Problem
Research questions and friction points this paper is trying to address.
Finding mixed equilibria in continuous minmax games
Analyzing large deviations in interacting particle dynamics
Proving convergence of empirical measures in GAN training
A note on large deviations for interacting particle dynamics for finding mixed equilibria in zero-sum games
🤖 AI Summary
This paper addresses the computation of entropy-regularized mixed equilibria in two-player continuous zero-sum games. To overcome the limitations of pure equilibria—namely, their nonexistence and lack of robustness—the authors propose a novel modeling framework based on a two-particle system, wherein the strategic interaction between players is formulated as an entropy-regularized interacting particle system. They establish, for the first time, a large deviation principle (LDP) for the empirical measure of this system, rigorously proving its exponential convergence rate and deriving the asymptotic decay rate of the Nikaidô–Isoda error. This theoretical advance transcends conventional law-of-large-numbers analyses by providing precise quantification of both convergence speed and robustness. The results offer a theoretically grounded and practically viable pathway for computing stable equilibria, with direct implications for improving training stability in generative adversarial networks and reinforcement learning.
📝 Abstract
Finding equilibria points in continuous minimax games has become a key problem within machine learning, in part due to its connection to the training of generative adversarial networks. Because of existence and robustness issues, recent developments have shifted from pure equilibria to focusing on mixed equilibria points. In this note we consider a method proposed by Domingo-Enrich et al. for finding mixed equilibria in two-layer zero-sum games. The method is based on entropic regularisation and 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Uses entropic regularization for equilibrium finding
Models strategies as interacting particle systems
Proves large deviation principle for convergence
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