  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.
