We consider the non-convex non-concave objective function in two-player zero-sum continuous games. The existence of pure Nash equilibrium requires stringent conditions, posing a major challenge for this problem. To circumvent this difficulty, we examine the problem of identifying a mixed Nash equilibrium, where strategies are randomized and characterized by probability distributions over continuous domains.To this end, we propose PArticle-based Primal-dual ALgorithm (PAPAL) tailored for a weakly entropy-regularized min-max optimization over probability distributions. This algorithm employs the stochastic movements of particles to represent the updates of random strategies for the $\epsilon$-mixed Nash equilibrium. We offer a comprehensive convergence analysis of the proposed algorithm, demonstrating its effectiveness. In contrast to prior research that attempted to update particle importance without movements, PAPAL is the first implementable particle-based algorithm accompanied by non-asymptotic quantitative convergence results, running time, and sample complexity guarantees. Our framework contributes novel insights into the particle-based algorithms for continuous min-max optimization in the general non-convex non-concave setting.

翻译：我们考虑两人零和连续博弈中的非凸非凹目标函数。纯纳什均衡的存在需要严格的条件，这对此问题构成了重大挑战。为规避这一困难，我们研究识别混合纳什均衡的问题，其中策略是随机的，并通过连续域上的概率分布表征。为此，我们提出了一种专为概率分布上弱熵正则化极小极大优化设计的粒子基原始-对偶算法（PAPAL）。该算法利用粒子的随机运动来表示$\epsilon$-混合纳什均衡中随机策略的更新。我们对该算法进行了全面的收敛性分析，证明了其有效性。与以往试图在不移动粒子情况下更新粒子重要性的研究不同，PAPAL是首个具备非渐近定量收敛结果、运行时间和样本复杂度保证的可实现粒子基算法。我们的框架为一般非凸非凹设定下连续极小极大优化的粒子基算法提供了新的见解。