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name="searchtype"><option value="all">All fields</option><option value="title">Title</option><option selected value="author">Author(s)</option><option value="abstract">Abstract</option><option value="comments">Comments</option><option value="journal_ref">Journal reference</option><option value="acm_class">ACM classification</option><option value="msc_class">MSC classification</option><option value="report_num">Report number</option><option value="paper_id">arXiv identifier</option><option value="doi">DOI</option><option value="orcid">ORCID</option><option value="license">License (URI)</option><option value="author_id">arXiv author ID</option><option value="help">Help pages</option><option value="full_text">Full text</option></select> <input id="query" name="query" type="text" value="Luo, L"> <ul id="abstracts"><li><input checked id="abstracts-0" name="abstracts" type="radio" value="show"> <label for="abstracts-0">Show abstracts</label></li><li><input id="abstracts-1" name="abstracts" 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is-link">Go</button> </div> </div> </form> </div> </div> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Luo%2C+L&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Luo%2C+L&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Luo%2C+L&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> </ul> </nav> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.09314">arXiv:2411.09314</a> <span> [<a href="https://arxiv.org/pdf/2411.09314">pdf</a>, <a href="https://arxiv.org/format/2411.09314">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Theory of the lattice Boltzmann method: discrete effects due to advection </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lallemand%2C+P">Pierre Lallemand</a>, <a href="/search/math?searchtype=author&query=Dubois%2C+F">Fran莽ois Dubois</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Li-shi Luo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.09314v1-abstract-short" style="display: inline;"> Lattice Boltzmann models are briefly introduced together with references to methods used to predict their ability for simulations of systems described by partial differential equations that are first order in time and low order in space derivatives. Several previous works have been devoted to analyzing the accuracy of these models with special emphasis on deviations from pure Newtonian viscous be… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.09314v1-abstract-full').style.display = 'inline'; document.getElementById('2411.09314v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.09314v1-abstract-full" style="display: none;"> Lattice Boltzmann models are briefly introduced together with references to methods used to predict their ability for simulations of systems described by partial differential equations that are first order in time and low order in space derivatives. Several previous works have been devoted to analyzing the accuracy of these models with special emphasis on deviations from pure Newtonian viscous behaviour, related to higher order space derivatives of even order. The presentcontribution concentrates on possible inaccuracies of the advection behaviour linked to space derivatives of odd order. Detailed properties of advection-diffusion and athermal fluids are presented for two-dimensional situations allowing to propose situations that are accurate to third order in space derivatives. Simulations of the advection of a gaussian dot or vortex are presented. Similar results are discussed in appendices for three-dimensional advection-diffusion. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.09314v1-abstract-full').style.display = 'none'; document.getElementById('2411.09314v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.10195">arXiv:2410.10195</a> <span> [<a href="https://arxiv.org/pdf/2410.10195">pdf</a>, <a href="https://arxiv.org/format/2410.10195">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> A fully-decoupled second-order-in-time and unconditionally energy stable scheme for a phase-field model of two phase flow with variable density </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jinpeng Zhang</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Li Luo</a>, <a href="/search/math?searchtype=author&query=Wang%2C+X">Xiaoping Wang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.10195v2-abstract-short" style="display: inline;"> In this paper, we develop a second-order, fully decoupled, and energy-stable numerical scheme for the Cahn-Hilliard-Navier-Stokes model for two phase flow with variable density and viscosity. We propose a new decoupling Constant Scalar Auxiliary Variable (D-CSAV) method which is easy to generalize to schemes with high order accuracy in time. The method is designed using the "zero-energy-contributi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.10195v2-abstract-full').style.display = 'inline'; document.getElementById('2410.10195v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.10195v2-abstract-full" style="display: none;"> In this paper, we develop a second-order, fully decoupled, and energy-stable numerical scheme for the Cahn-Hilliard-Navier-Stokes model for two phase flow with variable density and viscosity. We propose a new decoupling Constant Scalar Auxiliary Variable (D-CSAV) method which is easy to generalize to schemes with high order accuracy in time. The method is designed using the "zero-energy-contribution" property while maintaining conservative time discretization for the "non-zero-energy-contribution" terms. Our algorithm simplifies to solving three independent linear elliptic systems per time step, two of them with constant coefficients. The update of all scalar auxiliary variables is explicit and decoupled from solving the phase field variable and velocity field. We rigorously prove unconditional energy stability of the scheme and perform extensive benchmark simulations to demonstrate accuracy and efficiency of the method. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.10195v2-abstract-full').style.display = 'none'; document.getElementById('2410.10195v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.09392">arXiv:2410.09392</a> <span> [<a href="https://arxiv.org/pdf/2410.09392">pdf</a>, <a href="https://arxiv.org/ps/2410.09392">ps</a>, <a href="https://arxiv.org/format/2410.09392">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Finite-time stability of nonlinear conformable fractional-order delayed impulsive systems: Impulsive control and perturbation perspectives </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+L">L. Luo</a>, <a href="/search/math?searchtype=author&query=Li%2C+L">L. Li</a>, <a href="/search/math?searchtype=author&query=Liu%2C+Z">Z. Liu</a>, <a href="/search/math?searchtype=author&query=Shi%2C+J">J. Shi</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.09392v1-abstract-short" style="display: inline;"> This paper investigates the finite-time stability (FTS) of nonlinear conformable fractional-order delayed impulsive systems (CFODISs). Using the conformable fractional-order (CFO) derivative framework, we derive a novel FTS result by extending the existing works on continuous integer-order (IO) systems. This result highlights that the settling time of continuous CFO systems depends on the system o… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.09392v1-abstract-full').style.display = 'inline'; document.getElementById('2410.09392v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.09392v1-abstract-full" style="display: none;"> This paper investigates the finite-time stability (FTS) of nonlinear conformable fractional-order delayed impulsive systems (CFODISs). Using the conformable fractional-order (CFO) derivative framework, we derive a novel FTS result by extending the existing works on continuous integer-order (IO) systems. This result highlights that the settling time of continuous CFO systems depends on the system order and plays a crucial role in discussing FTS scenarios subject to delayed impulses. We establish Lyapunov-based FTS criteria for CFODISs, considering both impulsive control and impulsive perturbation. Additionally, we estimate the settling time for both cases, revealing distinct forms compared to the IO case. We apply the theoretical results to delayed impulsive conformable fractional-order memristive neural networks (CFOMNNs) under an elaborately designed controller. We present several simulations to illustrate the validity and applicability of our results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.09392v1-abstract-full').style.display = 'none'; document.getElementById('2410.09392v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.14873">arXiv:2408.14873</a> <span> [<a href="https://arxiv.org/pdf/2408.14873">pdf</a>, <a href="https://arxiv.org/format/2408.14873">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Robotics">cs.RO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Robo-GS: A Physics Consistent Spatial-Temporal Model for Robotic Arm with Hybrid Representation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lou%2C+H">Haozhe Lou</a>, <a href="/search/math?searchtype=author&query=Liu%2C+Y">Yurong Liu</a>, <a href="/search/math?searchtype=author&query=Pan%2C+Y">Yike Pan</a>, <a href="/search/math?searchtype=author&query=Geng%2C+Y">Yiran Geng</a>, <a href="/search/math?searchtype=author&query=Chen%2C+J">Jianteng Chen</a>, <a href="/search/math?searchtype=author&query=Ma%2C+W">Wenlong Ma</a>, <a href="/search/math?searchtype=author&query=Li%2C+C">Chenglong Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+L">Lin Wang</a>, <a href="/search/math?searchtype=author&query=Feng%2C+H">Hengzhen Feng</a>, <a href="/search/math?searchtype=author&query=Shi%2C+L">Lu Shi</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Liyi Luo</a>, <a href="/search/math?searchtype=author&query=Shi%2C+Y">Yongliang Shi</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2408.14873v2-abstract-short" style="display: inline;"> Real2Sim2Real plays a critical role in robotic arm control and reinforcement learning, yet bridging this gap remains a significant challenge due to the complex physical properties of robots and the objects they manipulate. Existing methods lack a comprehensive solution to accurately reconstruct real-world objects with spatial representations and their associated physics attributes. We propose a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.14873v2-abstract-full').style.display = 'inline'; document.getElementById('2408.14873v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.14873v2-abstract-full" style="display: none;"> Real2Sim2Real plays a critical role in robotic arm control and reinforcement learning, yet bridging this gap remains a significant challenge due to the complex physical properties of robots and the objects they manipulate. Existing methods lack a comprehensive solution to accurately reconstruct real-world objects with spatial representations and their associated physics attributes. We propose a Real2Sim pipeline with a hybrid representation model that integrates mesh geometry, 3D Gaussian kernels, and physics attributes to enhance the digital asset representation of robotic arms. This hybrid representation is implemented through a Gaussian-Mesh-Pixel binding technique, which establishes an isomorphic mapping between mesh vertices and Gaussian models. This enables a fully differentiable rendering pipeline that can be optimized through numerical solvers, achieves high-fidelity rendering via Gaussian Splatting, and facilitates physically plausible simulation of the robotic arm's interaction with its environment using mesh-based methods. The code,full presentation and datasets will be made publicly available at our website https://robostudioapp.com <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.14873v2-abstract-full').style.display = 'none'; document.getElementById('2408.14873v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.06351">arXiv:2408.06351</a> <span> [<a href="https://arxiv.org/pdf/2408.06351">pdf</a>, <a href="https://arxiv.org/format/2408.06351">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Signal Processing">eess.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> A Probabilistic Approach for Queue Length Estimation Using License Plate Recognition Data: Considering Overtaking in Multi-lane Scenarios </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+L">Lyuzhou Luo</a>, <a href="/search/math?searchtype=author&query=Wu%2C+H">Hao Wu</a>, <a href="/search/math?searchtype=author&query=Liu%2C+J">Jiahao Liu</a>, <a href="/search/math?searchtype=author&query=Tang%2C+K">Keshuang Tang</a>, <a href="/search/math?searchtype=author&query=Tan%2C+C">Chaopeng Tan</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2408.06351v1-abstract-short" style="display: inline;"> Multi-section license plate recognition (LPR) data provides input-output information and sampled travel times of the investigated link, serving as an ideal data source for lane-based queue length estimation in recent studies. However, most of these studies assumed the strict FIFO rule or a specific arrival process, thus ignoring the potential impact of overtaking and the variation of traffic flows… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.06351v1-abstract-full').style.display = 'inline'; document.getElementById('2408.06351v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.06351v1-abstract-full" style="display: none;"> Multi-section license plate recognition (LPR) data provides input-output information and sampled travel times of the investigated link, serving as an ideal data source for lane-based queue length estimation in recent studies. However, most of these studies assumed the strict FIFO rule or a specific arrival process, thus ignoring the potential impact of overtaking and the variation of traffic flows, especially in multi-lane scenarios. To address this issue, we propose a probabilistic approach to derive the stochastic queue length by constructing a conditional probability model of no-delay arrival time (NAT), i.e., the arrival time of vehicles without experiencing any delay, based on multi-section LPR data. First, the NAT conditions for all vehicles are established based on upstream and downstream vehicle departure times and sequences. To reduce the computational dimensionality and complexity, a DP-based algorithm is developed for vehicle group partitioning based on potential interactions between vehicles. Then, the conditional probability of NATs of each vehicle group is derived and an MCMC sampling method is employed for calculation. Subsequently, the stochastic queue profile and maximum queue length for each cycle can be derived based on the NATs of vehicles. Eventually, to leverage the LPR data sufficiently, we extend our approach to multi-lane scenarios, where the problem can be converted to a weighted general exact coverage problem and solved by a backtracking algorithm with heuristics. Empirical and simulation experiments have shown that the proposed approach outperforms the state-of-the-art method, demonstrating significant improvements in accuracy and robustness across various traffic conditions, including different V/C ratios, matching rates, and FIFO violation rates. In addition, the performance of the proposed approach can be further improved by utilizing multi-lane LPR data. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.06351v1-abstract-full').style.display = 'none'; document.getElementById('2408.06351v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">30 pages, 20 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.14839">arXiv:2407.14839</a> <span> [<a href="https://arxiv.org/pdf/2407.14839">pdf</a>, <a href="https://arxiv.org/ps/2407.14839">ps</a>, <a href="https://arxiv.org/format/2407.14839">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Optimizing over Multiple Distributions under Generalized Quasar-Convexity Condition </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ding%2C+S">Shihong Ding</a>, <a href="/search/math?searchtype=author&query=Yang%2C+L">Long Yang</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Luo Luo</a>, <a href="/search/math?searchtype=author&query=Fang%2C+C">Cong Fang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.14839v2-abstract-short" style="display: inline;"> We study a typical optimization model where the optimization variable is composed of multiple probability distributions. Though the model appears frequently in practice, such as for policy problems, it lacks specific analysis in the general setting. For this optimization problem, we propose a new structural condition/landscape description named generalized quasar-convexity (GQC) beyond the realms… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.14839v2-abstract-full').style.display = 'inline'; document.getElementById('2407.14839v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.14839v2-abstract-full" style="display: none;"> We study a typical optimization model where the optimization variable is composed of multiple probability distributions. Though the model appears frequently in practice, such as for policy problems, it lacks specific analysis in the general setting. For this optimization problem, we propose a new structural condition/landscape description named generalized quasar-convexity (GQC) beyond the realms of convexity. In contrast to original quasar-convexity \citep{hinder2020near}, GQC allows an individual quasar-convex parameter $纬_i$ for each variable block $i$ and the smaller of $纬_i$ implies less block-convexity. To minimize the objective function, we consider a generalized oracle termed as the internal function that includes the standard gradient oracle as a special case. We provide optimistic mirror descent (OMD) for multiple distributions and prove that the algorithm can achieve an adaptive $\tilde{\mathcal{O}}((\sum_{i=1}^d1/纬_i)蔚^{-1})$ iteration complexity to find an $epsilon$-suboptimal global solution without pre-known the exact values of $纬_i$ when the objective admits "polynomial-like" structural. Notably, it achieves iteration complexity that does not explicitly depend on the number of distributions and strictly faster $(\sum_{i=1}^d 1/纬_i \text{ v.s. } d\max_{i\in[1:d]} 1/纬_i)$ than mirror decent methods. We also extend GQC to the minimax optimization problem proposing the generalized quasar-convexity-concavity (GQCC) condition and a decentralized variant of OMD with regularization. Finally, we show the applications of our algorithmic framework on discounted Markov Decision Processes problem and Markov games, which bring new insights on the landscape analysis of reinforcement learning. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.14839v2-abstract-full').style.display = 'none'; document.getElementById('2407.14839v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.03195">arXiv:2407.03195</a> <span> [<a href="https://arxiv.org/pdf/2407.03195">pdf</a>, <a href="https://arxiv.org/format/2407.03195">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Incremental Gauss--Newton Methods with Superlinear Convergence Rates </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhou%2C+Z">Zhiling Zhou</a>, <a href="/search/math?searchtype=author&query=Liu%2C+Z">Zhuanghua Liu</a>, <a href="/search/math?searchtype=author&query=Liu%2C+C">Chengchang Liu</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Luo Luo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.03195v1-abstract-short" style="display: inline;"> This paper addresses the challenge of solving large-scale nonlinear equations with H枚lder continuous Jacobians. We introduce a novel Incremental Gauss--Newton (IGN) method within explicit superlinear convergence rate, which outperforms existing methods that only achieve linear convergence rate. In particular, we formulate our problem by the nonlinear least squares with finite-sum structure, and ou… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.03195v1-abstract-full').style.display = 'inline'; document.getElementById('2407.03195v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.03195v1-abstract-full" style="display: none;"> This paper addresses the challenge of solving large-scale nonlinear equations with H枚lder continuous Jacobians. We introduce a novel Incremental Gauss--Newton (IGN) method within explicit superlinear convergence rate, which outperforms existing methods that only achieve linear convergence rate. In particular, we formulate our problem by the nonlinear least squares with finite-sum structure, and our method incrementally iterates with the information of one component in each round. We also provide a mini-batch extension to our IGN method that obtains an even faster superlinear convergence rate. Furthermore, we conduct numerical experiments to show the advantages of the proposed methods. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.03195v1-abstract-full').style.display = 'none'; document.getElementById('2407.03195v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">37 pages, 9 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.16126">arXiv:2405.16126</a> <span> [<a href="https://arxiv.org/pdf/2405.16126">pdf</a>, <a href="https://arxiv.org/format/2405.16126">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Near-Optimal Distributed Minimax Optimization under the Second-Order Similarity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhou%2C+Q">Qihao Zhou</a>, <a href="/search/math?searchtype=author&query=Ye%2C+H">Haishan Ye</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Luo Luo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.16126v1-abstract-short" style="display: inline;"> This paper considers the distributed convex-concave minimax optimization under the second-order similarity. We propose stochastic variance-reduced optimistic gradient sliding (SVOGS) method, which takes the advantage of the finite-sum structure in the objective by involving the mini-batch client sampling and variance reduction. We prove SVOGS can achieve the $\varepsilon$-duality gap within commun… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.16126v1-abstract-full').style.display = 'inline'; document.getElementById('2405.16126v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.16126v1-abstract-full" style="display: none;"> This paper considers the distributed convex-concave minimax optimization under the second-order similarity. We propose stochastic variance-reduced optimistic gradient sliding (SVOGS) method, which takes the advantage of the finite-sum structure in the objective by involving the mini-batch client sampling and variance reduction. We prove SVOGS can achieve the $\varepsilon$-duality gap within communication rounds of ${\mathcal O}(未D^2/\varepsilon)$, communication complexity of ${\mathcal O}(n+\sqrt{n}未D^2/\varepsilon)$, and local gradient calls of $\tilde{\mathcal O}(n+(\sqrt{n}未+L)D^2/\varepsilon\log(1/\varepsilon))$, where $n$ is the number of nodes, $未$ is the degree of the second-order similarity, $L$ is the smoothness parameter and $D$ is the diameter of the constraint set. We can verify that all of above complexity (nearly) matches the corresponding lower bounds. For the specific $渭$-strongly-convex-$渭$-strongly-convex case, our algorithm has the upper bounds on communication rounds, communication complexity, and local gradient calls of $\mathcal O(未/渭\log(1/\varepsilon))$, ${\mathcal O}((n+\sqrt{n}未/渭)\log(1/\varepsilon))$, and $\tilde{\mathcal O}(n+(\sqrt{n}未+L)/渭)\log(1/\varepsilon))$ respectively, which are also nearly tight. Furthermore, we conduct the numerical experiments to show the empirical advantages of proposed method. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.16126v1-abstract-full').style.display = 'none'; document.getElementById('2405.16126v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2403.03687">arXiv:2403.03687</a> <span> [<a href="https://arxiv.org/pdf/2403.03687">pdf</a>, <a href="https://arxiv.org/format/2403.03687">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Precise upper deviation estimates for the maximum of a branching random walk </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+L">Lianghui Luo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2403.03687v1-abstract-short" style="display: inline;"> We consider the precise upper large deviations estimates for the maximal displacement of a branching random walk. In addition, we obtain a description of the extremal process of the branching random walk conditioned on this large deviations event. This introduces a family of point measure playing a role similar to the decoration measures introduced in [9] for branching Brownian motion. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.03687v1-abstract-full" style="display: none;"> We consider the precise upper large deviations estimates for the maximal displacement of a branching random walk. In addition, we obtain a description of the extremal process of the branching random walk conditioned on this large deviations event. This introduces a family of point measure playing a role similar to the decoration measures introduced in [9] for branching Brownian motion. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.03687v1-abstract-full').style.display = 'none'; document.getElementById('2403.03687v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">31 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.13976">arXiv:2402.13976</a> <span> [<a href="https://arxiv.org/pdf/2402.13976">pdf</a>, <a href="https://arxiv.org/ps/2402.13976">ps</a>, <a href="https://arxiv.org/format/2402.13976">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Non-Markovian maximal couplings and a vertical reflection principle on a class of sub-Riemannian manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+L">Liangbing Luo</a>, <a href="/search/math?searchtype=author&query=Neel%2C+R+W">Robert W. Neel</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2402.13976v1-abstract-short" style="display: inline;"> We develop non-Markovian, non-co-adapted couplings for sub-Riemannian Brownian motions in various sub-Riemannian manifolds, namely the three-dimensional Heisenberg group, higher-dimensional non-isotropic Heisenberg groups, SL(2,R) and its universal cover, and SU(2,C). Our primary focus is on the situation when the processes start from two points on the same vertical fiber, since in general co-adap… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.13976v1-abstract-full').style.display = 'inline'; document.getElementById('2402.13976v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.13976v1-abstract-full" style="display: none;"> We develop non-Markovian, non-co-adapted couplings for sub-Riemannian Brownian motions in various sub-Riemannian manifolds, namely the three-dimensional Heisenberg group, higher-dimensional non-isotropic Heisenberg groups, SL(2,R) and its universal cover, and SU(2,C). Our primary focus is on the situation when the processes start from two points on the same vertical fiber, since in general co-adapted couplings cannot give the sharp rate for the coupling time in this case. Then for general points, we use this vertical coupling as the second stage of a two-stage coupling. Non-Markovian couplings in this context were first introduced by Banerjee-Gordina-Mariano, for the three-dimensional Heisenberg group, and were more recently extended by B茅n茅fice to SL(2,R) and SU(2,C), using a detailed consideration of the Brownian bridge. In contrast, our couplings are based on global isometries of the space, giving couplings that are maximal, as well as making the construction relatively simple and uniform across different manifolds. Moreover, this construction gives a coupling time that equals a hitting time for the vertical component of one of the Brownian motions, and this vertical component satisfies a reflection principle, which is useful in explicitly bounding the tail probability of the coupling time. We estimate the coupling time in these various situations and give applications to inequalities for the heat semigroup. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.13976v1-abstract-full').style.display = 'none'; document.getElementById('2402.13976v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">37 pages; no figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 58J65 (Primary) 53C17; 58J35; 60J45 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.03872">arXiv:2402.03872</a> <span> [<a href="https://arxiv.org/pdf/2402.03872">pdf</a>, <a href="https://arxiv.org/format/2402.03872">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Upper deviation probabilities for level sets of a supercritical branching random walk </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+S">Shuxiong Zhang</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Lianghui Luo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2402.03872v1-abstract-short" style="display: inline;"> Given a supercritical branching random walk $\{Z_n\}_{n\geq 0}$ on $\mathbb{R}$, let $Z_n([y,\infty))$ be the number of particles located in $[y,\infty)\subset\mathbb{R}$ at generation $n$. Let $m$ be the mean of the offspring law of $\{Z_n\}_{n\geq 0}$ and $I(x)$ be the large deviation rate function of the underlying random walk of $\{Z_n\}_{n\geq 0}$. It is known from [6] that under some mild co… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.03872v1-abstract-full').style.display = 'inline'; document.getElementById('2402.03872v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.03872v1-abstract-full" style="display: none;"> Given a supercritical branching random walk $\{Z_n\}_{n\geq 0}$ on $\mathbb{R}$, let $Z_n([y,\infty))$ be the number of particles located in $[y,\infty)\subset\mathbb{R}$ at generation $n$. Let $m$ be the mean of the offspring law of $\{Z_n\}_{n\geq 0}$ and $I(x)$ be the large deviation rate function of the underlying random walk of $\{Z_n\}_{n\geq 0}$. It is known from [6] that under some mild conditions, for $x\in(0,x^*)$, $n^{-1}\log Z_n([nx,\infty))$ converges almost surely to $\log m- I(x)$ on the event of nonextinction as $n\to\infty$, where $x^*$ is the speed of maximal position of the branching random walk. In this work, we investigate its upper deviation probabilities, in other words, the convergence rates of \[\mathbb{P}(Z_n([xn,\infty))\geq e^{an})\] as $n\to\infty$, where $x>0$ and $a>(\log m- I(x))^+$. This paper is a counterpart work of the lower deviation probabilities [28] and also completes those results in [1] for the branching Brownian motion. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.03872v1-abstract-full').style.display = 'none'; document.getElementById('2402.03872v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">28 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.02569">arXiv:2402.02569</a> <span> [<a href="https://arxiv.org/pdf/2402.02569">pdf</a>, <a href="https://arxiv.org/format/2402.02569">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> On the Complexity of Finite-Sum Smooth Optimization under the Polyak-艁ojasiewicz Condition </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bai%2C+Y">Yunyan Bai</a>, <a href="/search/math?searchtype=author&query=Liu%2C+Y">Yuxing Liu</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Luo Luo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2402.02569v1-abstract-short" style="display: inline;"> This paper considers the optimization problem of the form $\min_{{\bf x}\in{\mathbb R}^d} f({\bf x})\triangleq \frac{1}{n}\sum_{i=1}^n f_i({\bf x})$, where $f(\cdot)$ satisfies the Polyak--艁ojasiewicz (PL) condition with parameter $渭$ and $\{f_i(\cdot)\}_{i=1}^n$ is $L$-mean-squared smooth. We show that any gradient method requires at least $惟(n+魏\sqrt{n}\log(1/蔚))$ incremental first-order oracle… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.02569v1-abstract-full').style.display = 'inline'; document.getElementById('2402.02569v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.02569v1-abstract-full" style="display: none;"> This paper considers the optimization problem of the form $\min_{{\bf x}\in{\mathbb R}^d} f({\bf x})\triangleq \frac{1}{n}\sum_{i=1}^n f_i({\bf x})$, where $f(\cdot)$ satisfies the Polyak--艁ojasiewicz (PL) condition with parameter $渭$ and $\{f_i(\cdot)\}_{i=1}^n$ is $L$-mean-squared smooth. We show that any gradient method requires at least $惟(n+魏\sqrt{n}\log(1/蔚))$ incremental first-order oracle (IFO) calls to find an $蔚$-suboptimal solution, where $魏\triangleq L/渭$ is the condition number of the problem. This result nearly matches upper bounds of IFO complexity for best-known first-order methods. We also study the problem of minimizing the PL function in the distributed setting such that the individuals $f_1(\cdot),\dots,f_n(\cdot)$ are located on a connected network of $n$ agents. We provide lower bounds of $惟(魏/\sqrt纬\,\log(1/蔚))$, $惟((魏+蟿魏/\sqrt纬\,)\log(1/蔚))$ and $惟\big(n+魏\sqrt{n}\log(1/蔚)\big)$ for communication rounds, time cost and local first-order oracle calls respectively, where $纬\in(0,1]$ is the spectral gap of the mixing matrix associated with the network and~$蟿>0$ is the time cost of per communication round. Furthermore, we propose a decentralized first-order method that nearly matches above lower bounds in expectation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.02569v1-abstract-full').style.display = 'none'; document.getElementById('2402.02569v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.02359">arXiv:2402.02359</a> <span> [<a href="https://arxiv.org/pdf/2402.02359">pdf</a>, <a href="https://arxiv.org/format/2402.02359">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Incremental Quasi-Newton Methods with Faster Superlinear Convergence Rates </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+Z">Zhuanghua Liu</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Luo Luo</a>, <a href="/search/math?searchtype=author&query=Low%2C+B+K+H">Bryan Kian Hsiang Low</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2402.02359v1-abstract-short" style="display: inline;"> We consider the finite-sum optimization problem, where each component function is strongly convex and has Lipschitz continuous gradient and Hessian. The recently proposed incremental quasi-Newton method is based on BFGS update and achieves a local superlinear convergence rate that is dependent on the condition number of the problem. This paper proposes a more efficient quasi-Newton method by incor… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.02359v1-abstract-full').style.display = 'inline'; document.getElementById('2402.02359v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.02359v1-abstract-full" style="display: none;"> We consider the finite-sum optimization problem, where each component function is strongly convex and has Lipschitz continuous gradient and Hessian. The recently proposed incremental quasi-Newton method is based on BFGS update and achieves a local superlinear convergence rate that is dependent on the condition number of the problem. This paper proposes a more efficient quasi-Newton method by incorporating the symmetric rank-1 update into the incremental framework, which results in the condition-number-free local superlinear convergence rate. Furthermore, we can boost our method by applying the block update on the Hessian approximation, which leads to an even faster local convergence rate. The numerical experiments show the proposed methods significantly outperform the baseline methods. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.02359v1-abstract-full').style.display = 'none'; document.getElementById('2402.02359v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.05438">arXiv:2311.05438</a> <span> [<a href="https://arxiv.org/pdf/2311.05438">pdf</a>, <a href="https://arxiv.org/format/2311.05438">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> A branch-and-price approach for the nurse rostering problem with multiple units </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Hu%2C+W">Wanzhe Hu</a>, <a href="/search/math?searchtype=author&query=He%2C+X">Xiaozhou He</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Li Luo</a>, <a href="/search/math?searchtype=author&query=Pardalos%2C+P+M">Panos M. Pardalos</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2311.05438v1-abstract-short" style="display: inline;"> In this paper, we study the nurse rostering problem that considers multiple units and many soft time-related constraints. An efficient branch and price solution approach that relies on a fast algorithm to solve the pricing subproblem of the column generation process is presented. For the nurse rostering problem, its pricing subproblem can be formulated as a shortest path problem with resource cons… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.05438v1-abstract-full').style.display = 'inline'; document.getElementById('2311.05438v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.05438v1-abstract-full" style="display: none;"> In this paper, we study the nurse rostering problem that considers multiple units and many soft time-related constraints. An efficient branch and price solution approach that relies on a fast algorithm to solve the pricing subproblem of the column generation process is presented. For the nurse rostering problem, its pricing subproblem can be formulated as a shortest path problem with resource constraints, which has been the backbone of several solutions for several classical problems like vehicle routing problems. However, approaches that perform well on these problems cannot be used since most constraints in the nurse rostering problem are soft. Based on ideas borrowed from global constraints in constraint programming to model rostering problems, an efficient dynamic programming algorithm with novel label definitions and dominating rules specific to soft time-related constraints is proposed. In addition, several acceleration strategies are employed to improve the branch and price algorithm. Computational results on instances of different sizes indicate that the proposed algorithm is a promising solution for the nurse rostering problem with multiple units. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.05438v1-abstract-full').style.display = 'none'; document.getElementById('2311.05438v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2310.14436">arXiv:2310.14436</a> <span> [<a href="https://arxiv.org/pdf/2310.14436">pdf</a>, <a href="https://arxiv.org/ps/2310.14436">ps</a>, <a href="https://arxiv.org/format/2310.14436">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Metric Geometry">math.MG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Construction of a Dirichlet form on metric measure spaces of controlled geometry </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Butaev%2C+A">Almaz Butaev</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Liangbing Luo</a>, <a href="/search/math?searchtype=author&query=Shanmugalingam%2C+N">Nageswari Shanmugalingam</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2310.14436v1-abstract-short" style="display: inline;"> Given a compact doubling metric measure space $X$ that supports a $2$-Poincar茅 inequality, we construct a Dirichlet form on $N^{1,2}(X)$ that is comparable to the upper gradient energy form on $N^{1,2}(X)$. Our approach is based on the approximation of $X$ by a family of graphs that is doubling and supports a $2$-Poincar茅 inequality. We construct a bilinear form on $N^{1,2}(X)$ using the Dirichlet… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.14436v1-abstract-full').style.display = 'inline'; document.getElementById('2310.14436v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2310.14436v1-abstract-full" style="display: none;"> Given a compact doubling metric measure space $X$ that supports a $2$-Poincar茅 inequality, we construct a Dirichlet form on $N^{1,2}(X)$ that is comparable to the upper gradient energy form on $N^{1,2}(X)$. Our approach is based on the approximation of $X$ by a family of graphs that is doubling and supports a $2$-Poincar茅 inequality. We construct a bilinear form on $N^{1,2}(X)$ using the Dirichlet form on the graph. We show that the $螕$-limit $\mathcal{E}$ of this family of bilinear forms (by taking a subsequence) exists and that $\mathcal{E}$ is a Dirichlet form on $X$. Properties of $\mathcal{E}$ are established. Moreover, we prove that $\mathcal{E}$ has the property of matching boundary values on a domain $惟\subseteq X$. This construction makes it possible to approximate harmonic functions (with respect to the Dirichlet form $\mathcal{E}$) on a domain in $X$ with a prescribed Lipschitz boundary data via a numerical scheme dictated by the approximating Dirichlet forms, which are discrete objects. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.14436v1-abstract-full').style.display = 'none'; document.getElementById('2310.14436v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 31C25; 46E35; 49Q20; 65N55 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2310.13470">arXiv:2310.13470</a> <span> [<a href="https://arxiv.org/pdf/2310.13470">pdf</a>, <a href="https://arxiv.org/ps/2310.13470">ps</a>, <a href="https://arxiv.org/format/2310.13470">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Logarithmic Sobolev inequalities on homogeneous spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gordina%2C+M">Maria Gordina</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Liangbing Luo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2310.13470v1-abstract-short" style="display: inline;"> We consider sub-Riemannian manifolds which are homogeneous spaces equipped with a natural sub-Riemannian structure induced by a transitive action by a Lie group. In such a setting, the corresponding sub-Laplacian is not an elliptic but a hypoelliptic operator. We study logarithmic Sobolev inequalities with respect to the hypoelliptic heat kernel measure on such homogeneous spaces. We show that the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.13470v1-abstract-full').style.display = 'inline'; document.getElementById('2310.13470v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2310.13470v1-abstract-full" style="display: none;"> We consider sub-Riemannian manifolds which are homogeneous spaces equipped with a natural sub-Riemannian structure induced by a transitive action by a Lie group. In such a setting, the corresponding sub-Laplacian is not an elliptic but a hypoelliptic operator. We study logarithmic Sobolev inequalities with respect to the hypoelliptic heat kernel measure on such homogeneous spaces. We show that the logarithmic Sobolev constant can be chosen to depend only on the Lie group acting transitively on such a homogeneous space but the constant is independent of the action of its isotropy group. This approach allows us to track the dependence of the logarithmic Sobolev constant on the geometry of the underlying space, in particular we are able to show that the logarithmic Sobolev constants is independent of the dimension of the underlying spaces in several examples. We illustrate the results by considering the Grushin plane, non-isotropic Heisenberg groups, Heisenberg-like groups, Hopf fibration, $\operatorname{SO}(3)$, $\operatorname{SO}(4)$, and compact Heisenberg manifolds. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.13470v1-abstract-full').style.display = 'none'; document.getElementById('2310.13470v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2310.12721">arXiv:2310.12721</a> <span> [<a href="https://arxiv.org/pdf/2310.12721">pdf</a>, <a href="https://arxiv.org/ps/2310.12721">ps</a>, <a href="https://arxiv.org/format/2310.12721">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Invariant theory of $\imath$quantum groups of type AIII </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+L">Li Luo</a>, <a href="/search/math?searchtype=author&query=Xu%2C+Z">Zheming Xu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2310.12721v2-abstract-short" style="display: inline;"> We develop an invariant theory of quasi-split $\imath$quantum groups $\mathbf{U}_n^\imath$ of type AIII on a tensor space associated to $\imath$Howe dualities. The first and second fundamental theorems for $\mathbf{U}_n^\imath$-invariants are derived. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2310.12721v2-abstract-full" style="display: none;"> We develop an invariant theory of quasi-split $\imath$quantum groups $\mathbf{U}_n^\imath$ of type AIII on a tensor space associated to $\imath$Howe dualities. The first and second fundamental theorems for $\mathbf{U}_n^\imath$-invariants are derived. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.12721v2-abstract-full').style.display = 'none'; document.getElementById('2310.12721v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">17 pages. Fixed some mild errors and added more details</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2310.11973">arXiv:2310.11973</a> <span> [<a href="https://arxiv.org/pdf/2310.11973">pdf</a>, <a href="https://arxiv.org/format/2310.11973">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Distributed, Parallel, and Cluster Computing">cs.DC</span> </div> </div> <p class="title is-5 mathjax"> Decentralized Gradient-Free Methods for Stochastic Non-Smooth Non-Convex Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lin%2C+Z">Zhenwei Lin</a>, <a href="/search/math?searchtype=author&query=Xia%2C+J">Jingfan Xia</a>, <a href="/search/math?searchtype=author&query=Deng%2C+Q">Qi Deng</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Luo Luo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2310.11973v1-abstract-short" style="display: inline;"> We consider decentralized gradient-free optimization of minimizing Lipschitz continuous functions that satisfy neither smoothness nor convexity assumption. We propose two novel gradient-free algorithms, the Decentralized Gradient-Free Method (DGFM) and its variant, the Decentralized Gradient-Free Method$^+$ (DGFM$^{+}$). Based on the techniques of randomized smoothing and gradient tracking, DGFM r… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.11973v1-abstract-full').style.display = 'inline'; document.getElementById('2310.11973v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2310.11973v1-abstract-full" style="display: none;"> We consider decentralized gradient-free optimization of minimizing Lipschitz continuous functions that satisfy neither smoothness nor convexity assumption. We propose two novel gradient-free algorithms, the Decentralized Gradient-Free Method (DGFM) and its variant, the Decentralized Gradient-Free Method$^+$ (DGFM$^{+}$). Based on the techniques of randomized smoothing and gradient tracking, DGFM requires the computation of the zeroth-order oracle of a single sample in each iteration, making it less demanding in terms of computational resources for individual computing nodes. Theoretically, DGFM achieves a complexity of $\mathcal O(d^{3/2}未^{-1}\varepsilon ^{-4})$ for obtaining an $(未,\varepsilon)$-Goldstein stationary point. DGFM$^{+}$, an advanced version of DGFM, incorporates variance reduction to further improve the convergence behavior. It samples a mini-batch at each iteration and periodically draws a larger batch of data, which improves the complexity to $\mathcal O(d^{3/2}未^{-1} \varepsilon^{-3})$. Moreover, experimental results underscore the empirical advantages of our proposed algorithms when applied to real-world datasets. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.11973v1-abstract-full').style.display = 'none'; document.getElementById('2310.11973v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2308.06446">arXiv:2308.06446</a> <span> [<a href="https://arxiv.org/pdf/2308.06446">pdf</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> The S-rule and 1-d representation for the traversal of a planar graph in AEC industry </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+L">Luxin Luo</a>, <a href="/search/math?searchtype=author&query=Kong%2C+S">Sida Kong</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2308.06446v1-abstract-short" style="display: inline;"> Based on two trivial observations of the AEC industry, this paper proposes a traversal method ("S-rule") and expression ("1-dimensional-graph") transformed from DFS. This traversal method conforms to the original cognitive logic of the AEC industry, and the 1-d expression has clear language characteristics while completely retaining the topological relationship of the planar graph : a sequence of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.06446v1-abstract-full').style.display = 'inline'; document.getElementById('2308.06446v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2308.06446v1-abstract-full" style="display: none;"> Based on two trivial observations of the AEC industry, this paper proposes a traversal method ("S-rule") and expression ("1-dimensional-graph") transformed from DFS. This traversal method conforms to the original cognitive logic of the AEC industry, and the 1-d expression has clear language characteristics while completely retaining the topological relationship of the planar graph : a sequence of finite symbols (vocabularies) under definite rules. Moreover, the language can be restored to a standard 2-d form that is isomorphic to the original planar graph, thus ensuring its visualization characteristics. Fragments of the 1-d language can be used as planar units for free combination and weighting, and as the data foundation to support advanced calculations including FEM and isomorphic matching. And after the 2-d graph is reduced to 1-d, any 3-d or higher-dimensional graphs can also be reduced to 1 or 2 dimensions. The first half of this paper (Chapter 1) takes the 4X4 standard grid as an example to introduce the prototype of S-rule and 1-d expression, and gives the mapping rule from 1-d expression to its editable text form. In the second half of this paper, the rule and expression are gradually extended to non-embedded planar graphs (Chapter 2) and embedded planar graphs (Chapter 3), and the "grammar" is finally summarized (Chapter 4). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.06446v1-abstract-full').style.display = 'none'; document.getElementById('2308.06446v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 August, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2307.15868">arXiv:2307.15868</a> <span> [<a href="https://arxiv.org/pdf/2307.15868">pdf</a>, <a href="https://arxiv.org/ps/2307.15868">ps</a>, <a href="https://arxiv.org/format/2307.15868">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Faster Stochastic Algorithms for Minimax Optimization under Polyak--艁ojasiewicz Conditions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Lesi Chen</a>, <a href="/search/math?searchtype=author&query=Yao%2C+B">Boyuan Yao</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Luo Luo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2307.15868v1-abstract-short" style="display: inline;"> This paper considers stochastic first-order algorithms for minimax optimization under Polyak--艁ojasiewicz (PL) conditions. We propose SPIDER-GDA for solving the finite-sum problem of the form $\min_x \max_y f(x,y)\triangleq \frac{1}{n} \sum_{i=1}^n f_i(x,y)$, where the objective function $f(x,y)$ is $渭_x$-PL in $x$ and $渭_y$-PL in $y$; and each $f_i(x,y)$ is $L$-smooth. We prove SPIDER-GDA could f… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.15868v1-abstract-full').style.display = 'inline'; document.getElementById('2307.15868v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2307.15868v1-abstract-full" style="display: none;"> This paper considers stochastic first-order algorithms for minimax optimization under Polyak--艁ojasiewicz (PL) conditions. We propose SPIDER-GDA for solving the finite-sum problem of the form $\min_x \max_y f(x,y)\triangleq \frac{1}{n} \sum_{i=1}^n f_i(x,y)$, where the objective function $f(x,y)$ is $渭_x$-PL in $x$ and $渭_y$-PL in $y$; and each $f_i(x,y)$ is $L$-smooth. We prove SPIDER-GDA could find an $蔚$-optimal solution within ${\mathcal O}\left((n + \sqrt{n}\,魏_x魏_y^2)\log (1/蔚)\right)$ stochastic first-order oracle (SFO) complexity, which is better than the state-of-the-art method whose SFO upper bound is ${\mathcal O}\big((n + n^{2/3}魏_x魏_y^2)\log (1/蔚)\big)$, where $魏_x\triangleq L/渭_x$ and $魏_y\triangleq L/渭_y$. For the ill-conditioned case, we provide an accelerated algorithm to reduce the computational cost further. It achieves $\tilde{\mathcal O}\big((n+\sqrt{n}\,魏_x魏_y)\log^2 (1/蔚)\big)$ SFO upper bound when $魏_y \gtrsim \sqrt{n}$. Our ideas also can be applied to the more general setting that the objective function only satisfies PL condition for one variable. Numerical experiments validate the superiority of proposed methods. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.15868v1-abstract-full').style.display = 'none'; document.getElementById('2307.15868v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 July, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">published in NeurIPS 2022; fix a mistake in the proof of Thm. 4.1 and polish the writing</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2307.00126">arXiv:2307.00126</a> <span> [<a href="https://arxiv.org/pdf/2307.00126">pdf</a>, <a href="https://arxiv.org/format/2307.00126">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Accelerating Inexact HyperGradient Descent for Bilevel Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+H">Haikuo Yang</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Luo Luo</a>, <a href="/search/math?searchtype=author&query=Li%2C+C+J">Chris Junchi Li</a>, <a href="/search/math?searchtype=author&query=Jordan%2C+M+I">Michael I. Jordan</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2307.00126v1-abstract-short" style="display: inline;"> We present a method for solving general nonconvex-strongly-convex bilevel optimization problems. Our method -- the \emph{Restarted Accelerated HyperGradient Descent} (\texttt{RAHGD}) method -- finds an $蔚$-first-order stationary point of the objective with $\tilde{\mathcal{O}}(魏^{3.25}蔚^{-1.75})$ oracle complexity, where $魏$ is the condition number of the lower-level objective and $蔚$ is the desir… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.00126v1-abstract-full').style.display = 'inline'; document.getElementById('2307.00126v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2307.00126v1-abstract-full" style="display: none;"> We present a method for solving general nonconvex-strongly-convex bilevel optimization problems. Our method -- the \emph{Restarted Accelerated HyperGradient Descent} (\texttt{RAHGD}) method -- finds an $蔚$-first-order stationary point of the objective with $\tilde{\mathcal{O}}(魏^{3.25}蔚^{-1.75})$ oracle complexity, where $魏$ is the condition number of the lower-level objective and $蔚$ is the desired accuracy. We also propose a perturbed variant of \texttt{RAHGD} for finding an $\big(蔚,\mathcal{O}(魏^{2.5}\sqrt蔚\,)\big)$-second-order stationary point within the same order of oracle complexity. Our results achieve the best-known theoretical guarantees for finding stationary points in bilevel optimization and also improve upon the existing upper complexity bound for finding second-order stationary points in nonconvex-strongly-concave minimax optimization problems, setting a new state-of-the-art benchmark. Empirical studies are conducted to validate the theoretical results in this paper. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.00126v1-abstract-full').style.display = 'none'; document.getElementById('2307.00126v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 June, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2305.17945">arXiv:2305.17945</a> <span> [<a href="https://arxiv.org/pdf/2305.17945">pdf</a>, <a href="https://arxiv.org/format/2305.17945">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Communication Efficient Distributed Newton Method with Fast Convergence Rates </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+C">Chengchang Liu</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Lesi Chen</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Luo Luo</a>, <a href="/search/math?searchtype=author&query=Lui%2C+J+C+S">John C. S. Lui</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2305.17945v1-abstract-short" style="display: inline;"> We propose a communication and computation efficient second-order method for distributed optimization. For each iteration, our method only requires $\mathcal{O}(d)$ communication complexity, where $d$ is the problem dimension. We also provide theoretical analysis to show the proposed method has the similar convergence rate as the classical second-order optimization algorithms. Concretely, our meth… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.17945v1-abstract-full').style.display = 'inline'; document.getElementById('2305.17945v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2305.17945v1-abstract-full" style="display: none;"> We propose a communication and computation efficient second-order method for distributed optimization. For each iteration, our method only requires $\mathcal{O}(d)$ communication complexity, where $d$ is the problem dimension. We also provide theoretical analysis to show the proposed method has the similar convergence rate as the classical second-order optimization algorithms. Concretely, our method can find~$\big(蔚, \sqrt{dL蔚}\,\big)$-second-order stationary points for nonconvex problem by $\mathcal{O}\big(\sqrt{dL}\,蔚^{-3/2}\big)$ iterations, where $L$ is the Lipschitz constant of Hessian. Moreover, it enjoys a local superlinear convergence under the strongly-convex assumption. Experiments on both convex and nonconvex problems show that our proposed method performs significantly better than baselines. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.17945v1-abstract-full').style.display = 'none'; document.getElementById('2305.17945v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 May, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Accepted in SIGKDD 2023</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2305.14633">arXiv:2305.14633</a> <span> [<a href="https://arxiv.org/pdf/2305.14633">pdf</a>, <a href="https://arxiv.org/ps/2305.14633">ps</a>, <a href="https://arxiv.org/format/2305.14633">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Asymptotic Schur algebras and cellularity of q-Schur algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cui%2C+W">Weideng Cui</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Li Luo</a>, <a href="/search/math?searchtype=author&query=Xu%2C+Z">Zheming Xu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2305.14633v1-abstract-short" style="display: inline;"> We prove that the q-Schur algebras of finite type introduced in [LW22] are cellular in the sense of Graham and Lehrer, which is a generalization of Geck's theorem on the cellularity of Hecke algebras of finite type. Moreover, we study special modules of the associated asymptotic Schur algebras and left cell representations of Schur algebras, which generalize Lusztig's work about special modules of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.14633v1-abstract-full').style.display = 'inline'; document.getElementById('2305.14633v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2305.14633v1-abstract-full" style="display: none;"> We prove that the q-Schur algebras of finite type introduced in [LW22] are cellular in the sense of Graham and Lehrer, which is a generalization of Geck's theorem on the cellularity of Hecke algebras of finite type. Moreover, we study special modules of the associated asymptotic Schur algebras and left cell representations of Schur algebras, which generalize Lusztig's work about special modules of asymptotic Hecke algebras and left cell representations of Weyl groups, respectively. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.14633v1-abstract-full').style.display = 'none'; document.getElementById('2305.14633v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 May, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">28 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20G43 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2304.10818">arXiv:2304.10818</a> <span> [<a href="https://arxiv.org/pdf/2304.10818">pdf</a>, <a href="https://arxiv.org/ps/2304.10818">ps</a>, <a href="https://arxiv.org/format/2304.10818">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Rings and Algebras">math.RA</span> </div> </div> <p class="title is-5 mathjax"> On some derivations of Lie conformal superalgebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+L">Lipeng Luo</a>, <a href="/search/math?searchtype=author&query=Asif%2C+S">Sania Asif</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2304.10818v1-abstract-short" style="display: inline;"> Let $\mathcal{R}$ be a Lie conformal superalgebra. In this paper, we first investigate the conformal derivation algebra $CDer(\mathcal{R})$, the conformal triple derivation algebra $CTDer(\mathcal{R})$, and the generalized conformal triple derivation algebra $GCTDer(\mathcal{R})$. Moreover, we determine the connection of these derivation algebras. Next, we give a complete classification of the (ge… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2304.10818v1-abstract-full').style.display = 'inline'; document.getElementById('2304.10818v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2304.10818v1-abstract-full" style="display: none;"> Let $\mathcal{R}$ be a Lie conformal superalgebra. In this paper, we first investigate the conformal derivation algebra $CDer(\mathcal{R})$, the conformal triple derivation algebra $CTDer(\mathcal{R})$, and the generalized conformal triple derivation algebra $GCTDer(\mathcal{R})$. Moreover, we determine the connection of these derivation algebras. Next, we give a complete classification of the (generalized) conformal triple derivation algebra on all finite simple Lie conformal superalgebras. More specifically, $CTDer(\mathcal{R})=CDer(\mathcal{R})$, where $\mathcal{R}$ is a finite simple Lie conformal superalgebra, but for $GCTDer(\mathcal{R})$, we obtain a conclusion that is closely related to $CDer(\mathcal{R})$. Furthermore, we evaluate the $(\varPhi, \varPsi)$-Lie triple derivations on Lie conformal superalgebra, where $\varPhi$ and $\varPsi$ are associated automorphism of $蠁_{x}\in gc(\mathcal R)$. We evaluated some fundamental properties of $(\varPhi, \varPsi)$- Lie triple derivations. Later, we introduce the definition of $(A, B, C, D)$-derivation on Lie conformal superalgebra. We obtain the relationships between the generalized conformal triple derivations and the conformal $(A, B, C, D)$-derivations on Lie conformal superalgebra. Finally, we have presented the triple homomorphism of Lie conformal superalgebras. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2304.10818v1-abstract-full').style.display = 'none'; document.getElementById('2304.10818v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 April, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">arXiv admin note: substantial text overlap with arXiv:2304.08067</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 15A99; 17B67; 17B10; 16G30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2304.08067">arXiv:2304.08067</a> <span> [<a href="https://arxiv.org/pdf/2304.08067">pdf</a>, <a href="https://arxiv.org/ps/2304.08067">ps</a>, <a href="https://arxiv.org/format/2304.08067">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Rings and Algebras">math.RA</span> </div> </div> <p class="title is-5 mathjax"> Conformal triple derivations and triple homomorphisms of Lie conformal algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Asif%2C+S">Sania Asif</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Lipeng Luo</a>, <a href="/search/math?searchtype=author&query=Hong%2C+Y">Yanyong Hong</a>, <a href="/search/math?searchtype=author&query=Wu%2C+Z">Zhixiang Wu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2304.08067v1-abstract-short" style="display: inline;"> Let $\mathcal{R}$ be a finite Lie conformal algebra. In this paper, we first investigate the conformal derivation algebra $CDer(\mathcal{R})$, the conformal triple derivation algebra $CTDer(\mathcal{R})$ and the generalized conformal triple derivation algebra $GCTDer(\mathcal{R})$. Mainly, we focus on the connections among these derivation algebras. Next, we give a complete classification of (gene… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2304.08067v1-abstract-full').style.display = 'inline'; document.getElementById('2304.08067v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2304.08067v1-abstract-full" style="display: none;"> Let $\mathcal{R}$ be a finite Lie conformal algebra. In this paper, we first investigate the conformal derivation algebra $CDer(\mathcal{R})$, the conformal triple derivation algebra $CTDer(\mathcal{R})$ and the generalized conformal triple derivation algebra $GCTDer(\mathcal{R})$. Mainly, we focus on the connections among these derivation algebras. Next, we give a complete classification of (generalized) conformal triple derivation algebras on all finite simple Lie conformal algebras. In particular, $CTDer(\mathcal{R})= CDer(\mathcal{R})$, where $\mathcal{R}$ is a finite simple Lie conformal algebra. But for $GCDer(\mathcal{R})$, we obtain a conclusion that is closely related to $CDer(\mathcal{R})$. Finally, we introduce the definition of triple homomorphism of a Lie conformal algebra. Furthermore, triple homomorphisms of all finite simple Lie conformal algebras are also characterized. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2304.08067v1-abstract-full').style.display = 'none'; document.getElementById('2304.08067v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 April, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 11R52; 15A99; 17B67; 17B10; 16G30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2303.16188">arXiv:2303.16188</a> <span> [<a href="https://arxiv.org/pdf/2303.16188">pdf</a>, <a href="https://arxiv.org/format/2303.16188">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Symmetric Rank-$k$ Methods </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+C">Chengchang Liu</a>, <a href="/search/math?searchtype=author&query=Chen%2C+C">Cheng Chen</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Luo Luo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2303.16188v6-abstract-short" style="display: inline;"> This paper proposes a novel class of block quasi-Newton methods for convex optimization which we call symmetric rank-$k$ (SR-$k$) methods. Each iteration of SR-$k$ incorporates the curvature information with~$k$ Hessian-vector products achieved from the greedy or random strategy. We prove that SR-$k$ methods have the local superlinear convergence rate of $\mathcal{O}\big((1-k/d)^{t(t-1)/2}\big)$ f… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.16188v6-abstract-full').style.display = 'inline'; document.getElementById('2303.16188v6-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2303.16188v6-abstract-full" style="display: none;"> This paper proposes a novel class of block quasi-Newton methods for convex optimization which we call symmetric rank-$k$ (SR-$k$) methods. Each iteration of SR-$k$ incorporates the curvature information with~$k$ Hessian-vector products achieved from the greedy or random strategy. We prove that SR-$k$ methods have the local superlinear convergence rate of $\mathcal{O}\big((1-k/d)^{t(t-1)/2}\big)$ for minimizing smooth and strongly convex function, where $d$ is the problem dimension and $t$ is the iteration counter. This is the first explicit superlinear convergence rate for block quasi-Newton methods, and it successfully explains why block quasi-Newton methods converge faster than ordinary quasi-Newton methods in practice. We also leverage the idea of SR-$k$ methods to study the block BFGS and block DFP methods, showing their superior convergence rates. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.16188v6-abstract-full').style.display = 'none'; document.getElementById('2303.16188v6-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 March, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Contain new results on block DFP method and faster block BFGS method</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2301.06428">arXiv:2301.06428</a> <span> [<a href="https://arxiv.org/pdf/2301.06428">pdf</a>, <a href="https://arxiv.org/ps/2301.06428">ps</a>, <a href="https://arxiv.org/format/2301.06428">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Faster Gradient-Free Algorithms for Nonsmooth Nonconvex Stochastic Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Lesi Chen</a>, <a href="/search/math?searchtype=author&query=Xu%2C+J">Jing Xu</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Luo Luo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2301.06428v3-abstract-short" style="display: inline;"> We consider the optimization problem of the form $\min_{x \in \mathbb{R}^d} f(x) \triangleq \mathbb{E}_尉 [F(x; 尉)]$, where the component $F(x;尉)$ is $L$-mean-squared Lipschitz but possibly nonconvex and nonsmooth. The recently proposed gradient-free method requires at most $\mathcal{O}( L^4 d^{3/2} 蔚^{-4} + 螖L^3 d^{3/2} 未^{-1} 蔚^{-4})$ stochastic zeroth-order oracle complexity to find a $(未,蔚)$-Go… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2301.06428v3-abstract-full').style.display = 'inline'; document.getElementById('2301.06428v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2301.06428v3-abstract-full" style="display: none;"> We consider the optimization problem of the form $\min_{x \in \mathbb{R}^d} f(x) \triangleq \mathbb{E}_尉 [F(x; 尉)]$, where the component $F(x;尉)$ is $L$-mean-squared Lipschitz but possibly nonconvex and nonsmooth. The recently proposed gradient-free method requires at most $\mathcal{O}( L^4 d^{3/2} 蔚^{-4} + 螖L^3 d^{3/2} 未^{-1} 蔚^{-4})$ stochastic zeroth-order oracle complexity to find a $(未,蔚)$-Goldstein stationary point of objective function, where $螖= f(x_0) - \inf_{x \in \mathbb{R}^d} f(x)$ and $x_0$ is the initial point of the algorithm. This paper proposes a more efficient algorithm using stochastic recursive gradient estimators, which improves the complexity to $\mathcal{O}(L^3 d^{3/2} 蔚^{-3}+ 螖L^2 d^{3/2} 未^{-1} 蔚^{-3})$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2301.06428v3-abstract-full').style.display = 'none'; document.getElementById('2301.06428v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 January, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">ICML 2023</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2212.07824">arXiv:2212.07824</a> <span> [<a href="https://arxiv.org/pdf/2212.07824">pdf</a>, <a href="https://arxiv.org/ps/2212.07824">ps</a>, <a href="https://arxiv.org/format/2212.07824">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Regularized Newton Methods for Monotone Variational Inequalities with H枚lder Continuous Jacobians </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+C">Chengchang Liu</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Luo Luo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2212.07824v2-abstract-short" style="display: inline;"> This paper considers the problems of solving monotone variational inequalities with H枚lder continuous Jacobians. By employing the knowledge of H枚lder parameter $谓$, we propose the $谓$-regularized extra-Newton method within at most $\mathcal O(蔚^{-2/(2+谓)})$ iterations to obtain an $蔚$-accurate solution. In the case of $谓$ is unknown, we propose the universal regularized extra-Newton method within… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.07824v2-abstract-full').style.display = 'inline'; document.getElementById('2212.07824v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2212.07824v2-abstract-full" style="display: none;"> This paper considers the problems of solving monotone variational inequalities with H枚lder continuous Jacobians. By employing the knowledge of H枚lder parameter $谓$, we propose the $谓$-regularized extra-Newton method within at most $\mathcal O(蔚^{-2/(2+谓)})$ iterations to obtain an $蔚$-accurate solution. In the case of $谓$ is unknown, we propose the universal regularized extra-Newton method within $\mathcal O(蔚^{-4/(3(1+谓))})$ iteration complexity. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.07824v2-abstract-full').style.display = 'none'; document.getElementById('2212.07824v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 December, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 December, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2212.02387">arXiv:2212.02387</a> <span> [<a href="https://arxiv.org/pdf/2212.02387">pdf</a>, <a href="https://arxiv.org/ps/2212.02387">ps</a>, <a href="https://arxiv.org/format/2212.02387">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> An Efficient Stochastic Algorithm for Decentralized Nonconvex-Strongly-Concave Minimax Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Lesi Chen</a>, <a href="/search/math?searchtype=author&query=Ye%2C+H">Haishan Ye</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Luo Luo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2212.02387v4-abstract-short" style="display: inline;"> This paper studies the stochastic nonconvex-strongly-concave minimax optimization over a multi-agent network. We propose an efficient algorithm, called Decentralized Recursive gradient descEnt Ascent Method (DREAM), which achieves the best-known theoretical guarantee for finding the $蔚$-stationary points. Concretely, it requires $\mathcal{O}(\min (魏^3蔚^{-3},魏^2 \sqrt{N} 蔚^{-2} ))$ stochastic first… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.02387v4-abstract-full').style.display = 'inline'; document.getElementById('2212.02387v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2212.02387v4-abstract-full" style="display: none;"> This paper studies the stochastic nonconvex-strongly-concave minimax optimization over a multi-agent network. We propose an efficient algorithm, called Decentralized Recursive gradient descEnt Ascent Method (DREAM), which achieves the best-known theoretical guarantee for finding the $蔚$-stationary points. Concretely, it requires $\mathcal{O}(\min (魏^3蔚^{-3},魏^2 \sqrt{N} 蔚^{-2} ))$ stochastic first-order oracle (SFO) calls and $\tilde{\mathcal{O}}(魏^2 蔚^{-2})$ communication rounds, where $魏$ is the condition number and $N$ is the total number of individual functions. Our numerical experiments also validate the superiority of DREAM over previous methods. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.02387v4-abstract-full').style.display = 'none'; document.getElementById('2212.02387v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 5 December, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2210.13931">arXiv:2210.13931</a> <span> [<a href="https://arxiv.org/pdf/2210.13931">pdf</a>, <a href="https://arxiv.org/ps/2210.13931">ps</a>, <a href="https://arxiv.org/format/2210.13931">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> An Optimal Stochastic Algorithm for Decentralized Nonconvex Finite-sum Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+L">Luo Luo</a>, <a href="/search/math?searchtype=author&query=Ye%2C+H">Haishan Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2210.13931v3-abstract-short" style="display: inline;"> This paper studies the decentralized nonconvex optimization problem $\min_{x\in{\mathbb R}^d} f(x)\triangleq \frac{1}{m}\sum_{i=1}^m f_i(x)$, where $f_i(x)\triangleq \frac{1}{n}\sum_{j=1}^n f_{i,j}(x)$ is the local function on the $i$-th agent of the network. We propose a novel stochastic algorithm called DEcentralized probAbilistic Recursive gradiEnt deScenT (\DEAREST), which integrates the techn… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2210.13931v3-abstract-full').style.display = 'inline'; document.getElementById('2210.13931v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2210.13931v3-abstract-full" style="display: none;"> This paper studies the decentralized nonconvex optimization problem $\min_{x\in{\mathbb R}^d} f(x)\triangleq \frac{1}{m}\sum_{i=1}^m f_i(x)$, where $f_i(x)\triangleq \frac{1}{n}\sum_{j=1}^n f_{i,j}(x)$ is the local function on the $i$-th agent of the network. We propose a novel stochastic algorithm called DEcentralized probAbilistic Recursive gradiEnt deScenT (\DEAREST), which integrates the techniques of variance reduction, gradient tracking and multi-consensus. We construct a Lyapunov function that simultaneously characterizes the function value, the gradient estimation error and the consensus error for the convergence analysis. Based on this measure, we provide a concise proof to show DEAREST requires at most ${\mathcal O}(mn+\sqrt{mn}L\varepsilon^{-2})$ incremental first-order oracle (IFO) calls and ${\mathcal O}({L\varepsilon^{-2}}/{\sqrt{1-位_2(W)}}\,)$ communication rounds to find an $\varepsilon$-stationary point in expectation, where $L$ is the smoothness parameter and $位_2(W)$ is the second-largest eigenvalue of the gossip matrix $W$. We can verify both of the IFO complexity and communication complexity match the lower bounds. To the best of our knowledge, DEAREST is the first optimal algorithm for decentralized nonconvex finite-sum optimization. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2210.13931v3-abstract-full').style.display = 'none'; document.getElementById('2210.13931v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 November, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 25 October, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">We correct the definition of $桅_t$ on page 5, which does not affect the result</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2208.02890">arXiv:2208.02890</a> <span> [<a href="https://arxiv.org/pdf/2208.02890">pdf</a>, <a href="https://arxiv.org/format/2208.02890">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1093/biomet/asad010.">10.1093/biomet/asad010. <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Statistical Inference for Streamed Longitudinal Data </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+L">Lan Luo</a>, <a href="/search/math?searchtype=author&query=Wang%2C+J">Jingshen Wang</a>, <a href="/search/math?searchtype=author&query=Hector%2C+E+C">Emily C. Hector</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2208.02890v1-abstract-short" style="display: inline;"> Modern longitudinal data, for example from wearable devices, measures biological signals on a fixed set of participants at a diverging number of time points. Traditional statistical methods are not equipped to handle the computational burden of repeatedly analyzing the cumulatively growing dataset each time new data is collected. We propose a new estimation and inference framework for dynamic upda… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.02890v1-abstract-full').style.display = 'inline'; document.getElementById('2208.02890v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2208.02890v1-abstract-full" style="display: none;"> Modern longitudinal data, for example from wearable devices, measures biological signals on a fixed set of participants at a diverging number of time points. Traditional statistical methods are not equipped to handle the computational burden of repeatedly analyzing the cumulatively growing dataset each time new data is collected. We propose a new estimation and inference framework for dynamic updating of point estimates and their standard errors across serially collected dependent datasets. The key technique is a decomposition of the extended score function of the quadratic inference function constructed over the cumulative longitudinal data into a sum of summary statistics over data batches. We show how this sum can be recursively updated without the need to access the whole dataset, resulting in a computationally efficient streaming procedure with minimal loss of statistical efficiency. We prove consistency and asymptotic normality of our streaming estimator as the number of data batches diverges, even as the number of independent participants remains fixed. Simulations highlight the advantages of our approach over traditional statistical methods that assume independence between data batches. Finally, we investigate the relationship between physical activity and several diseases through the analysis of accelerometry data from the National Health and Nutrition Examination Survey. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.02890v1-abstract-full').style.display = 'none'; document.getElementById('2208.02890v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 August, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages, 2 figures. Biometrika (2023)</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2206.00854">arXiv:2206.00854</a> <span> [<a href="https://arxiv.org/pdf/2206.00854">pdf</a>, <a href="https://arxiv.org/ps/2206.00854">ps</a>, <a href="https://arxiv.org/format/2206.00854">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> A class of graded conformal algebras which is induced by Heisenberg-Virasoro conformal algebra </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+L">Lipeng Luo</a>, <a href="/search/math?searchtype=author&query=Su%2C+Y">Yucai Su</a>, <a href="/search/math?searchtype=author&query=Yue%2C+X">Xiaoqing Yue</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2206.00854v1-abstract-short" style="display: inline;"> In this paper, we obtain a class of $\mathbb{Z}$-graded conformal algebras which is induced by Heisenberg-Virasoro conformal algebra. More precisely, we classify $\mathbb{Z}$-graded conformal algebras $\mathcal{A} = \oplus^\infty_{i=-1}\mathcal{A}_i$ satisfying the following conditions, (C1) $\mathcal{A}_0$ is the Heisenberg-Virasoro conformal algebra; C2) Each $\mathcal{A}_i$ for… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2206.00854v1-abstract-full').style.display = 'inline'; document.getElementById('2206.00854v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2206.00854v1-abstract-full" style="display: none;"> In this paper, we obtain a class of $\mathbb{Z}$-graded conformal algebras which is induced by Heisenberg-Virasoro conformal algebra. More precisely, we classify $\mathbb{Z}$-graded conformal algebras $\mathcal{A} = \oplus^\infty_{i=-1}\mathcal{A}_i$ satisfying the following conditions, (C1) $\mathcal{A}_0$ is the Heisenberg-Virasoro conformal algebra; C2) Each $\mathcal{A}_i$ for $i\in\mathbb{Z}_{\ge-1}^*$ is an $\mathcal{A}_0$-module of rank one; (C3) $[{X_{-1}}_位X_i]\neq 0$ for $i\ge 0$, where $X_i$ is any one of $\mathbb{C}[\partial]$-generators of $\mathcal{A}_i$ for $i\in \mathbb{Z}_{\ge -1}$. Further, we prove that all finite nontrivial irreducible modules of these algebras under some special conditions are free of rank one as a $\mathbb{C}[\partial]$-module. The conformal derivations of this class of graded Lie conformal algebras are also determined. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2206.00854v1-abstract-full').style.display = 'none'; document.getElementById('2206.00854v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 June, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2204.13503">arXiv:2204.13503</a> <span> [<a href="https://arxiv.org/pdf/2204.13503">pdf</a>, <a href="https://arxiv.org/ps/2204.13503">ps</a>, <a href="https://arxiv.org/format/2204.13503">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Rings and Algebras">math.RA</span> </div> </div> <p class="title is-5 mathjax"> Cohomology of the extended Schr枚dinger-Virasoro conformal algebra </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+L">Lipeng Luo</a>, <a href="/search/math?searchtype=author&query=Wu%2C+H">Henan Wu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2204.13503v1-abstract-short" style="display: inline;"> All the basic cohomology groups and reduced cohomology groups of the extended Schr枚dinger-Virasoro conformal algebra with trivial coefficients are completely determined. In particular, we introduce the notion of the relative cohomology of Lie conformal algebra, and then we can express our main results in a more concise manner. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2204.13503v1-abstract-full" style="display: none;"> All the basic cohomology groups and reduced cohomology groups of the extended Schr枚dinger-Virasoro conformal algebra with trivial coefficients are completely determined. In particular, we introduce the notion of the relative cohomology of Lie conformal algebra, and then we can express our main results in a more concise manner. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2204.13503v1-abstract-full').style.display = 'none'; document.getElementById('2204.13503v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 April, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">arXiv admin note: substantial text overlap with arXiv:2112.12915</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 17B56; 17B65; 17B68 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2203.14226">arXiv:2203.14226</a> <span> [<a href="https://arxiv.org/pdf/2203.14226">pdf</a>, <a href="https://arxiv.org/ps/2203.14226">ps</a>, <a href="https://arxiv.org/format/2203.14226">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="K-Theory and Homology">math.KT</span> </div> </div> <p class="title is-5 mathjax"> $n$-Lie conformal algebras and its associated infinite-dimensional $n$-Lie algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+M">Mengjun Wang</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Lipeng Luo</a>, <a href="/search/math?searchtype=author&query=Wu%2C+Z">Zhixiang Wu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2203.14226v1-abstract-short" style="display: inline;"> In this paper, we introduce a $\{位_{1\to n-1}\}$-bracket and a distribution notion of an $n$-Lie conformal algebra. For any $n$-Lie conformal algebra $R$, there exists a series of associated infinite-dimensional linearly compact $n$-Lie algebras $\{(\mathscr{L}ie_p\mbox{ }R)_\_\}_{(p\ge1)}$. We show that torsionless finite $n$-Lie conformal algebras $R$ and $S$ are isomorphic if and only if… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2203.14226v1-abstract-full').style.display = 'inline'; document.getElementById('2203.14226v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2203.14226v1-abstract-full" style="display: none;"> In this paper, we introduce a $\{位_{1\to n-1}\}$-bracket and a distribution notion of an $n$-Lie conformal algebra. For any $n$-Lie conformal algebra $R$, there exists a series of associated infinite-dimensional linearly compact $n$-Lie algebras $\{(\mathscr{L}ie_p\mbox{ }R)_\_\}_{(p\ge1)}$. We show that torsionless finite $n$-Lie conformal algebras $R$ and $S$ are isomorphic if and only if $(\mathscr{L}ie_p\mbox{ }R)_\_\simeq (\mathscr{L}ie_p\mbox{ }S)_\_$ as linearly compact $n$-Lie algebras with $\partial_{t_i}$-action for any $p\ge1$. Moreover, the representation and cohomology theory of $n$-Lie conformal algebras are established. In particular, the complex of $R$ is isomorphic to a subcomplex of $n$-Lie algebra $(\mathscr{L}ie_p\mbox{ }R)_\_$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2203.14226v1-abstract-full').style.display = 'none'; document.getElementById('2203.14226v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 March, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2202.02564">arXiv:2202.02564</a> <span> [<a href="https://arxiv.org/pdf/2202.02564">pdf</a>, <a href="https://arxiv.org/ps/2202.02564">ps</a>, <a href="https://arxiv.org/format/2202.02564">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> </div> <p class="title is-5 mathjax"> Multiplication formulas and isomorphism theorem of $\imath$Schur superalgebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+J">Jian Chen</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Li Luo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2202.02564v2-abstract-short" style="display: inline;"> We introduce the notion of $\imath$Schur superalgebra, which can be regarded as a type B/C counterpart of the $q$-Schur superalgebra (of type A) formulated as centralizer algebras of certain signed $q$-permutation modules over Hecke algebras. Some multiplication formulas for $\imath$Schur superalgebra are obtained to construct their canonical bases. Furthermore, we established an isomorphism theor… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.02564v2-abstract-full').style.display = 'inline'; document.getElementById('2202.02564v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2202.02564v2-abstract-full" style="display: none;"> We introduce the notion of $\imath$Schur superalgebra, which can be regarded as a type B/C counterpart of the $q$-Schur superalgebra (of type A) formulated as centralizer algebras of certain signed $q$-permutation modules over Hecke algebras. Some multiplication formulas for $\imath$Schur superalgebra are obtained to construct their canonical bases. Furthermore, we established an isomorphism theorem between the $\imath$Scuhr superalgebras and the $q$-Schur superalgebras of type A, which helps us derive a semisimplicity criteria of the $\imath$Schur superalgebras. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.02564v2-abstract-full').style.display = 'none'; document.getElementById('2202.02564v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 September, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 5 February, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">v2, 39 pages, minor edit</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2202.00509">arXiv:2202.00509</a> <span> [<a href="https://arxiv.org/pdf/2202.00509">pdf</a>, <a href="https://arxiv.org/ps/2202.00509">ps</a>, <a href="https://arxiv.org/format/2202.00509">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Decentralized Stochastic Variance Reduced Extragradient Method </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+L">Luo Luo</a>, <a href="/search/math?searchtype=author&query=Ye%2C+H">Haishan Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2202.00509v2-abstract-short" style="display: inline;"> This paper studies decentralized convex-concave minimax optimization problems of the form $\min_x\max_y f(x,y) \triangleq\frac{1}{m}\sum_{i=1}^m f_i(x,y)$, where $m$ is the number of agents and each local function can be written as $f_i(x,y)=\frac{1}{n}\sum_{j=1}^n f_{i,j}(x,y)$. We propose a novel decentralized optimization algorithm, called multi-consensus stochastic variance reduced extragradie… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.00509v2-abstract-full').style.display = 'inline'; document.getElementById('2202.00509v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2202.00509v2-abstract-full" style="display: none;"> This paper studies decentralized convex-concave minimax optimization problems of the form $\min_x\max_y f(x,y) \triangleq\frac{1}{m}\sum_{i=1}^m f_i(x,y)$, where $m$ is the number of agents and each local function can be written as $f_i(x,y)=\frac{1}{n}\sum_{j=1}^n f_{i,j}(x,y)$. We propose a novel decentralized optimization algorithm, called multi-consensus stochastic variance reduced extragradient, which achieves the best known stochastic first-order oracle (SFO) complexity for this problem. Specifically, each agent requires $\mathcal O((n+魏\sqrt{n})\log(1/\varepsilon))$ SFO calls for strongly-convex-strongly-concave problem and $\mathcal O((n+\sqrt{n}L/\varepsilon)\log(1/\varepsilon))$ SFO call for general convex-concave problem to achieve $\varepsilon$-accurate solution in expectation, where $魏$ is the condition number and $L$ is the smoothness parameter. The numerical experiments show the proposed method performs better than baselines. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.00509v2-abstract-full').style.display = 'none'; document.getElementById('2202.00509v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 February, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 1 February, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2112.12915">arXiv:2112.12915</a> <span> [<a href="https://arxiv.org/pdf/2112.12915">pdf</a>, <a href="https://arxiv.org/ps/2112.12915">ps</a>, <a href="https://arxiv.org/format/2112.12915">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Rings and Algebras">math.RA</span> </div> </div> <p class="title is-5 mathjax"> Cohomology of the Schr枚dinger-Virasoro conformal algebra </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wu%2C+H">Henan Wu</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Lipeng Luo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2112.12915v1-abstract-short" style="display: inline;"> Both the basic cohomology groups and the reduced cohomology groups of the Schr枚dinger-Virasoro conformal algebra with trivial coefficients are completely determined. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2112.12915v1-abstract-full" style="display: none;"> Both the basic cohomology groups and the reduced cohomology groups of the Schr枚dinger-Virasoro conformal algebra with trivial coefficients are completely determined. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2112.12915v1-abstract-full').style.display = 'none'; document.getElementById('2112.12915v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 December, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">13 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 17B65; 17B68 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2111.02708">arXiv:2111.02708</a> <span> [<a href="https://arxiv.org/pdf/2111.02708">pdf</a>, <a href="https://arxiv.org/format/2111.02708">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Quasi-Newton Methods for Saddle Point Problems and Beyond </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+C">Chengchang Liu</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Luo Luo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2111.02708v5-abstract-short" style="display: inline;"> This paper studies quasi-Newton methods for solving strongly-convex-strongly-concave saddle point problems (SPP). We propose greedy and random Broyden family updates for SPP, which have explicit local superlinear convergence rate of ${\mathcal O}\big(\big(1-\frac{1}{n魏^2}\big)^{k(k-1)/2}\big)$, where $n$ is dimensions of the problem, $魏$ is the condition number and $k$ is the number of iterations.… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2111.02708v5-abstract-full').style.display = 'inline'; document.getElementById('2111.02708v5-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2111.02708v5-abstract-full" style="display: none;"> This paper studies quasi-Newton methods for solving strongly-convex-strongly-concave saddle point problems (SPP). We propose greedy and random Broyden family updates for SPP, which have explicit local superlinear convergence rate of ${\mathcal O}\big(\big(1-\frac{1}{n魏^2}\big)^{k(k-1)/2}\big)$, where $n$ is dimensions of the problem, $魏$ is the condition number and $k$ is the number of iterations. The design and analysis of proposed algorithm are based on estimating the square of indefinite Hessian matrix, which is different from classical quasi-Newton methods in convex optimization. We also present two specific Broyden family algorithms with BFGS-type and SR1-type updates, which enjoy the faster local convergence rate of $\mathcal O\big(\big(1-\frac{1}{n}\big)^{k(k-1)/2}\big)$. Additionally, we extend our algorithms to solve general nonlinear equations and prove it enjoys the similar convergence rate. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2111.02708v5-abstract-full').style.display = 'none'; document.getElementById('2111.02708v5-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 April, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 November, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">We use $位_k=\|\nabla f({\bf z}_k)\|$ as the measure for the convergence analysis in this version and fix some mistakes in the original analysis. The modification does not change the convergence rates shown in the last version.$ $</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2110.04814">arXiv:2110.04814</a> <span> [<a href="https://arxiv.org/pdf/2110.04814">pdf</a>, <a href="https://arxiv.org/format/2110.04814">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Finding Second-Order Stationary Points in Nonconvex-Strongly-Concave Minimax Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+L">Luo Luo</a>, <a href="/search/math?searchtype=author&query=Li%2C+Y">Yujun Li</a>, <a href="/search/math?searchtype=author&query=Chen%2C+C">Cheng Chen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2110.04814v3-abstract-short" style="display: inline;"> We study the smooth minimax optimization problem $\min_{\bf x}\max_{\bf y} f({\bf x},{\bf y})$, where $f$ is $\ell$-smooth, strongly-concave in ${\bf y}$ but possibly nonconvex in ${\bf x}$. Most of existing works focus on finding the first-order stationary points of the function $f({\bf x},{\bf y})$ or its primal function $P({\bf x})\triangleq \max_{\bf y} f({\bf x},{\bf y})$, but few of them foc… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2110.04814v3-abstract-full').style.display = 'inline'; document.getElementById('2110.04814v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2110.04814v3-abstract-full" style="display: none;"> We study the smooth minimax optimization problem $\min_{\bf x}\max_{\bf y} f({\bf x},{\bf y})$, where $f$ is $\ell$-smooth, strongly-concave in ${\bf y}$ but possibly nonconvex in ${\bf x}$. Most of existing works focus on finding the first-order stationary points of the function $f({\bf x},{\bf y})$ or its primal function $P({\bf x})\triangleq \max_{\bf y} f({\bf x},{\bf y})$, but few of them focus on achieving second-order stationary points. In this paper, we propose a novel approach for minimax optimization, called Minimax Cubic Newton (MCN), which could find an $\big(\varepsilon,魏^{1.5}\sqrt{蟻\varepsilon}\,\big)$-second-order stationary point of $P({\bf x})$ with calling ${\mathcal O}\big(魏^{1.5}\sqrt蟻\varepsilon^{-1.5}\big)$ times of second-order oracles and $\tilde{\mathcal O}\big(魏^{2}\sqrt蟻\varepsilon^{-1.5}\big)$ times of first-order oracles, where $魏$ is the condition number and $蟻$ is the Lipschitz continuous constant for the Hessian of $f({\bf x},{\bf y})$. In addition, we propose an inexact variant of MCN for high-dimensional problems to avoid calling expensive second-order oracles. Instead, our method solves the cubic sub-problem inexactly via gradient descent and matrix Chebyshev expansion. This strategy still obtains the desired approximate second-order stationary point with high probability but only requires $\tilde{\mathcal O}\big(魏^{1.5}\ell\varepsilon^{-2}\big)$ Hessian-vector oracle calls and $\tilde{\mathcal O}\big(魏^{2}\sqrt蟻\varepsilon^{-1.5}\big)$ first-order oracle calls. To the best of our knowledge, this is the first work that considers the non-asymptotic convergence behavior of finding second-order stationary points for minimax problems without the convex-concave assumptions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2110.04814v3-abstract-full').style.display = 'none'; document.getElementById('2110.04814v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 November, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 October, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2109.10537">arXiv:2109.10537</a> <span> [<a href="https://arxiv.org/pdf/2109.10537">pdf</a>, <a href="https://arxiv.org/ps/2109.10537">ps</a>, <a href="https://arxiv.org/format/2109.10537">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> </div> <p class="title is-5 mathjax"> Geometric Howe dualities of finite type </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+L">Li Luo</a>, <a href="/search/math?searchtype=author&query=Xu%2C+Z">Zheming Xu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2109.10537v2-abstract-short" style="display: inline;"> We develop a geometric approach toward an interplay between a pair of quantum Schur algebras of arbitrary finite type. Then by Beilinson-Lusztig-MacPherson's stabilization procedure in the setting of partial flag varieties of type A (resp. type B/C), the Howe duality between a pair of quantum general linear groups (resp. a pair of $\imath$quantum groups of type AIII/IV) is established. The Howe du… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2109.10537v2-abstract-full').style.display = 'inline'; document.getElementById('2109.10537v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2109.10537v2-abstract-full" style="display: none;"> We develop a geometric approach toward an interplay between a pair of quantum Schur algebras of arbitrary finite type. Then by Beilinson-Lusztig-MacPherson's stabilization procedure in the setting of partial flag varieties of type A (resp. type B/C), the Howe duality between a pair of quantum general linear groups (resp. a pair of $\imath$quantum groups of type AIII/IV) is established. The Howe duality for quantum general linear groups has been provided via quantum coordinate algebras in [Z02]. We also generalize this algebraic approach to $\imath$quantum groups of type AIII/IV, and prove that the quantum Howe duality derived from partial flag varieties coincides with the one constructed by quantum coordinate (co)algebras. Moreover, the explicit multiplicity-free decompositions for these Howe dualities are obtained. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2109.10537v2-abstract-full').style.display = 'none'; document.getElementById('2109.10537v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 October, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 September, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">v2. 34 pages. Section 2.5 was rewritten to fix some errors. Accepted by Adv. Math</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2108.04437">arXiv:2108.04437</a> <span> [<a href="https://arxiv.org/pdf/2108.04437">pdf</a>, <a href="https://arxiv.org/format/2108.04437">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> Statistical Inference in High-dimensional Generalized Linear Models with Streaming Data </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+L">Lan Luo</a>, <a href="/search/math?searchtype=author&query=Han%2C+R">Ruijian Han</a>, <a href="/search/math?searchtype=author&query=Lin%2C+Y">Yuanyuan Lin</a>, <a href="/search/math?searchtype=author&query=Huang%2C+J">Jian Huang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2108.04437v1-abstract-short" style="display: inline;"> In this paper we develop an online statistical inference approach for high-dimensional generalized linear models with streaming data for real-time estimation and inference. We propose an online debiased lasso (ODL) method to accommodate the special structure of streaming data. ODL differs from offline debiased lasso in two important aspects. First, in computing the estimate at the current stage, i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2108.04437v1-abstract-full').style.display = 'inline'; document.getElementById('2108.04437v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2108.04437v1-abstract-full" style="display: none;"> In this paper we develop an online statistical inference approach for high-dimensional generalized linear models with streaming data for real-time estimation and inference. We propose an online debiased lasso (ODL) method to accommodate the special structure of streaming data. ODL differs from offline debiased lasso in two important aspects. First, in computing the estimate at the current stage, it only uses summary statistics of the historical data. Second, in addition to debiasing an online lasso estimator, ODL corrects an approximation error term arising from nonlinear online updating with streaming data. We show that the proposed online debiased estimators for the GLMs are consistent and asymptotically normal. This result provides a theoretical basis for carrying out real-time interim statistical inference with streaming data. Extensive numerical experiments are conducted to evaluate the performance of the proposed ODL method. These experiments demonstrate the effectiveness of our algorithm and support the theoretical results. A streaming dataset from the National Automotive Sampling System-Crashworthiness Data System is analyzed to illustrate the application of the proposed method. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2108.04437v1-abstract-full').style.display = 'none'; document.getElementById('2108.04437v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 August, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Lan Luo and Ruijian Han contributed equally to this work. Co-corresponding authors: Yuanyuan Lin (email: ylin@sta.cuhk.edu.hk) and Jian Huang (email: jian-huang@uiowa.edu)</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2107.04753">arXiv:2107.04753</a> <span> [<a href="https://arxiv.org/pdf/2107.04753">pdf</a>, <a href="https://arxiv.org/ps/2107.04753">ps</a>, <a href="https://arxiv.org/format/2107.04753">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> </div> <p class="title is-5 mathjax"> Hopf action on vertex algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+L">Lipeng Luo</a>, <a href="/search/math?searchtype=author&query=Wu%2C+Z">Zhixiang Wu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2107.04753v1-abstract-short" style="display: inline;"> In present paper, some properties of Hopf action on vertex algebras will be given. We prove that $V#H$ is an $\mathcal{S}$-local vertex algebra but it is not a vertex algebra when $V$ is an $H$-module vertex algebra. In addition, we extend the $\mathcal{S}$-locality to the quantum vertex algebras. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2107.04753v1-abstract-full" style="display: none;"> In present paper, some properties of Hopf action on vertex algebras will be given. We prove that $V#H$ is an $\mathcal{S}$-local vertex algebra but it is not a vertex algebra when $V$ is an $H$-module vertex algebra. In addition, we extend the $\mathcal{S}$-locality to the quantum vertex algebras. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2107.04753v1-abstract-full').style.display = 'none'; document.getElementById('2107.04753v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 July, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">20 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2106.05925">arXiv:2106.05925</a> <span> [<a href="https://arxiv.org/pdf/2106.05925">pdf</a>, <a href="https://arxiv.org/format/2106.05925">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> Online Debiased Lasso for Streaming Data </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+R">Ruijian Han</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Lan Luo</a>, <a href="/search/math?searchtype=author&query=Lin%2C+Y">Yuanyuan Lin</a>, <a href="/search/math?searchtype=author&query=Huang%2C+J">Jian Huang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2106.05925v2-abstract-short" style="display: inline;"> We propose an online debiased lasso (ODL) method for statistical inference in high-dimensional linear models with streaming data. The proposed ODL consists of an efficient computational algorithm for streaming data and approximately normal estimators for the regression coefficients. Its implementation only requires the availability of the current data batch in the data stream and sufficient statis… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.05925v2-abstract-full').style.display = 'inline'; document.getElementById('2106.05925v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2106.05925v2-abstract-full" style="display: none;"> We propose an online debiased lasso (ODL) method for statistical inference in high-dimensional linear models with streaming data. The proposed ODL consists of an efficient computational algorithm for streaming data and approximately normal estimators for the regression coefficients. Its implementation only requires the availability of the current data batch in the data stream and sufficient statistics of the historical data at each stage of the analysis. A dynamic procedure is developed to select and update the tuning parameters upon the arrival of each new data batch so that we can adjust the amount of regularization adaptively along the data stream. The asymptotic normality of the ODL estimator is established under the conditions similar to those in an offline setting and mild conditions on the size of data batches in the stream, which provides theoretical justification for the proposed online statistical inference procedure. We conduct extensive numerical experiments to evaluate the performance of ODL. These experiments demonstrate the effectiveness of our algorithm and support the theoretical results. An air quality dataset and an index fund dataset from Hong Kong Stock Exchange are analyzed to illustrate the application of the proposed method. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.05925v2-abstract-full').style.display = 'none'; document.getElementById('2106.05925v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 August, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 June, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Ruijian Han and Lan Luo contributed equally to this work. Co-corresponding authors: Yuanyuan Lin (Email: ylin@sta.cuhk.edu.hk) and Jian Huang (Email: jian-huang@uiowa.edu)</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2106.01761">arXiv:2106.01761</a> <span> [<a href="https://arxiv.org/pdf/2106.01761">pdf</a>, <a href="https://arxiv.org/format/2106.01761">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Near Optimal Stochastic Algorithms for Finite-Sum Unbalanced Convex-Concave Minimax Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+L">Luo Luo</a>, <a href="/search/math?searchtype=author&query=Xie%2C+G">Guangzeng Xie</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+T">Tong Zhang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Z">Zhihua Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2106.01761v2-abstract-short" style="display: inline;"> This paper considers stochastic first-order algorithms for convex-concave minimax problems of the form $\min_{\bf x}\max_{\bf y}f(\bf x, \bf y)$, where $f$ can be presented by the average of $n$ individual components which are $L$-average smooth. For $渭_x$-strongly-convex-$渭_y$-strongly-concave setting, we propose a new method which could find a $\varepsilon$-saddle point of the problem in… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.01761v2-abstract-full').style.display = 'inline'; document.getElementById('2106.01761v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2106.01761v2-abstract-full" style="display: none;"> This paper considers stochastic first-order algorithms for convex-concave minimax problems of the form $\min_{\bf x}\max_{\bf y}f(\bf x, \bf y)$, where $f$ can be presented by the average of $n$ individual components which are $L$-average smooth. For $渭_x$-strongly-convex-$渭_y$-strongly-concave setting, we propose a new method which could find a $\varepsilon$-saddle point of the problem in $\tilde{\mathcal O} \big(\sqrt{n(\sqrt{n}+魏_x)(\sqrt{n}+魏_y)}\log(1/\varepsilon)\big)$ stochastic first-order complexity, where $魏_x\triangleq L/渭_x$ and $魏_y\triangleq L/渭_y$. This upper bound is near optimal with respect to $\varepsilon$, $n$, $魏_x$ and $魏_y$ simultaneously. In addition, the algorithm is easily implemented and works well in practical. Our methods can be extended to solve more general unbalanced convex-concave minimax problems and the corresponding upper complexity bounds are also near optimal. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.01761v2-abstract-full').style.display = 'none'; document.getElementById('2106.01761v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 January, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 June, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2105.03163">arXiv:2105.03163</a> <span> [<a href="https://arxiv.org/pdf/2105.03163">pdf</a>, <a href="https://arxiv.org/ps/2105.03163">ps</a>, <a href="https://arxiv.org/format/2105.03163">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Logarithmic Sobolev inequalities on non-isotropic Heisenberg groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gordina%2C+M">Maria Gordina</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Liangbing Luo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2105.03163v2-abstract-short" style="display: inline;"> We study logarithmic Sobolev inequalities with respect to a heat kernel measure on finite-dimensional and infinite-dimensional Heisenberg groups. Such a group is the simplest non-trivial example of a sub-Riemannian manifold. First we consider logarithmic Sobolev inequalities on non-isotropic Heisenberg groups. These inequalities are considered with respect to the hypoelliptic heat kernel measure,… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.03163v2-abstract-full').style.display = 'inline'; document.getElementById('2105.03163v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2105.03163v2-abstract-full" style="display: none;"> We study logarithmic Sobolev inequalities with respect to a heat kernel measure on finite-dimensional and infinite-dimensional Heisenberg groups. Such a group is the simplest non-trivial example of a sub-Riemannian manifold. First we consider logarithmic Sobolev inequalities on non-isotropic Heisenberg groups. These inequalities are considered with respect to the hypoelliptic heat kernel measure, and we show that the logarithmic Sobolev constants can be chosen to be independent of the dimension of the underlying space. In this setting, a natural Laplacian is not an elliptic but a hypoelliptic operator. The argument relies on comparing logarithmic Sobolev constants for the three-dimensional non-isotropic and isotropic Heisenberg groups, and tensorization of logarithmic Sobolev inequalities in the sub-Riemannian setting. Furthermore, we apply these results in an infinite-dimensional setting and prove a logarithmic Sobolev inequality on an infinite-dimensional Heisenberg group modelled on an abstract Wiener space. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.03163v2-abstract-full').style.display = 'none'; document.getElementById('2105.03163v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 December, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 May, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">minor edits</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 58J35; 22E30; 22E66; 35A23; 35K08; 35R03; 60J65 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2101.04937">arXiv:2101.04937</a> <span> [<a href="https://arxiv.org/pdf/2101.04937">pdf</a>, <a href="https://arxiv.org/ps/2101.04937">ps</a>, <a href="https://arxiv.org/format/2101.04937">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Supersingular $j$-invariants and the Class Number of $\mathbb{Q}(\sqrt{-p})$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xiao%2C+G">Guanju Xiao</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Lixia Luo</a>, <a href="/search/math?searchtype=author&query=Deng%2C+Y">Yingpu Deng</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2101.04937v2-abstract-short" style="display: inline;"> For a prime $p>3$, let $D$ be the discriminant of an imaginary quadratic order with $|D|< \frac{4}{\sqrt{3}}\sqrt{p}$. We research the solutions of the class polynomial $H_D(X)$ mod $p$ in $\mathbb{F}_p$ if $D$ is not a quadratic residue in $\mathbb{F}_p$. We also discuss the common roots of different class polynomials in $\mathbb{F}_p$. As a result, we get a deterministic algorithm (Algorithm 3)… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.04937v2-abstract-full').style.display = 'inline'; document.getElementById('2101.04937v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2101.04937v2-abstract-full" style="display: none;"> For a prime $p>3$, let $D$ be the discriminant of an imaginary quadratic order with $|D|< \frac{4}{\sqrt{3}}\sqrt{p}$. We research the solutions of the class polynomial $H_D(X)$ mod $p$ in $\mathbb{F}_p$ if $D$ is not a quadratic residue in $\mathbb{F}_p$. We also discuss the common roots of different class polynomials in $\mathbb{F}_p$. As a result, we get a deterministic algorithm (Algorithm 3) for computing the class number of $\mathbb{Q}(\sqrt{-p})$. The time complexity of Algorithm 3 is $O(p^{3/4+蔚})$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.04937v2-abstract-full').style.display = 'none'; document.getElementById('2101.04937v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 March, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 January, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2012.07203">arXiv:2012.07203</a> <span> [<a href="https://arxiv.org/pdf/2012.07203">pdf</a>, <a href="https://arxiv.org/ps/2012.07203">ps</a>, <a href="https://arxiv.org/format/2012.07203">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> </div> <p class="title is-5 mathjax"> Lectures on dualities ABC in representation theory </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+L">Li Luo</a>, <a href="/search/math?searchtype=author&query=Wang%2C+W">Weiqiang Wang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2012.07203v2-abstract-short" style="display: inline;"> In these lecture notes for a summer mini-course, we provide an exposition on quantum groups and Hecke algebras, including (quasi) R-matrix, canonical basis, and $q$-Schur duality. Then we formulate their counterparts in the setting of $\imath$quantum groups arising from quantum symmetric pairs, including (quasi) K-matrix, $\imath$-canonical basis, and $\imath$Schur duality. As an application, the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.07203v2-abstract-full').style.display = 'inline'; document.getElementById('2012.07203v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2012.07203v2-abstract-full" style="display: none;"> In these lecture notes for a summer mini-course, we provide an exposition on quantum groups and Hecke algebras, including (quasi) R-matrix, canonical basis, and $q$-Schur duality. Then we formulate their counterparts in the setting of $\imath$quantum groups arising from quantum symmetric pairs, including (quasi) K-matrix, $\imath$-canonical basis, and $\imath$Schur duality. As an application, the ($\imath$-)canonical bases are used to formulate Kazhdan-Lusztig theories and character formulas in the BGG categories for Lie (super)algebras of type A-D. Finally, geometric constructions for $q$-Schur and $\imath$Schur dualities are provided. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.07203v2-abstract-full').style.display = 'none'; document.getElementById('2012.07203v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 January, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 December, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">57 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2012.00998">arXiv:2012.00998</a> <span> [<a href="https://arxiv.org/pdf/2012.00998">pdf</a>, <a href="https://arxiv.org/ps/2012.00998">ps</a>, <a href="https://arxiv.org/format/2012.00998">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Blocks and characters of $G(3)$-modules of non-integral weights </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+C">Chih-Whi Chen</a>, <a href="/search/math?searchtype=author&query=Cheng%2C+S">Shun-Jen Cheng</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Li Luo</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2012.00998v1-abstract-short" style="display: inline;"> We classify blocks in the BGG category $\mathcal O$ of modules of non-integral weights for the exceptional Lie superalgebra $G(3)$. We compute the characters for tilting modules of non-integral weights in $\mathcal O$. Reduction methods are established to connect non-integral blocks of $G(3)$ with blocks of the special linear Lie algebra $\mathfrak{sl}(2)$, the exceptional Lie algebra $G_2$, the g… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.00998v1-abstract-full').style.display = 'inline'; document.getElementById('2012.00998v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2012.00998v1-abstract-full" style="display: none;"> We classify blocks in the BGG category $\mathcal O$ of modules of non-integral weights for the exceptional Lie superalgebra $G(3)$. We compute the characters for tilting modules of non-integral weights in $\mathcal O$. Reduction methods are established to connect non-integral blocks of $G(3)$ with blocks of the special linear Lie algebra $\mathfrak{sl}(2)$, the exceptional Lie algebra $G_2$, the general linear Lie superalgebras $\mathfrak{gl}(1|1)$, $\mathfrak{gl}(2|1)$ and the ortho-symplectic Lie superalgebra $\mathfrak{osp}(3|2)$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.00998v1-abstract-full').style.display = 'none'; document.getElementById('2012.00998v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 December, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">33 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 17B10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2011.03929">arXiv:2011.03929</a> <span> [<a href="https://arxiv.org/pdf/2011.03929">pdf</a>, <a href="https://arxiv.org/ps/2011.03929">ps</a>, <a href="https://arxiv.org/format/2011.03929">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Connectivity keeping paths in $k$-connected bipartite graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+L">Lian Luo</a>, <a href="/search/math?searchtype=author&query=Tian%2C+Y">Yingzhi Tian</a>, <a href="/search/math?searchtype=author&query=Wu%2C+L">Liyun Wu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2011.03929v3-abstract-short" style="display: inline;"> In 2010, Mader [W. Mader, Connectivity keeping paths in $k$-connected graphs, J. Graph Theory 65 (2010) 61-69.] proved that every $k$-connected graph $G$ with minimum degree at least $\lfloor\frac{3k}{2}\rfloor+m-1$ contains a path $P$ of order $m$ such that $G-V(P)$ is still $k$-connected. In this paper, we consider similar problem for bipartite graphs, and prove that every $k$-connected bipartit… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2011.03929v3-abstract-full').style.display = 'inline'; document.getElementById('2011.03929v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2011.03929v3-abstract-full" style="display: none;"> In 2010, Mader [W. Mader, Connectivity keeping paths in $k$-connected graphs, J. Graph Theory 65 (2010) 61-69.] proved that every $k$-connected graph $G$ with minimum degree at least $\lfloor\frac{3k}{2}\rfloor+m-1$ contains a path $P$ of order $m$ such that $G-V(P)$ is still $k$-connected. In this paper, we consider similar problem for bipartite graphs, and prove that every $k$-connected bipartite graph $G$ with minimum degree at least $k+m$ contains a path $P$ of order $m$ such that $G-V(P)$ is still $k$-connected. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2011.03929v3-abstract-full').style.display = 'none'; document.getElementById('2011.03929v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 October, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 November, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2020. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2010.11737">arXiv:2010.11737</a> <span> [<a href="https://arxiv.org/pdf/2010.11737">pdf</a>, <a href="https://arxiv.org/format/2010.11737">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Efficient Projection-Free Algorithms for Saddle Point Problems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+C">Cheng Chen</a>, <a href="/search/math?searchtype=author&query=Luo%2C+L">Luo Luo</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+W">Weinan Zhang</a>, <a href="/search/math?searchtype=author&query=Yu%2C+Y">Yong Yu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2010.11737v1-abstract-short" style="display: inline;"> The Frank-Wolfe algorithm is a classic method for constrained optimization problems. It has recently been popular in many machine learning applications because its projection-free property leads to more efficient iterations. In this paper, we study projection-free algorithms for convex-strongly-concave saddle point problems with complicated constraints. Our method combines Conditional Gradient Sli… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2010.11737v1-abstract-full').style.display = 'inline'; document.getElementById('2010.11737v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2010.11737v1-abstract-full" style="display: none;"> The Frank-Wolfe algorithm is a classic method for constrained optimization problems. It has recently been popular in many machine learning applications because its projection-free property leads to more efficient iterations. In this paper, we study projection-free algorithms for convex-strongly-concave saddle point problems with complicated constraints. Our method combines Conditional Gradient Sliding with Mirror-Prox and shows that it only requires $\tilde{O}(1/\sqrt蔚)$ gradient evaluations and $\tilde{O}(1/蔚^2)$ linear optimizations in the batch setting. We also extend our method to the stochastic setting and propose first stochastic projection-free algorithms for saddle point problems. Experimental results demonstrate the effectiveness of our algorithms and verify our theoretical guarantees. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2010.11737v1-abstract-full').style.display = 'none'; document.getElementById('2010.11737v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 October, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2020. </p> </li> </ol> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Luo%2C+L&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Luo%2C+L&start=0" class="pagination-link is-current" aria-label="Goto 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