Large deviations for interacting particle dynamics for finding mixed equilibria in zero-sum games

TL;DR

This paper introduces a particle-based method with entropic regularisation for finding mixed equilibria in zero-sum games, demonstrating large deviation principles that ensure convergence as particle numbers increase.

Contribution

It establishes a large deviation principle for the empirical measures of interacting particles in zero-sum games, advancing understanding of convergence to mixed equilibria.

Findings

Empirical measures satisfy a large deviation principle as particles grow large

Convergence of empirical measures and Nikaid extsuperscript{o}-Isoda error is demonstrated

Supports the use of entropic regularisation in equilibrium-finding algorithms

Abstract

Finding equilibrium points in continuous minmax games has become a key problem within machine learning, in part due to its connection to the training of generative adversarial networks and reinforcement learning. Because of existence and robustness issues, recent developments have shifted from pure equilibria to focusing on mixed equilibrium points. In this work we consider a method for finding mixed equilibria in two-layer zero-sum games based on entropic regularisation, where the two competing strategies are represented by two sets of interacting particles. We show that the sequence of empirical measures of the particle system satisfies a large deviation principle as the number of particles grows to infinity, and how this implies convergence of the empirical measure and the associated Nikaid\^o-Isoda error, complementing existing law of large numbers results.