Convergence of the Min-Max Langevin Dynamics and Algorithm for Zero-Sum Games

2024-12-29

AI Summary

This work addresses entropy-regularized equilibrium computation for smooth strongly convex-concave zero-sum games in Euclidean space. Methodologically, it introduces the mean-field minimax Langevin dynamics (MMLD) and establishes, for the first time, its exponential convergence to the entropy-regularized Nash equilibrium. Two particle-free biased finite-particle algorithms—operating in continuous and discrete time—are proposed, both exhibiting iteration complexity bounds independent of particle count. Through rigorous mean-field analysis, stochastic differential equation (SDE) characterization, and bias control, the work quantifies the approximation error between finite-particle implementations and the mean-field equilibrium distribution, yielding provably biased convergence guarantees with explicit, computable bias upper bounds. This constitutes the first Langevin-type algorithmic framework for distributed zero-sum game equilibrium computation that simultaneously ensures theoretical convergence guarantees and practical computational efficiency.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Machine Learning: Learning with Manifolds

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

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-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 estimate 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.

Problem

• Zero-sum Games
• Minimax Langevin Dynamics
• Error Analysis

Innovation

• Mean-Field Minimax Langevin Dynamics
• Nash Equilibrium
• Error Quantification