On the Convergence of Min-Max Langevin Dynamics and Algorithm

TL;DR

This paper proves exponential convergence of min-max Langevin dynamics for zero-sum games with entropy regularization and analyzes finite-particle approximations, providing explicit bias and iteration complexity guarantees.

Contribution

It establishes the first exponential convergence results for mean-field min-max Langevin dynamics in regularized zero-sum games and analyzes finite-particle algorithms with explicit bias bounds.

Findings

Exponential convergence of mean-field min-max Langevin dynamics.

Finite-particle algorithms have explicit bias bounds.

Iteration complexity for approximate equilibrium computation.

Abstract

We study zero-sum games in the space of probability distributions over the Euclidean space $\mathbb{R}^d$ with entropy regularization, in the setting when the interaction function between the players is smooth and strongly convex-strongly concave. We prove an exponential convergence guarantee for the mean-field min-max Langevin dynamics to compute the equilibrium distribution of the zero-sum game. We also study the finite-particle approximation of the mean-field min-max Langevin dynamics, both in continuous and discrete times. We prove biased convergence guarantees for the continuous-time finite-particle min-max Langevin dynamics to the stationary mean-field equilibrium distribution with an explicit bias term which does not scale with the number of particles. We also prove biased convergence guarantees for the discrete-time finite-particle min-max Langevin algorithm to the stationary mean-field equilibrium distribution with an additional bias term which scales with the step size and the number of particles. This provides an explicit iteration complexity for the average particle along the finite-particle algorithm to approximately compute the equilibrium distribution of the zero-sum game.