[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85762-en":3,"doc-seo-85762-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85762,2336464648322,"Aria","https://ap-avatar.wpscdn.com/avatar/2200025388227c56fec?_k=1778556882303663488",8,"Research & Report","Implicit Midpoint Gradient Descent Fast And Learning Rate Free Convergence For Zero-Sum Games","Unconstrained bilinear zero-sum games are studied through the introduction of implicit midpoint gradient descent. The update rule is derived from continuous-time follow-the-regularized leader dynamics using symplectic integration, enabling the algorithm to inherit properties of the original system. Results establish bounded orbits, fast ergodic convergence to Nash equilibria, and stability guarantees independent of the learning rate, in contrast to prior traditional online optimization methods. Experiments confirm strong empirical gains over optimistic and alternating gradient descent.","arXiv :2607 .09950v 1 [ cs .GT] 10 Jul 2026  \nIMPLICIT MIDPOINT GRADIENT DESCENT: FAST AND LEARNING RATE FREE CONVERGENCE FOR ZERO-SUM GAMES  \nGaoqi Xue  \nIndustrial and Systems Engineering  \nRensselaer Polytechnic Institute  \n[xueg@rpi.edu](xueg@rpi.edu)  \n[James P. Bailey](James P. Bailey)  \nIndustrial and Systems Engineering  \nRensselaer Polytechnic Institute  \n[bailej6@rpi.edu](bailej6@rpi.edu)  \nABSTRACT  \nWe study unconstrained bilinear zero-sum games, a fundamental model in online learning, adversarial optimization, and multi-agent decision-making. We introduce the implicit midpoint gradient descent rule, which we derive from continuous-time follow-the-regularized leader dynamics via symplectic integration methods. We prove that implicit midpoint gradient descent inherits several powerful properties from the continuous-time dynamics, including bounded orbits, fast ergodic convergence to Nash equilibria, and learning-rate-independent stability guarantees. This is the first traditional online optimization approach to simultaneously achieve these properties in unconstrained bilinear zero-sum games. Finally, computational experiments demonstrate that the proposed method significantly outperforms the standard methods, optimistic and alternating gradient descent.  \nKeywords Zero-sum Games · Efficient Algorithm for Nash Equilibria · Online Optimization  \n1 Introduction  \nZero-sum games provide a fundamental framework for modeling competitive interactions in which the objectives of two players are perfectly opposed. The mathematical formulation is the saddle-point problem, where one player minimizes a payoff function while the other maximizes it. The study of zero-sum games is grounded in the minimax theorem, first established by von Neumann (1928) [27], which guarantees the existence of an equilibrium under mild conditions.  \nThe mathematical framework given by zero-sum games has important applications of modeling adversarial decisionmaking in economics, machine learning, and control systems. In economics, the pioneering work of von Neumann and Morgenstern [28] established the foundation of modern game theory through the minimax principle, with important applications in market competition, bargaining, auctions, and decision-making under uncertainty, while further developments by Kuhn and Tucker [21], Blackwell and Girshick [8], and Fudenberg and Tirole [13] strengthened its role in economic analysis and strategic behavior. In modern machine learning, Goodfellow et al. [16] introduced Generative Adversarial Networks (GANs), one of the most influential applications of zero-sum games, where a generator and a discriminator are trained through a minimax optimization process; this adversarial formulation has become central to deep generative modeling and has significantly advanced image generation, representation learning, and robust optimization [15] . In control theory and engineering, Isaacs [19] introduced zero-sum differential games as a foundation for robust control. These ideas have major applications in robotics, missile guidance, cyber-physical systems, and autonomous decision systems.  \nFrom a computational perspective, solving bilinear zero-sum games reduces to finding a saddle point of a structured objective; first-order methods are widely used due to their simplicity and scalability. Among the classical methods,  \nA PREPRINT-JULY 14, 2026  \nthe Arrow–Hurwicz and Uzawa methods proposed in [1] are the earliest and most fundamental ones, which simultaneously execute gradient descent on the primal minimizing variables and gradient ascent on the dual maximizing variables. We also have Optimistic Gradient Descent (OGD), which improves stability by incorporating predictive correction terms that reduce the cycling behavior common in standard gradient methods, and the extragradient method, which introduces an additional intermediate step to better handle saddle-point structure and is particularly effective for monotone variation","cbCaiop471GPIgrr","https://ap.wps.com/l/cbCaiop471GPIgrr","pdf",788066,1,21,"English","en",105,"# Abstract\n# Introduction\n## Our Contributions","[{\"question\":\"What problem does the paper address in online learning and optimization?\",\"answer\":\"It studies unconstrained bilinear zero-sum games, a foundational model for online learning, adversarial optimization, and multi-agent decision-making.\"},{\"question\":\"How is implicit midpoint gradient descent derived and what does it aim to preserve?\",\"answer\":\"It is derived from continuous-time follow-the-regularized leader dynamics using symplectic integration, aiming to preserve the energy/geometric structure of the underlying dynamics.\"},{\"question\":\"What convergence and stability properties are proved for the proposed method?\",\"answer\":\"The method inherits bounded orbits, fast ergodic convergence to Nash equilibria, and learning-rate-independent stability guarantees from the continuous-time dynamics.\"}]",1784206104,53,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"implicit-midpoint-gradient-descent-fast-and-learning-rate-free-convergence-for-zero-sum-games","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/implicit-midpoint-gradient-descent-fast-and-learning-rate-free-convergence-for-zero-sum-games/85762/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the paper address in online learning and optimization?","Question",{"text":75,"@type":76},"It studies unconstrained bilinear zero-sum games, a foundational model for online learning, adversarial optimization, and multi-agent decision-making.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is implicit midpoint gradient descent derived and what does it aim to preserve?",{"text":80,"@type":76},"It is derived from continuous-time follow-the-regularized leader dynamics using symplectic integration, aiming to preserve the energy/geometric structure of the underlying dynamics.",{"name":82,"@type":73,"acceptedAnswer":83},"What convergence and stability properties are proved for the proposed method?",{"text":84,"@type":76},"The method inherits bounded orbits, fast ergodic convergence to Nash equilibria, and learning-rate-independent stability guarantees from the continuous-time 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