{"title":"接近非凸多人游戏的全局纳什均衡","authors":"Guanpu Chen, Gehui Xu, Fengxiang He, Yiguang Hong, Leszek Rutkowski, Dacheng Tao","doi":"10.1109/TPAMI.2024.3445666","DOIUrl":null,"url":null,"abstract":"<p><p>Many machine learning problems can be formulated as non-convex multi-player games. Due to non-convexity, it is challenging to obtain the existence condition of the global Nash equilibrium (NE) and design theoretically guaranteed algorithms. This paper studies a class of non-convex multi-player games, where players' payoff functions consist of canonical functions and quadratic operators. We leverage conjugate properties to transform the complementary problem into a variational inequality (VI) problem using a continuous pseudo-gradient mapping. We prove the existence condition of the global NE as the solution to the VI problem satisfies a duality relation. We then design an ordinary differential equation to approach the global NE with an exponential convergence rate. For practical implementation, we derive a discretized algorithm and apply it to two scenarios: multi-player games with generalized monotonicity and multi-player potential games. In the two settings, step sizes are required to be O(1/k) and O(1/√k) to yield the convergence rates of O(1/ k) and O(1/√k), respectively. Extensive experiments on robust neural network training and sensor network localization validate our theory. Our code is available at https://github.com/GuanpuChen/Global-NE.</p>","PeriodicalId":94034,"journal":{"name":"IEEE transactions on pattern analysis and machine intelligence","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Approaching the Global Nash Equilibrium of Non-convex Multi-player Games.\",\"authors\":\"Guanpu Chen, Gehui Xu, Fengxiang He, Yiguang Hong, Leszek Rutkowski, Dacheng Tao\",\"doi\":\"10.1109/TPAMI.2024.3445666\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><p>Many machine learning problems can be formulated as non-convex multi-player games. Due to non-convexity, it is challenging to obtain the existence condition of the global Nash equilibrium (NE) and design theoretically guaranteed algorithms. This paper studies a class of non-convex multi-player games, where players' payoff functions consist of canonical functions and quadratic operators. We leverage conjugate properties to transform the complementary problem into a variational inequality (VI) problem using a continuous pseudo-gradient mapping. We prove the existence condition of the global NE as the solution to the VI problem satisfies a duality relation. We then design an ordinary differential equation to approach the global NE with an exponential convergence rate. For practical implementation, we derive a discretized algorithm and apply it to two scenarios: multi-player games with generalized monotonicity and multi-player potential games. In the two settings, step sizes are required to be O(1/k) and O(1/√k) to yield the convergence rates of O(1/ k) and O(1/√k), respectively. Extensive experiments on robust neural network training and sensor network localization validate our theory. Our code is available at https://github.com/GuanpuChen/Global-NE.</p>\",\"PeriodicalId\":94034,\"journal\":{\"name\":\"IEEE transactions on pattern analysis and machine intelligence\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"IEEE transactions on pattern analysis and machine intelligence\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/TPAMI.2024.3445666\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"IEEE transactions on pattern analysis and machine intelligence","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/TPAMI.2024.3445666","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
许多机器学习问题都可以表述为非凸多玩家博弈。由于非凸性,获得全局纳什均衡(NE)的存在条件和设计理论上有保证的算法是一项挑战。本文研究了一类非凸多玩家博弈,其中玩家的报酬函数由典型函数和二次算子组成。我们利用共轭特性,使用连续伪梯度映射将互补问题转化为变不等式(VI)问题。我们证明了全局 NE 的存在条件,因为 VI 问题的解满足对偶关系。然后,我们设计了一个常微分方程,以指数收敛速度逼近全局近似值。在实际应用中,我们推导出一种离散化算法,并将其应用于两种情况:具有广义单调性的多人博弈和多人潜在博弈。在这两种情况下,步长要求分别为 O(1/k) 和 O(1/√k),收敛率分别为 O(1/k) 和 O(1/√k)。鲁棒神经网络训练和传感器网络定位的大量实验验证了我们的理论。我们的代码见 https://github.com/GuanpuChen/Global-NE。
Approaching the Global Nash Equilibrium of Non-convex Multi-player Games.
Many machine learning problems can be formulated as non-convex multi-player games. Due to non-convexity, it is challenging to obtain the existence condition of the global Nash equilibrium (NE) and design theoretically guaranteed algorithms. This paper studies a class of non-convex multi-player games, where players' payoff functions consist of canonical functions and quadratic operators. We leverage conjugate properties to transform the complementary problem into a variational inequality (VI) problem using a continuous pseudo-gradient mapping. We prove the existence condition of the global NE as the solution to the VI problem satisfies a duality relation. We then design an ordinary differential equation to approach the global NE with an exponential convergence rate. For practical implementation, we derive a discretized algorithm and apply it to two scenarios: multi-player games with generalized monotonicity and multi-player potential games. In the two settings, step sizes are required to be O(1/k) and O(1/√k) to yield the convergence rates of O(1/ k) and O(1/√k), respectively. Extensive experiments on robust neural network training and sensor network localization validate our theory. Our code is available at https://github.com/GuanpuChen/Global-NE.