From Poincaré Recurrence to Convergence in Imperfect Information Games: Finding Equilibrium via Regularization

Poincaré Recurrence, Cycles and Spurious Equilibria in Gradient-Descent-Ascent for Non-Convex Non-Concave Zero-Sum Games

We study a wide class of non-convex non-concave min-max games that generalizes over standard bilinear zero-sum games. In this class, players control the inputs of a smooth function whose output is being applied to a bilinear zero-sum game. This class of games is motivated by the indirect nature of the competition in Generative Adversarial Networks, where players control the parameters of a neural network while the actual competition happens between the distributions that the generator and discriminator capture. We establish theoretically, that depending on the specific instance of the problem …

From Darwin to Poincaré and von Neumann: Recurrence and Cycles in Evolutionary and Algorithmic Game Theory

Replicator dynamics, the continuous-time analogue of Multiplicative Weights Updates, is the main dynamic in evolutionary game theory. In simple evolutionary zero-sum games, such as Rock-Paper-Scissors, replicator dynamic is periodic \cite{zeeman1980population}, however, its behavior in higher dimensions is not well understood. We provide a complete characterization of its behavior in zero-sum evolutionary games. We prove that, if and only if, the system has an interior Nash equilibrium, the dynamics exhibit Poincaré recurrence, i.e., almost all orbits come arbitrary close to their initial cond…

Online Optimization in Games via Control Theory: Connecting Regret, Passivity and Poincaré Recurrence

We present a novel control-theoretic understanding of online optimization and learning in games, via the notion of passivity. Passivity is a fundamental concept in control theory, which abstracts energy conservation and dissipation in physical systems. It has become a standard tool in analysis of general feedback systems, to which game dynamics belong. Our starting point is to show that all continuous-time Follow-the-Regularized-Leader (FTRL) dynamics, which include the well-known Replicator Dynamic, are lossless, i.e. it is passive with no energy dissipation. Interestingly, we prove that pass…

有限博弈的几何分解：指数权重下的收敛与回复

A geometric decomposition of finite games: Convergence vs. recurrence under exponential weights

本研究针对博弈学习动力学的复杂性，基于Shahshahani度量的黎曼框架提出不可压缩博弈类，证明其与谐波博弈等价，且连续时间指数权重动力学在这类博弈中具有庞加莱回复性，揭示了博弈几何分解与学习动力学长期行为的关联。

量子零和博弈中的矩阵乘法权重更新：守恒定律与回归

Matrix Multiplicative Weights Updates in Quantum Zero-Sum Games: Conservation Laws & Recurrence

本文研究量子零和博弈中的矩阵乘法权重更新与量子复制者动力学，证明系统具有量子信息论守恒定律和Poincaré回归，并推广经典博弈结果。

不完美感知下的移动目标搜索：一种部分可观测随机博弈论方法

Mobile Target Search with Imperfect Perception: A Partially Observable Stochastic Game Theoretical Approach

针对传感器限制、恶意干扰或通信噪声导致的不完美感知，采用部分可观测随机博弈（POSG）框架建模搜索者与目标间的对抗互动，提出可检测性概念和基于随机递归分析的充分判据，并开发服务器辅助分布式算法。