We consider the general nonconvex nonconcave minimax problem over continuous variables. A major challenge for this problem is that a saddle point may not exist. In order to resolve this difficulty, we consider the related problem of finding a Mixed Nash Equilibrium, which is a randomized strategy represented by probability distributions over the continuous variables. We propose a Particle-based Primal-Dual Algorithm (PPDA) for a weakly entropy-regularized min-max optimization procedure over the probability distributions, which employs the stochastic movements of particles to represent the updates of random strategies for the mixed Nash Equilibrium. A rigorous convergence analysis of the proposed algorithm is provided. Compared to previous works that try to update particle weights without movements, PPDA is the first implementable particle-based algorithm with non-asymptotic quantitative convergence results, running time, and sample complexity guarantees. Our framework gives new insights into the design of particle-based algorithms for continuous min-max optimization in the general nonconvex nonconcave setting.

翻译：我们考虑连续变量上的一般非凸非凹极小极大问题。该问题的一个主要挑战是可能不存在鞍点。为解决这一困难，我们考虑寻找混合纳什均衡的相关问题，这是一种由连续变量上的概率分布表示的随机策略。我们提出了一种基于粒子的原始-对偶算法（PPDA），用于概率分布上的弱熵正则化极小极大优化过程，该算法利用粒子的随机运动来更新混合纳什均衡的随机策略。我们提供了所提算法的严格收敛性分析。与先前试图在不移动粒子的情况下更新粒子权重的工作相比，PPDA是首个具有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的基于粒子算法。我们的框架为一般非凸非凹背景下连续极小极大优化中基于粒子算法的设计提供了新见解。