscript</csymbol><ci id="S5.SS4.SSS1.p1.2.m2.1.1.2.cmml" xref="S5.SS4.SSS1.p1.2.m2.1.1.2">𝑒</ci><ci id="S5.SS4.SSS1.p1.2.m2.1.1.3.cmml" xref="S5.SS4.SSS1.p1.2.m2.1.1.3">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.SSS1.p1.2.m2.1c">e_{q}</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.SSS1.p1.2.m2.1d">italic_e start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="e_{a}" class="ltx_Math" display="inline" id="S5.SS4.SSS1.p1.3.m3.1"><semantics id="S5.SS4.SSS1.p1.3.m3.1a"><msub id="S5.SS4.SSS1.p1.3.m3.1.1" xref="S5.SS4.SSS1.p1.3.m3.1.1.cmml"><mi id="S5.SS4.SSS1.p1.3.m3.1.1.2" xref="S5.SS4.SSS1.p1.3.m3.1.1.2.cmml">e</mi><mi id="S5.SS4.SSS1.p1.3.m3.1.1.3" xref="S5.SS4.SSS1.p1.3.m3.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS4.SSS1.p1.3.m3.1b"><apply id="S5.SS4.SSS1.p1.3.m3.1.1.cmml" xref="S5.SS4.SSS1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.SS4.SSS1.p1.3.m3.1.1.1.cmml" xref="S5.SS4.SSS1.p1.3.m3.1.1">subscript</csymbol><ci id="S5.SS4.SSS1.p1.3.m3.1.1.2.cmml" xref="S5.SS4.SSS1.p1.3.m3.1.1.2">𝑒</ci><ci id="S5.SS4.SSS1.p1.3.m3.1.1.3.cmml" xref="S5.SS4.SSS1.p1.3.m3.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.SSS1.p1.3.m3.1c">e_{a}</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.SSS1.p1.3.m3.1d">italic_e start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> under the query’s context. We utilize GPT-4 to mimic the user and extract the relations to construct <math alttext="\mathcal{H}\left(q,a,K\right)" class="ltx_Math" display="inline" id="S5.SS4.SSS1.p1.4.m4.3"><semantics id="S5.SS4.SSS1.p1.4.m4.3a"><mrow id="S5.SS4.SSS1.p1.4.m4.3.4" xref="S5.SS4.SSS1.p1.4.m4.3.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS4.SSS1.p1.4.m4.3.4.2" xref="S5.SS4.SSS1.p1.4.m4.3.4.2.cmml">ℋ</mi><mo id="S5.SS4.SSS1.p1.4.m4.3.4.1" xref="S5.SS4.SSS1.p1.4.m4.3.4.1.cmml">⁢</mo><mrow id="S5.SS4.SSS1.p1.4.m4.3.4.3.2" xref="S5.SS4.SSS1.p1.4.m4.3.4.3.1.cmml"><mo id="S5.SS4.SSS1.p1.4.m4.3.4.3.2.1" xref="S5.SS4.SSS1.p1.4.m4.3.4.3.1.cmml">(</mo><mi id="S5.SS4.SSS1.p1.4.m4.1.1" xref="S5.SS4.SSS1.p1.4.m4.1.1.cmml">q</mi><mo id="S5.SS4.SSS1.p1.4.m4.3.4.3.2.2" xref="S5.SS4.SSS1.p1.4.m4.3.4.3.1.cmml">,</mo><mi id="S5.SS4.SSS1.p1.4.m4.2.2" xref="S5.SS4.SSS1.p1.4.m4.2.2.cmml">a</mi><mo id="S5.SS4.SSS1.p1.4.m4.3.4.3.2.3" xref="S5.SS4.SSS1.p1.4.m4.3.4.3.1.cmml">,</mo><mi id="S5.SS4.SSS1.p1.4.m4.3.3" xref="S5.SS4.SSS1.p1.4.m4.3.3.cmml">K</mi><mo id="S5.SS4.SSS1.p1.4.m4.3.4.3.2.4" xref="S5.SS4.SSS1.p1.4.m4.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.SSS1.p1.4.m4.3b"><apply id="S5.SS4.SSS1.p1.4.m4.3.4.cmml" xref="S5.SS4.SSS1.p1.4.m4.3.4"><times id="S5.SS4.SSS1.p1.4.m4.3.4.1.cmml" xref="S5.SS4.SSS1.p1.4.m4.3.4.1"></times><ci id="S5.SS4.SSS1.p1.4.m4.3.4.2.cmml" xref="S5.SS4.SSS1.p1.4.m4.3.4.2">ℋ</ci><vector id="S5.SS4.SSS1.p1.4.m4.3.4.3.1.cmml" xref="S5.SS4.SSS1.p1.4.m4.3.4.3.2"><ci id="S5.SS4.SSS1.p1.4.m4.1.1.cmml" xref="S5.SS4.SSS1.p1.4.m4.1.1">𝑞</ci><ci id="S5.SS4.SSS1.p1.4.m4.2.2.cmml" xref="S5.SS4.SSS1.p1.4.m4.2.2">𝑎</ci><ci id="S5.SS4.SSS1.p1.4.m4.3.3.cmml" xref="S5.SS4.SSS1.p1.4.m4.3.3">𝐾</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.SSS1.p1.4.m4.3c">\mathcal{H}\left(q,a,K\right)</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.SSS1.p1.4.m4.3d">caligraphic_H ( italic_q , italic_a , italic_K )</annotation></semantics></math>. We use the same instruction template in <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S4.SS2" title="4.2 Objective Formulation ‣ 4 Method ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4.2</span></a> for the GPT-4 to conduct relation extraction. The compared results with and without the user’s feedback of relations are shown in <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S5.F3" title="In 5.4.1 Does User Need to Provide Feedback of Relations? ‣ 5.4 Ablation Study ‣ 5 Experiments ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">3</span></a>. It is shown that KGT achieves similar performance if the user provides feedback on the relations in addition to the answer. Notably, utilizing the relations extracted by the LLM <math alttext="\theta" class="ltx_Math" display="inline" id="S5.SS4.SSS1.p1.5.m5.1"><semantics id="S5.SS4.SSS1.p1.5.m5.1a"><mi id="S5.SS4.SSS1.p1.5.m5.1.1" xref="S5.SS4.SSS1.p1.5.m5.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S5.SS4.SSS1.p1.5.m5.1b"><ci id="S5.SS4.SSS1.p1.5.m5.1.1.cmml" xref="S5.SS4.SSS1.p1.5.m5.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.SSS1.p1.5.m5.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.SSS1.p1.5.m5.1d">italic_θ</annotation></semantics></math> can even achieve higher performance on efficacy and paraphrase rates in most cases. Our analysis is that when extracting relations using the LLM <math alttext="\theta" class="ltx_Math" display="inline" id="S5.SS4.SSS1.p1.6.m6.1"><semantics id="S5.SS4.SSS1.p1.6.m6.1a"><mi id="S5.SS4.SSS1.p1.6.m6.1.1" xref="S5.SS4.SSS1.p1.6.m6.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S5.SS4.SSS1.p1.6.m6.1b"><ci id="S5.SS4.SSS1.p1.6.m6.1.1.cmml" xref="S5.SS4.SSS1.p1.6.m6.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.SSS1.p1.6.m6.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.SSS1.p1.6.m6.1d">italic_θ</annotation></semantics></math>, KGT implicitly distills knowledge from the model to the knowledge triples, which might benefit the model inference more than human feedback on relations. Thus, in practice, the user will only need to provide the personalized answer to a query as feedback to KGT.</p> </div> <figure class="ltx_figure" id="S5.F3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="118" id="S5.F3.g1" src="x3.png" width="479"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span>Compared results on <span class="ltx_text ltx_font_italic" id="S5.F3.2.1">CounterFact</span> dataset with and without user feedback on relations.</figcaption> </figure> </section> <section class="ltx_subsubsection" id="S5.SS4.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">5.4.2 </span>Effect of the Size of Query Set</h4> <div class="ltx_para" id="S5.SS4.SSS2.p1"> <p class="ltx_p" id="S5.SS4.SSS2.p1.1">We also evaluate the effect of the query set size on the personalization performance. 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The results are shown in <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S5.F4" title="In 5.4.2 Effect of the Size of Query Set ‣ 5.4 Ablation Study ‣ 5 Experiments ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">4</span></a>. It is shown that when the size of the query set increases, the performance of baselines degrades dramatically, while KGT can preserve high performance. This compared result illustrates that KGT can be scaled to a large amount of personalized knowledge. This scalability is crucial for meeting the long-term needs of users who require the accumulation of extensive personalized knowledge when using LLMs.</p> </div> <figure class="ltx_figure" id="S5.F4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="129" id="S5.F4.g1" src="x4.png" width="399"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 4: </span>Compared results on <span class="ltx_text ltx_font_italic" id="S5.F4.2.1">CounterFact</span> dataset using Llama3-8B with different query set sizes.</figcaption> </figure> </section> </section> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">6 </span>Conclusion and Limitation</h2> <div class="ltx_para" id="S6.p1"> <p class="ltx_p" id="S6.p1.3">We propose an approach, KGT, that personalizes models by optimizing external knowledge graph rather than model parameters. Our method and experimental results demonstrate that our approach offers benefits in terms of performance and efficiency, as supported by empirical studies. By addressing the critical challenges of efficiency and interpretability in model personalization, KGT offers a promising direction for future research and application in enhancing user interactions with LLMs, which has a positive societal impact. 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By approximating the <math alttext="Q(z)" class="ltx_Math" display="inline" id="A1.p1.1.m1.1"><semantics id="A1.p1.1.m1.1a"><mrow id="A1.p1.1.m1.1.2" xref="A1.p1.1.m1.1.2.cmml"><mi id="A1.p1.1.m1.1.2.2" xref="A1.p1.1.m1.1.2.2.cmml">Q</mi><mo id="A1.p1.1.m1.1.2.1" xref="A1.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="A1.p1.1.m1.1.2.3.2" xref="A1.p1.1.m1.1.2.cmml"><mo id="A1.p1.1.m1.1.2.3.2.1" stretchy="false" xref="A1.p1.1.m1.1.2.cmml">(</mo><mi id="A1.p1.1.m1.1.1" xref="A1.p1.1.m1.1.1.cmml">z</mi><mo id="A1.p1.1.m1.1.2.3.2.2" stretchy="false" xref="A1.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.1.m1.1b"><apply id="A1.p1.1.m1.1.2.cmml" xref="A1.p1.1.m1.1.2"><times id="A1.p1.1.m1.1.2.1.cmml" xref="A1.p1.1.m1.1.2.1"></times><ci id="A1.p1.1.m1.1.2.2.cmml" xref="A1.p1.1.m1.1.2.2">𝑄</ci><ci id="A1.p1.1.m1.1.1.cmml" xref="A1.p1.1.m1.1.1">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.1.m1.1c">Q(z)</annotation><annotation encoding="application/x-llamapun" id="A1.p1.1.m1.1d">italic_Q ( italic_z )</annotation></semantics></math> with <math alttext="Q(z|q,a)" class="ltx_Math" display="inline" id="A1.p1.2.m2.3"><semantics id="A1.p1.2.m2.3a"><mrow id="A1.p1.2.m2.3.3" xref="A1.p1.2.m2.3.3.cmml"><mi id="A1.p1.2.m2.3.3.3" xref="A1.p1.2.m2.3.3.3.cmml">Q</mi><mo id="A1.p1.2.m2.3.3.2" xref="A1.p1.2.m2.3.3.2.cmml">⁢</mo><mrow id="A1.p1.2.m2.3.3.1.1" xref="A1.p1.2.m2.3.3.1.1.1.cmml"><mo id="A1.p1.2.m2.3.3.1.1.2" stretchy="false" xref="A1.p1.2.m2.3.3.1.1.1.cmml">(</mo><mrow id="A1.p1.2.m2.3.3.1.1.1" xref="A1.p1.2.m2.3.3.1.1.1.cmml"><mi id="A1.p1.2.m2.3.3.1.1.1.2" xref="A1.p1.2.m2.3.3.1.1.1.2.cmml">z</mi><mo fence="false" id="A1.p1.2.m2.3.3.1.1.1.1" xref="A1.p1.2.m2.3.3.1.1.1.1.cmml">|</mo><mrow id="A1.p1.2.m2.3.3.1.1.1.3.2" xref="A1.p1.2.m2.3.3.1.1.1.3.1.cmml"><mi id="A1.p1.2.m2.1.1" xref="A1.p1.2.m2.1.1.cmml">q</mi><mo id="A1.p1.2.m2.3.3.1.1.1.3.2.1" xref="A1.p1.2.m2.3.3.1.1.1.3.1.cmml">,</mo><mi id="A1.p1.2.m2.2.2" xref="A1.p1.2.m2.2.2.cmml">a</mi></mrow></mrow><mo id="A1.p1.2.m2.3.3.1.1.3" stretchy="false" xref="A1.p1.2.m2.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.2.m2.3b"><apply id="A1.p1.2.m2.3.3.cmml" xref="A1.p1.2.m2.3.3"><times id="A1.p1.2.m2.3.3.2.cmml" xref="A1.p1.2.m2.3.3.2"></times><ci id="A1.p1.2.m2.3.3.3.cmml" xref="A1.p1.2.m2.3.3.3">𝑄</ci><apply id="A1.p1.2.m2.3.3.1.1.1.cmml" xref="A1.p1.2.m2.3.3.1.1"><csymbol cd="latexml" id="A1.p1.2.m2.3.3.1.1.1.1.cmml" xref="A1.p1.2.m2.3.3.1.1.1.1">conditional</csymbol><ci id="A1.p1.2.m2.3.3.1.1.1.2.cmml" xref="A1.p1.2.m2.3.3.1.1.1.2">𝑧</ci><list id="A1.p1.2.m2.3.3.1.1.1.3.1.cmml" xref="A1.p1.2.m2.3.3.1.1.1.3.2"><ci id="A1.p1.2.m2.1.1.cmml" xref="A1.p1.2.m2.1.1">𝑞</ci><ci id="A1.p1.2.m2.2.2.cmml" xref="A1.p1.2.m2.2.2">𝑎</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.2.m2.3c">Q(z|q,a)</annotation><annotation encoding="application/x-llamapun" id="A1.p1.2.m2.3d">italic_Q ( italic_z | italic_q , italic_a )</annotation></semantics></math>, the KL divergence is calculated as</p> </div> <div class="ltx_para" id="A1.p2"> <table class="ltx_equationgroup ltx_eqn_table" id="A1.E8"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="A1.E8X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{L}_{\text{retrieve}}=D_{KL}\left(Q(z)||P_{\theta,% \mathcal{G}}\left(z|q\right)\right)" class="ltx_math_unparsed" display="inline" id="A1.E8X.2.1.1.m1.3"><semantics id="A1.E8X.2.1.1.m1.3a"><mrow id="A1.E8X.2.1.1.m1.3b"><msub id="A1.E8X.2.1.1.m1.3.4"><mi class="ltx_font_mathcaligraphic" id="A1.E8X.2.1.1.m1.3.4.2">ℒ</mi><mtext id="A1.E8X.2.1.1.m1.3.4.3">retrieve</mtext></msub><mo id="A1.E8X.2.1.1.m1.3.5">=</mo><msub id="A1.E8X.2.1.1.m1.3.6"><mi id="A1.E8X.2.1.1.m1.3.6.2">D</mi><mrow id="A1.E8X.2.1.1.m1.3.6.3"><mi id="A1.E8X.2.1.1.m1.3.6.3.2">K</mi><mo id="A1.E8X.2.1.1.m1.3.6.3.1">⁢</mo><mi id="A1.E8X.2.1.1.m1.3.6.3.3">L</mi></mrow></msub><mrow id="A1.E8X.2.1.1.m1.3.7"><mo id="A1.E8X.2.1.1.m1.3.7.1">(</mo><mi id="A1.E8X.2.1.1.m1.3.7.2">Q</mi><mrow id="A1.E8X.2.1.1.m1.3.7.3"><mo id="A1.E8X.2.1.1.m1.3.7.3.1" stretchy="false">(</mo><mi id="A1.E8X.2.1.1.m1.3.3">z</mi><mo id="A1.E8X.2.1.1.m1.3.7.3.2" stretchy="false">)</mo></mrow><mo fence="false" id="A1.E8X.2.1.1.m1.3.7.4" rspace="0.167em" stretchy="false">|</mo><mo fence="false" id="A1.E8X.2.1.1.m1.3.7.5" rspace="0.167em" stretchy="false">|</mo><msub id="A1.E8X.2.1.1.m1.3.7.6"><mi id="A1.E8X.2.1.1.m1.3.7.6.2">P</mi><mrow id="A1.E8X.2.1.1.m1.2.2.2.4"><mi id="A1.E8X.2.1.1.m1.1.1.1.1">θ</mi><mo id="A1.E8X.2.1.1.m1.2.2.2.4.1">,</mo><mi class="ltx_font_mathcaligraphic" id="A1.E8X.2.1.1.m1.2.2.2.2">𝒢</mi></mrow></msub><mrow id="A1.E8X.2.1.1.m1.3.7.7"><mo id="A1.E8X.2.1.1.m1.3.7.7.1">(</mo><mi id="A1.E8X.2.1.1.m1.3.7.7.2">z</mi><mo fence="false" 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id="A1.E8X.3.2.2.m1.4a"><mrow id="A1.E8X.3.2.2.m1.4b"><mo id="A1.E8X.3.2.2.m1.4.5">=</mo><msub id="A1.E8X.3.2.2.m1.4.6"><mi id="A1.E8X.3.2.2.m1.4.6.2">D</mi><mrow id="A1.E8X.3.2.2.m1.4.6.3"><mi id="A1.E8X.3.2.2.m1.4.6.3.2">K</mi><mo id="A1.E8X.3.2.2.m1.4.6.3.1">⁢</mo><mi id="A1.E8X.3.2.2.m1.4.6.3.3">L</mi></mrow></msub><mrow id="A1.E8X.3.2.2.m1.4.7"><mo id="A1.E8X.3.2.2.m1.4.7.1">(</mo><mi id="A1.E8X.3.2.2.m1.4.7.2">Q</mi><mrow id="A1.E8X.3.2.2.m1.4.7.3"><mo id="A1.E8X.3.2.2.m1.4.7.3.1" stretchy="false">(</mo><mi id="A1.E8X.3.2.2.m1.4.7.3.2">z</mi><mo fence="false" id="A1.E8X.3.2.2.m1.4.7.3.3" rspace="0.167em" stretchy="false">|</mo><mi id="A1.E8X.3.2.2.m1.3.3">q</mi><mo id="A1.E8X.3.2.2.m1.4.7.3.4">,</mo><mi id="A1.E8X.3.2.2.m1.4.4">a</mi><mo id="A1.E8X.3.2.2.m1.4.7.3.5" stretchy="false">)</mo></mrow><mo fence="false" id="A1.E8X.3.2.2.m1.4.7.4" rspace="0.167em" stretchy="false">|</mo><mo fence="false" id="A1.E8X.3.2.2.m1.4.7.5" rspace="0.167em" stretchy="false">|</mo><msub id="A1.E8X.3.2.2.m1.4.7.6"><mi id="A1.E8X.3.2.2.m1.4.7.6.2">P</mi><mrow id="A1.E8X.3.2.2.m1.2.2.2.4"><mi id="A1.E8X.3.2.2.m1.1.1.1.1">θ</mi><mo id="A1.E8X.3.2.2.m1.2.2.2.4.1">,</mo><mi class="ltx_font_mathcaligraphic" id="A1.E8X.3.2.2.m1.2.2.2.2">𝒢</mi></mrow></msub><mrow id="A1.E8X.3.2.2.m1.4.7.7"><mo id="A1.E8X.3.2.2.m1.4.7.7.1">(</mo><mi id="A1.E8X.3.2.2.m1.4.7.7.2">z</mi><mo fence="false" id="A1.E8X.3.2.2.m1.4.7.7.3" rspace="0.167em" stretchy="false">|</mo><mi id="A1.E8X.3.2.2.m1.4.7.7.4">q</mi><mo id="A1.E8X.3.2.2.m1.4.7.7.5">)</mo></mrow><mo id="A1.E8X.3.2.2.m1.4.7.8">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="A1.E8X.3.2.2.m1.4c">\displaystyle=D_{KL}\left(Q(z|q,a)||P_{\theta,\mathcal{G}}\left(z|q\right)\right)</annotation><annotation encoding="application/x-llamapun" id="A1.E8X.3.2.2.m1.4d">= italic_D start_POSTSUBSCRIPT italic_K italic_L end_POSTSUBSCRIPT ( italic_Q ( italic_z | italic_q , italic_a ) | | italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_z | italic_q ) )</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="4"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(8)</span></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="A1.E8Xa"> <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=\mathbb{E}_{z\sim Q(z|q,a)}\left[\log Q(z|q,a)-\log P_{\theta,% \mathcal{G}}(z|q)\right]" class="ltx_Math" display="inline" id="A1.E8Xa.2.1.1.m1.8"><semantics id="A1.E8Xa.2.1.1.m1.8a"><mrow id="A1.E8Xa.2.1.1.m1.8.8" xref="A1.E8Xa.2.1.1.m1.8.8.cmml"><mi id="A1.E8Xa.2.1.1.m1.8.8.3" xref="A1.E8Xa.2.1.1.m1.8.8.3.cmml"></mi><mo id="A1.E8Xa.2.1.1.m1.8.8.2" xref="A1.E8Xa.2.1.1.m1.8.8.2.cmml">=</mo><mrow id="A1.E8Xa.2.1.1.m1.8.8.1" 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xref="A1.E8Xa.2.1.1.m1.5.5.2.2">𝒢</ci></list></apply></apply><apply id="A1.E8Xa.2.1.1.m1.8.8.1.1.1.1.2.1.1.1.cmml" xref="A1.E8Xa.2.1.1.m1.8.8.1.1.1.1.2.1.1"><csymbol cd="latexml" id="A1.E8Xa.2.1.1.m1.8.8.1.1.1.1.2.1.1.1.1.cmml" xref="A1.E8Xa.2.1.1.m1.8.8.1.1.1.1.2.1.1.1.1">conditional</csymbol><ci id="A1.E8Xa.2.1.1.m1.8.8.1.1.1.1.2.1.1.1.2.cmml" xref="A1.E8Xa.2.1.1.m1.8.8.1.1.1.1.2.1.1.1.2">𝑧</ci><ci id="A1.E8Xa.2.1.1.m1.8.8.1.1.1.1.2.1.1.1.3.cmml" xref="A1.E8Xa.2.1.1.m1.8.8.1.1.1.1.2.1.1.1.3">𝑞</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E8Xa.2.1.1.m1.8c">\displaystyle=\mathbb{E}_{z\sim Q(z|q,a)}\left[\log Q(z|q,a)-\log P_{\theta,% \mathcal{G}}(z|q)\right]</annotation><annotation encoding="application/x-llamapun" id="A1.E8Xa.2.1.1.m1.8d">= blackboard_E start_POSTSUBSCRIPT italic_z ∼ italic_Q ( italic_z | italic_q , italic_a ) end_POSTSUBSCRIPT [ roman_log italic_Q ( italic_z | italic_q , italic_a ) - roman_log italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_z | italic_q ) ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="A1.E8Xb"> <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=-\mathbb{E}_{z\sim Q(z|q,a)}\log P_{\theta,\mathcal{G}}(z|q)+% \mathbb{E}_{z\sim Q(z|q,a)}\log Q(z|q,a)" class="ltx_Math" display="inline" id="A1.E8Xb.2.1.1.m1.12"><semantics id="A1.E8Xb.2.1.1.m1.12a"><mrow id="A1.E8Xb.2.1.1.m1.12.12" xref="A1.E8Xb.2.1.1.m1.12.12.cmml"><mi id="A1.E8Xb.2.1.1.m1.12.12.4" xref="A1.E8Xb.2.1.1.m1.12.12.4.cmml"></mi><mo id="A1.E8Xb.2.1.1.m1.12.12.3" xref="A1.E8Xb.2.1.1.m1.12.12.3.cmml">=</mo><mrow id="A1.E8Xb.2.1.1.m1.12.12.2" xref="A1.E8Xb.2.1.1.m1.12.12.2.cmml"><mrow id="A1.E8Xb.2.1.1.m1.11.11.1.1" 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</tr> </tbody> </table> </div> <div class="ltx_para" id="A1.p3"> <p class="ltx_p" id="A1.p3.3">where the term <math alttext="\mathbb{E}_{z\sim Q(z|q,a)}\log Q(z|q,a)" class="ltx_Math" display="inline" id="A1.p3.1.m1.6"><semantics id="A1.p3.1.m1.6a"><mrow id="A1.p3.1.m1.6.6" xref="A1.p3.1.m1.6.6.cmml"><msub id="A1.p3.1.m1.6.6.3" xref="A1.p3.1.m1.6.6.3.cmml"><mi id="A1.p3.1.m1.6.6.3.2" xref="A1.p3.1.m1.6.6.3.2.cmml">𝔼</mi><mrow id="A1.p3.1.m1.3.3.3" xref="A1.p3.1.m1.3.3.3.cmml"><mi id="A1.p3.1.m1.3.3.3.5" xref="A1.p3.1.m1.3.3.3.5.cmml">z</mi><mo id="A1.p3.1.m1.3.3.3.4" xref="A1.p3.1.m1.3.3.3.4.cmml">∼</mo><mrow id="A1.p3.1.m1.3.3.3.3" xref="A1.p3.1.m1.3.3.3.3.cmml"><mi id="A1.p3.1.m1.3.3.3.3.3" xref="A1.p3.1.m1.3.3.3.3.3.cmml">Q</mi><mo id="A1.p3.1.m1.3.3.3.3.2" xref="A1.p3.1.m1.3.3.3.3.2.cmml">⁢</mo><mrow id="A1.p3.1.m1.3.3.3.3.1.1" xref="A1.p3.1.m1.3.3.3.3.1.1.1.cmml"><mo id="A1.p3.1.m1.3.3.3.3.1.1.2" stretchy="false" xref="A1.p3.1.m1.3.3.3.3.1.1.1.cmml">(</mo><mrow 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id="A1.p3.1.m1.3.3.3.3.1.1.1.3.1.cmml" xref="A1.p3.1.m1.3.3.3.3.1.1.1.3.2"><ci id="A1.p3.1.m1.1.1.1.1.cmml" xref="A1.p3.1.m1.1.1.1.1">𝑞</ci><ci id="A1.p3.1.m1.2.2.2.2.cmml" xref="A1.p3.1.m1.2.2.2.2">𝑎</ci></list></apply></apply></apply></apply><apply id="A1.p3.1.m1.6.6.4.cmml" xref="A1.p3.1.m1.6.6.4"><log id="A1.p3.1.m1.6.6.4.1.cmml" xref="A1.p3.1.m1.6.6.4.1"></log><ci id="A1.p3.1.m1.6.6.4.2.cmml" xref="A1.p3.1.m1.6.6.4.2">𝑄</ci></apply><apply id="A1.p3.1.m1.6.6.1.1.1.cmml" xref="A1.p3.1.m1.6.6.1.1"><csymbol cd="latexml" id="A1.p3.1.m1.6.6.1.1.1.1.cmml" xref="A1.p3.1.m1.6.6.1.1.1.1">conditional</csymbol><ci id="A1.p3.1.m1.6.6.1.1.1.2.cmml" xref="A1.p3.1.m1.6.6.1.1.1.2">𝑧</ci><list id="A1.p3.1.m1.6.6.1.1.1.3.1.cmml" xref="A1.p3.1.m1.6.6.1.1.1.3.2"><ci id="A1.p3.1.m1.4.4.cmml" xref="A1.p3.1.m1.4.4">𝑞</ci><ci id="A1.p3.1.m1.5.5.cmml" xref="A1.p3.1.m1.5.5">𝑎</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p3.1.m1.6c">\mathbb{E}_{z\sim Q(z|q,a)}\log Q(z|q,a)</annotation><annotation encoding="application/x-llamapun" id="A1.p3.1.m1.6d">blackboard_E start_POSTSUBSCRIPT italic_z ∼ italic_Q ( italic_z | italic_q , italic_a ) end_POSTSUBSCRIPT roman_log italic_Q ( italic_z | italic_q , italic_a )</annotation></semantics></math> is not related to <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="A1.p3.2.m2.1"><semantics id="A1.p3.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="A1.p3.2.m2.1.1" xref="A1.p3.2.m2.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="A1.p3.2.m2.1b"><ci id="A1.p3.2.m2.1.1.cmml" xref="A1.p3.2.m2.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.p3.2.m2.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="A1.p3.2.m2.1d">caligraphic_G</annotation></semantics></math> and can be treated as a constant. With <math alttext="Q(z|q,a)" class="ltx_Math" display="inline" id="A1.p3.3.m3.3"><semantics id="A1.p3.3.m3.3a"><mrow id="A1.p3.3.m3.3.3" xref="A1.p3.3.m3.3.3.cmml"><mi id="A1.p3.3.m3.3.3.3" xref="A1.p3.3.m3.3.3.3.cmml">Q</mi><mo id="A1.p3.3.m3.3.3.2" xref="A1.p3.3.m3.3.3.2.cmml">⁢</mo><mrow id="A1.p3.3.m3.3.3.1.1" xref="A1.p3.3.m3.3.3.1.1.1.cmml"><mo id="A1.p3.3.m3.3.3.1.1.2" stretchy="false" xref="A1.p3.3.m3.3.3.1.1.1.cmml">(</mo><mrow id="A1.p3.3.m3.3.3.1.1.1" xref="A1.p3.3.m3.3.3.1.1.1.cmml"><mi id="A1.p3.3.m3.3.3.1.1.1.2" xref="A1.p3.3.m3.3.3.1.1.1.2.cmml">z</mi><mo fence="false" id="A1.p3.3.m3.3.3.1.1.1.1" xref="A1.p3.3.m3.3.3.1.1.1.1.cmml">|</mo><mrow id="A1.p3.3.m3.3.3.1.1.1.3.2" xref="A1.p3.3.m3.3.3.1.1.1.3.1.cmml"><mi id="A1.p3.3.m3.1.1" xref="A1.p3.3.m3.1.1.cmml">q</mi><mo id="A1.p3.3.m3.3.3.1.1.1.3.2.1" xref="A1.p3.3.m3.3.3.1.1.1.3.1.cmml">,</mo><mi id="A1.p3.3.m3.2.2" xref="A1.p3.3.m3.2.2.cmml">a</mi></mrow></mrow><mo id="A1.p3.3.m3.3.3.1.1.3" stretchy="false" xref="A1.p3.3.m3.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p3.3.m3.3b"><apply id="A1.p3.3.m3.3.3.cmml" xref="A1.p3.3.m3.3.3"><times id="A1.p3.3.m3.3.3.2.cmml" xref="A1.p3.3.m3.3.3.2"></times><ci id="A1.p3.3.m3.3.3.3.cmml" xref="A1.p3.3.m3.3.3.3">𝑄</ci><apply id="A1.p3.3.m3.3.3.1.1.1.cmml" xref="A1.p3.3.m3.3.3.1.1"><csymbol cd="latexml" id="A1.p3.3.m3.3.3.1.1.1.1.cmml" xref="A1.p3.3.m3.3.3.1.1.1.1">conditional</csymbol><ci id="A1.p3.3.m3.3.3.1.1.1.2.cmml" xref="A1.p3.3.m3.3.3.1.1.1.2">𝑧</ci><list id="A1.p3.3.m3.3.3.1.1.1.3.1.cmml" xref="A1.p3.3.m3.3.3.1.1.1.3.2"><ci id="A1.p3.3.m3.1.1.cmml" xref="A1.p3.3.m3.1.1">𝑞</ci><ci id="A1.p3.3.m3.2.2.cmml" xref="A1.p3.3.m3.2.2">𝑎</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p3.3.m3.3c">Q(z|q,a)</annotation><annotation encoding="application/x-llamapun" id="A1.p3.3.m3.3d">italic_Q ( italic_z | italic_q , italic_a )</annotation></semantics></math> formulated in <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#S4.E3" title="In Knowledge Retrieval ‣ 4.2 Objective Formulation ‣ 4 Method ‣ Knowledge Graph Tuning: Real-time Large Language Model Personalization based on Human Feedback"><span class="ltx_text ltx_ref_tag">Equation</span> <span class="ltx_text ltx_ref_tag">3</span></a>, the loss can be calculated as</p> </div> <div class="ltx_para" id="A1.p4"> <table class="ltx_equationgroup ltx_eqn_table" id="A1.E9"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="A1.E9X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{L}_{\text{retrieve}}" class="ltx_Math" display="inline" id="A1.E9X.2.1.1.m1.1"><semantics id="A1.E9X.2.1.1.m1.1a"><msub id="A1.E9X.2.1.1.m1.1.1" xref="A1.E9X.2.1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A1.E9X.2.1.1.m1.1.1.2" xref="A1.E9X.2.1.1.m1.1.1.2.cmml">ℒ</mi><mtext id="A1.E9X.2.1.1.m1.1.1.3" xref="A1.E9X.2.1.1.m1.1.1.3a.cmml">retrieve</mtext></msub><annotation-xml encoding="MathML-Content" id="A1.E9X.2.1.1.m1.1b"><apply id="A1.E9X.2.1.1.m1.1.1.cmml" xref="A1.E9X.2.1.1.m1.1.1"><csymbol cd="ambiguous" id="A1.E9X.2.1.1.m1.1.1.1.cmml" xref="A1.E9X.2.1.1.m1.1.1">subscript</csymbol><ci id="A1.E9X.2.1.1.m1.1.1.2.cmml" xref="A1.E9X.2.1.1.m1.1.1.2">ℒ</ci><ci id="A1.E9X.2.1.1.m1.1.1.3a.cmml" xref="A1.E9X.2.1.1.m1.1.1.3"><mtext id="A1.E9X.2.1.1.m1.1.1.3.cmml" mathsize="70%" xref="A1.E9X.2.1.1.m1.1.1.3">retrieve</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E9X.2.1.1.m1.1c">\displaystyle\mathcal{L}_{\text{retrieve}}</annotation><annotation encoding="application/x-llamapun" id="A1.E9X.2.1.1.m1.1d">caligraphic_L start_POSTSUBSCRIPT retrieve end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=-\mathbb{E}_{z\sim Q(z|q,a)}\log P_{\theta,\mathcal{G}}(z|q)+% \text{CONST}" 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id="A1.E9X.3.2.2.m1.6.6.1.3" xref="A1.E9X.3.2.2.m1.6.6.1.3a.cmml">CONST</mtext></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.E9X.3.2.2.m1.6b"><apply id="A1.E9X.3.2.2.m1.6.6.cmml" xref="A1.E9X.3.2.2.m1.6.6"><eq id="A1.E9X.3.2.2.m1.6.6.2.cmml" xref="A1.E9X.3.2.2.m1.6.6.2"></eq><csymbol cd="latexml" id="A1.E9X.3.2.2.m1.6.6.3.cmml" xref="A1.E9X.3.2.2.m1.6.6.3">absent</csymbol><apply id="A1.E9X.3.2.2.m1.6.6.1.cmml" xref="A1.E9X.3.2.2.m1.6.6.1"><plus id="A1.E9X.3.2.2.m1.6.6.1.2.cmml" xref="A1.E9X.3.2.2.m1.6.6.1.2"></plus><apply id="A1.E9X.3.2.2.m1.6.6.1.1.cmml" xref="A1.E9X.3.2.2.m1.6.6.1.1"><minus id="A1.E9X.3.2.2.m1.6.6.1.1.2.cmml" xref="A1.E9X.3.2.2.m1.6.6.1.1"></minus><apply id="A1.E9X.3.2.2.m1.6.6.1.1.1.cmml" xref="A1.E9X.3.2.2.m1.6.6.1.1.1"><times id="A1.E9X.3.2.2.m1.6.6.1.1.1.2.cmml" xref="A1.E9X.3.2.2.m1.6.6.1.1.1.2"></times><apply id="A1.E9X.3.2.2.m1.6.6.1.1.1.3.cmml" xref="A1.E9X.3.2.2.m1.6.6.1.1.1.3"><csymbol cd="ambiguous" id="A1.E9X.3.2.2.m1.6.6.1.1.1.3.1.cmml" 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xref="A1.E9Xb.2.1.1.m1.5.5.2.2">𝒢</ci></list></apply></apply><apply id="A1.E9Xb.2.1.1.m1.6.6.1.1.1.1.1.1.1.1.1.1.cmml" xref="A1.E9Xb.2.1.1.m1.6.6.1.1.1.1.1.1.1.1.1"><csymbol cd="latexml" id="A1.E9Xb.2.1.1.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A1.E9Xb.2.1.1.m1.6.6.1.1.1.1.1.1.1.1.1.1.1">conditional</csymbol><ci id="A1.E9Xb.2.1.1.m1.6.6.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="A1.E9Xb.2.1.1.m1.6.6.1.1.1.1.1.1.1.1.1.1.2">𝑧</ci><ci id="A1.E9Xb.2.1.1.m1.6.6.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A1.E9Xb.2.1.1.m1.6.6.1.1.1.1.1.1.1.1.1.1.3">𝑞</ci></apply></apply></apply></apply></apply><ci id="A1.E9Xb.2.1.1.m1.6.6.1.1.1.3a.cmml" xref="A1.E9Xb.2.1.1.m1.6.6.1.1.1.3"><mtext id="A1.E9Xb.2.1.1.m1.6.6.1.1.1.3.cmml" xref="A1.E9Xb.2.1.1.m1.6.6.1.1.1.3">CONST</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E9Xb.2.1.1.m1.6c">\displaystyle=-\frac{1}{K}\sum_{z\in\mathcal{H}(q,a,K)}\log P_{\theta,\mathcal% {G}}\left(z|q\right)+\text{CONST},</annotation><annotation encoding="application/x-llamapun" id="A1.E9Xb.2.1.1.m1.6d">= - divide start_ARG 1 end_ARG start_ARG italic_K end_ARG ∑ start_POSTSUBSCRIPT italic_z ∈ caligraphic_H ( italic_q , italic_a , italic_K ) end_POSTSUBSCRIPT roman_log italic_P start_POSTSUBSCRIPT italic_θ , caligraphic_G end_POSTSUBSCRIPT ( italic_z | italic_q ) + CONST ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> </div> <div class="ltx_para" id="A1.p5"> <p class="ltx_p" id="A1.p5.1">where the CONST is omitted in the optimization.</p> </div> </section> <section class="ltx_appendix" id="A2"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix B </span>Details on the <span class="ltx_text ltx_font_italic" id="A2.1.1">CounterFactExtension</span> Dataset</h2> <div class="ltx_para" id="A2.p1"> <p class="ltx_p" id="A2.p1.1"><span class="ltx_text ltx_font_italic" id="A2.p1.1.1">CounterFactExtension</span> is a dataset that extends the <span class="ltx_text ltx_font_italic" id="A2.p1.1.2">CounterFact</span> with 27737 additional samples. Specifically, <span class="ltx_text ltx_font_italic" id="A2.p1.1.3">CounterFactExtension</span> is on the task of knowledge editing for <span class="ltx_text ltx_font_italic" id="A2.p1.1.4">Efficacy</span> and <span class="ltx_text ltx_font_italic" id="A2.p1.1.5">Paraphrase</span> evaluation. Likewise, <span class="ltx_text ltx_font_italic" id="A2.p1.1.6">CounterFactExtension</span> is crafted to identify the differences between minor adjustments in the vocabulary used by the model and substantial, widespread shifts in the foundational factual knowledge.</p> </div> <div class="ltx_para" id="A2.p2"> <p class="ltx_p" id="A2.p2.2"><span class="ltx_text ltx_font_italic" id="A2.p2.2.1">CounterFactExtension</span> is built on knowledge pairs processed by Dai et al. <cite class="ltx_cite ltx_citemacro_citep">[Dai et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib5" title="">2022</a>]</cite> based on PARAREL dataset <cite class="ltx_cite ltx_citemacro_citep">[Elazar et al., <a class="ltx_ref" href="https://arxiv.org/html/2405.19686v1#bib.bib9" title="">2021</a>]</cite>. Specifically, for each knowledge pair, we employ GPT-4 to extract the subject, relation, and the true target, <math alttext="t=(s,r,o^{*})" class="ltx_Math" display="inline" id="A2.p2.1.m1.3"><semantics id="A2.p2.1.m1.3a"><mrow id="A2.p2.1.m1.3.3" xref="A2.p2.1.m1.3.3.cmml"><mi id="A2.p2.1.m1.3.3.3" xref="A2.p2.1.m1.3.3.3.cmml">t</mi><mo id="A2.p2.1.m1.3.3.2" xref="A2.p2.1.m1.3.3.2.cmml">=</mo><mrow id="A2.p2.1.m1.3.3.1.1" xref="A2.p2.1.m1.3.3.1.2.cmml"><mo id="A2.p2.1.m1.3.3.1.1.2" stretchy="false" xref="A2.p2.1.m1.3.3.1.2.cmml">(</mo><mi id="A2.p2.1.m1.1.1" xref="A2.p2.1.m1.1.1.cmml">s</mi><mo id="A2.p2.1.m1.3.3.1.1.3" xref="A2.p2.1.m1.3.3.1.2.cmml">,</mo><mi id="A2.p2.1.m1.2.2" xref="A2.p2.1.m1.2.2.cmml">r</mi><mo id="A2.p2.1.m1.3.3.1.1.4" xref="A2.p2.1.m1.3.3.1.2.cmml">,</mo><msup id="A2.p2.1.m1.3.3.1.1.1" xref="A2.p2.1.m1.3.3.1.1.1.cmml"><mi id="A2.p2.1.m1.3.3.1.1.1.2" xref="A2.p2.1.m1.3.3.1.1.1.2.cmml">o</mi><mo id="A2.p2.1.m1.3.3.1.1.1.3" xref="A2.p2.1.m1.3.3.1.1.1.3.cmml">∗</mo></msup><mo id="A2.p2.1.m1.3.3.1.1.5" stretchy="false" xref="A2.p2.1.m1.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.p2.1.m1.3b"><apply id="A2.p2.1.m1.3.3.cmml" xref="A2.p2.1.m1.3.3"><eq id="A2.p2.1.m1.3.3.2.cmml" xref="A2.p2.1.m1.3.3.2"></eq><ci id="A2.p2.1.m1.3.3.3.cmml" xref="A2.p2.1.m1.3.3.3">𝑡</ci><vector id="A2.p2.1.m1.3.3.1.2.cmml" xref="A2.p2.1.m1.3.3.1.1"><ci id="A2.p2.1.m1.1.1.cmml" xref="A2.p2.1.m1.1.1">𝑠</ci><ci id="A2.p2.1.m1.2.2.cmml" xref="A2.p2.1.m1.2.2">𝑟</ci><apply id="A2.p2.1.m1.3.3.1.1.1.cmml" xref="A2.p2.1.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="A2.p2.1.m1.3.3.1.1.1.1.cmml" xref="A2.p2.1.m1.3.3.1.1.1">superscript</csymbol><ci id="A2.p2.1.m1.3.3.1.1.1.2.cmml" xref="A2.p2.1.m1.3.3.1.1.1.2">𝑜</ci><times id="A2.p2.1.m1.3.3.1.1.1.3.cmml" xref="A2.p2.1.m1.3.3.1.1.1.3"></times></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p2.1.m1.3c">t=(s,r,o^{*})</annotation><annotation encoding="application/x-llamapun" id="A2.p2.1.m1.3d">italic_t = ( italic_s , italic_r , italic_o start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math>, and generate a counterfactual target, <math alttext="o^{c}" class="ltx_Math" display="inline" id="A2.p2.2.m2.1"><semantics id="A2.p2.2.m2.1a"><msup id="A2.p2.2.m2.1.1" xref="A2.p2.2.m2.1.1.cmml"><mi id="A2.p2.2.m2.1.1.2" xref="A2.p2.2.m2.1.1.2.cmml">o</mi><mi id="A2.p2.2.m2.1.1.3" xref="A2.p2.2.m2.1.1.3.cmml">c</mi></msup><annotation-xml encoding="MathML-Content" id="A2.p2.2.m2.1b"><apply id="A2.p2.2.m2.1.1.cmml" xref="A2.p2.2.m2.1.1"><csymbol cd="ambiguous" id="A2.p2.2.m2.1.1.1.cmml" xref="A2.p2.2.m2.1.1">superscript</csymbol><ci id="A2.p2.2.m2.1.1.2.cmml" xref="A2.p2.2.m2.1.1.2">𝑜</ci><ci id="A2.p2.2.m2.1.1.3.cmml" xref="A2.p2.2.m2.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p2.2.m2.1c">o^{c}</annotation><annotation encoding="application/x-llamapun" id="A2.p2.2.m2.1d">italic_o start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT</annotation></semantics></math>. Further, a generation prompt (generated by GPT-4) is also provided for the <span class="ltx_text ltx_font_italic" id="A2.p2.2.2">Paraphrase</span> evaluation. Finally, we manually check each sample and the generation to ensure the correctness.</p> </div> <section class="ltx_subsection" id="A2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">B.1 </span>Dataset Samples</h3> <div class="ltx_para" id="A2.SS1.p1"> <div class="ltx_listing ltx_lst_language_json ltx_lst_numbers_left ltx_lstlisting ltx_framed ltx_framed_topbottom ltx_listing" id="A2.SS1.p1.1" style="background-color:#EEEEEE;"> <div class="ltx_listing_data"><a download="" href="data:text/plain;base64,ICAgIHsKICAgICAgICAiY2FzZV9pZCI6IDAsCiAgICAgICAgInJlcXVlc3RlZF9yZXdyaXRlIjogewogICAgICAgICAgICAicHJvbXB0IjogInt9IHdvcmtzIGluIHRoZSBmaWVsZCBvZiAiLAogICAgICAgICAgICAicmVsYXRpb25faWQiOiAiUDEwMShmaWVsZCBvZiB3b3JrKSIsCiAgICAgICAgICAgICJ0YXJnZXRfbmV3IjogewogICAgICAgICAgICAgICAgInN0ciI6ICJtZWNoYW5pY2FsIGVuZ2luZWVyaW5nIiwKICAgICAgICAgICAgfSwKICAgICAgICAgICAgInRhcmdldF90cnVlIjogewogICAgICAgICAgICAgICAgInN0ciI6ICJsb2dpYyIsCiAgICAgICAgICAgIH0sCiAgICAgICAgICAgICJzdWJqZWN0IjogIkFsYW4gVHVyaW5nIgogICAgICAgIH0sCiAgICAgICAgImdlbmVyYXRpb25fcHJvbXB0cyI6IFsKICAgICAgICAgICAgIlRoZSBmaWVsZCB0aGF0IEFsYW4gVHVyaW5nIGlzIGFzc29jaWF0ZWQgd2l0aCBpcyIKICAgICAgICBdCiAgICB9LAogICAgewogICAgICAgICJjYXNlX2lkIjogMjU1ODUsCiAgICAgICAgInJlcXVlc3RlZF9yZXdyaXRlIjogewogICAgICAgICAgICAicHJvbXB0IjogInt9IG1haW50YWlucyBkaXBsb21hdGljIHJlbGF0aW9ucyB3aXRoICIsCiAgICAgICAgICAgICJyZWxhdGlvbl9pZCI6ICJQNTMwKGRpcGxvbWF0aWMgcmVsYXRpb24pIiwKICAgICAgICAgICAgInRhcmdldF9uZXciOiB7CiAgICAgICAgICAgICAgICAic3RyIjogIlVtYm9yaWEiLAogICAgICAgICAgICB9LAogICAgICAgICAgICAidGFyZ2V0X3RydWUiOiB7CiAgICAgICAgICAgICAgICAic3RyIjogIk1hbGF5c2lhIiwKICAgICAgICAgICAgfSwKICAgICAgICAgICAgInN1YmplY3QiOiAiVWtyYWluZSIKICAgICAgICB9LAogICAgICAgICJnZW5lcmF0aW9uX3Byb21wdHMiOiBbCiAgICAgICAgICAgICJEaXBsb21hdGljIHJlbGF0aW9ucyBhcmUgZXN0YWJsaXNoZWQgYmV0d2VlbiBVa3JhaW5lIGFuZCIKICAgICAgICBdCiAgICB9LA==">⬇</a></div> <div class="ltx_listingline" id="lstnumberx1"> <span class="ltx_tag ltx_tag_listingline">1</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx1.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx1.2">{</span> </div> <div class="ltx_listingline" id="lstnumberx2"> <span class="ltx_tag ltx_tag_listingline">2</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx2.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx2.2">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx2.3">case_id</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx2.4">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx2.5" style="color:#990000;">:</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx2.6"> </span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx2.7" style="color:#990099;">0</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx2.8" style="color:#990000;">,</span> </div> <div class="ltx_listingline" id="lstnumberx3"> <span class="ltx_tag ltx_tag_listingline">3</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx3.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx3.2">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx3.3">requested_rewrite</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx3.4">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx3.5" style="color:#990000;">:</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx3.6"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx3.7">{</span> </div> <div class="ltx_listingline" id="lstnumberx4"> <span class="ltx_tag ltx_tag_listingline">4</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx4.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx4.2">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx4.3">prompt</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx4.4">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx4.5" style="color:#990000;">:</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx4.6"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx4.7">"{}</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx4.8"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx4.9">works</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx4.10"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx4.11">in</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx4.12"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx4.13">the</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx4.14"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx4.15">field</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx4.16"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx4.17">of</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx4.18"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx4.19">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx4.20" style="color:#990000;">,</span> </div> <div class="ltx_listingline" id="lstnumberx5"> <span class="ltx_tag ltx_tag_listingline">5</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx5.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx5.2">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx5.3">relation_id</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx5.4">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx5.5" style="color:#990000;">:</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx5.6"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx5.7">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx5.8">P101</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx5.9">(</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx5.10">field</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx5.11"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx5.12">of</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx5.13"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx5.14">work</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx5.15">)"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx5.16" style="color:#990000;">,</span> </div> <div class="ltx_listingline" id="lstnumberx6"> <span class="ltx_tag ltx_tag_listingline">6</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx6.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx6.2">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx6.3">target_new</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx6.4">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx6.5" style="color:#990000;">:</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx6.6"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx6.7">{</span> </div> <div class="ltx_listingline" id="lstnumberx7"> <span class="ltx_tag ltx_tag_listingline">7</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx7.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx7.2">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx7.3">str</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx7.4">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx7.5" style="color:#990000;">:</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx7.6"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx7.7">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx7.8">mechanical</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx7.9"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx7.10">engineering</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx7.11">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx7.12" style="color:#990000;">,</span> </div> <div class="ltx_listingline" id="lstnumberx8"> <span class="ltx_tag ltx_tag_listingline">8</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx8.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx8.2">}</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx8.3" style="color:#990000;">,</span> </div> <div class="ltx_listingline" id="lstnumberx9"> <span class="ltx_tag ltx_tag_listingline">9</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx9.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx9.2">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx9.3">target_true</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx9.4">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx9.5" style="color:#990000;">:</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx9.6"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx9.7">{</span> </div> <div class="ltx_listingline" id="lstnumberx10"> <span class="ltx_tag ltx_tag_listingline">10</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx10.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx10.2">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx10.3">str</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx10.4">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx10.5" style="color:#990000;">:</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx10.6"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx10.7">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx10.8">logic</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx10.9">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx10.10" style="color:#990000;">,</span> </div> <div class="ltx_listingline" id="lstnumberx11"> <span class="ltx_tag ltx_tag_listingline">11</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx11.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx11.2">}</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx11.3" style="color:#990000;">,</span> </div> <div class="ltx_listingline" id="lstnumberx12"> <span class="ltx_tag ltx_tag_listingline">12</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx12.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx12.2">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx12.3">subject</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx12.4">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx12.5" style="color:#990000;">:</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx12.6"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx12.7">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx12.8">Alan</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx12.9"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx12.10">Turing</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx12.11">"</span> </div> <div class="ltx_listingline" id="lstnumberx13"> <span class="ltx_tag ltx_tag_listingline">13</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx13.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx13.2">}</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx13.3" style="color:#990000;">,</span> </div> <div class="ltx_listingline" id="lstnumberx14"> <span class="ltx_tag ltx_tag_listingline">14</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx14.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx14.2">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx14.3">generation_prompts</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx14.4">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx14.5" style="color:#990000;">:</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx14.6"> </span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx14.7" style="color:#1469B0;">[</span> </div> <div class="ltx_listingline" id="lstnumberx15"> <span class="ltx_tag ltx_tag_listingline">15</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx15.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx15.2">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx15.3">The</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx15.4"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx15.5">field</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx15.6"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx15.7">that</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx15.8"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx15.9">Alan</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx15.10"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx15.11">Turing</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx15.12"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx15.13">is</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx15.14"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx15.15">associated</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx15.16"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx15.17">with</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx15.18"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx15.19">is</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx15.20">"</span> </div> <div class="ltx_listingline" id="lstnumberx16"> <span class="ltx_tag ltx_tag_listingline">16</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx16.1"> </span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx16.2" style="color:#1469B0;">]</span> </div> <div class="ltx_listingline" id="lstnumberx17"> <span class="ltx_tag ltx_tag_listingline">17</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx17.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx17.2">}</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx17.3" style="color:#990000;">,</span> </div> <div class="ltx_listingline" id="lstnumberx18"> <span class="ltx_tag ltx_tag_listingline">18</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx18.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx18.2">{</span> </div> <div class="ltx_listingline" id="lstnumberx19"> <span class="ltx_tag ltx_tag_listingline">19</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx19.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx19.2">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx19.3">case_id</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx19.4">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx19.5" style="color:#990000;">:</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx19.6"> </span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx19.7" style="color:#990099;">2</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx19.8" style="color:#990099;">5</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx19.9" style="color:#990099;">5</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx19.10" style="color:#990099;">8</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx19.11" style="color:#990099;">5</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx19.12" style="color:#990000;">,</span> </div> <div class="ltx_listingline" id="lstnumberx20"> <span class="ltx_tag ltx_tag_listingline">20</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx20.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx20.2">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx20.3">requested_rewrite</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx20.4">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx20.5" style="color:#990000;">:</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx20.6"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx20.7">{</span> </div> <div class="ltx_listingline" id="lstnumberx21"> <span class="ltx_tag ltx_tag_listingline">21</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx21.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx21.2">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx21.3">prompt</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx21.4">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx21.5" style="color:#990000;">:</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx21.6"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx21.7">"{}</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx21.8"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx21.9">maintains</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx21.10"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx21.11">diplomatic</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx21.12"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx21.13">relations</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx21.14"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx21.15">with</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx21.16"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx21.17">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx21.18" style="color:#990000;">,</span> </div> <div class="ltx_listingline" id="lstnumberx22"> <span class="ltx_tag ltx_tag_listingline">22</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx22.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx22.2">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx22.3">relation_id</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx22.4">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx22.5" style="color:#990000;">:</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx22.6"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx22.7">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx22.8">P530</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx22.9">(</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx22.10">diplomatic</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx22.11"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx22.12">relation</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx22.13">)"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx22.14" style="color:#990000;">,</span> </div> <div class="ltx_listingline" id="lstnumberx23"> <span class="ltx_tag ltx_tag_listingline">23</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx23.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx23.2">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx23.3">target_new</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx23.4">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx23.5" style="color:#990000;">:</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx23.6"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx23.7">{</span> </div> <div class="ltx_listingline" id="lstnumberx24"> <span class="ltx_tag ltx_tag_listingline">24</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx24.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx24.2">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx24.3">str</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx24.4">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx24.5" style="color:#990000;">:</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx24.6"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx24.7">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx24.8">Umboria</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx24.9">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx24.10" style="color:#990000;">,</span> </div> <div class="ltx_listingline" id="lstnumberx25"> <span class="ltx_tag ltx_tag_listingline">25</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx25.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx25.2">}</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx25.3" style="color:#990000;">,</span> </div> <div class="ltx_listingline" id="lstnumberx26"> <span class="ltx_tag ltx_tag_listingline">26</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx26.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx26.2">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx26.3">target_true</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx26.4">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx26.5" style="color:#990000;">:</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx26.6"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx26.7">{</span> </div> <div class="ltx_listingline" id="lstnumberx27"> <span class="ltx_tag ltx_tag_listingline">27</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx27.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx27.2">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx27.3">str</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx27.4">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx27.5" style="color:#990000;">:</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx27.6"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx27.7">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx27.8">Malaysia</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx27.9">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx27.10" style="color:#990000;">,</span> </div> <div class="ltx_listingline" id="lstnumberx28"> <span class="ltx_tag ltx_tag_listingline">28</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx28.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx28.2">}</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx28.3" style="color:#990000;">,</span> </div> <div class="ltx_listingline" id="lstnumberx29"> <span class="ltx_tag ltx_tag_listingline">29</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx29.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx29.2">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx29.3">subject</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx29.4">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx29.5" style="color:#990000;">:</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx29.6"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx29.7">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx29.8">Ukraine</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx29.9">"</span> </div> <div class="ltx_listingline" id="lstnumberx30"> <span class="ltx_tag ltx_tag_listingline">30</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx30.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx30.2">}</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx30.3" style="color:#990000;">,</span> </div> <div class="ltx_listingline" id="lstnumberx31"> <span class="ltx_tag ltx_tag_listingline">31</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx31.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx31.2">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx31.3">generation_prompts</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx31.4">"</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx31.5" style="color:#990000;">:</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx31.6"> </span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx31.7" style="color:#1469B0;">[</span> </div> <div class="ltx_listingline" id="lstnumberx32"> <span class="ltx_tag ltx_tag_listingline">32</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx32.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx32.2">"</span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx32.3">Diplomatic</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx32.4"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx32.5">relations</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx32.6"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx32.7">are</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx32.8"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx32.9">established</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx32.10"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx32.11">between</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx32.12"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx32.13">Ukraine</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx32.14"> </span><span class="ltx_text ltx_lst_identifier ltx_font_typewriter" id="lstnumberx32.15">and</span><span class="ltx_text ltx_font_typewriter" id="lstnumberx32.16">"</span> </div> <div class="ltx_listingline" id="lstnumberx33"> <span class="ltx_tag ltx_tag_listingline">33</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx33.1"> </span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx33.2" style="color:#1469B0;">]</span> </div> <div class="ltx_listingline" id="lstnumberx34"> <span class="ltx_tag ltx_tag_listingline">34</span><span class="ltx_text ltx_lst_space ltx_font_typewriter" id="lstnumberx34.1"> </span><span class="ltx_text ltx_font_typewriter" id="lstnumberx34.2">}</span><span class="ltx_text ltx_lst_literate ltx_font_typewriter" id="lstnumberx34.3" style="color:#990000;">,</span> </div> </div> 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