Local Exponential Stability of Mean-Field Langevin Descent-Ascent in Wasserstein Space

2026-02-02

AI Summary

This work investigates the local convergence of mean-field Langevin descent-ascent dynamics in the Wasserstein space for entropy-regularized two-player zero-sum games with nonconvex–nonconcave payoff functions. By conducting a spectral analysis of the linearized operator, the authors establish a coercivity estimate for the entropy functional in the vicinity of mixed Nash equilibria, thereby revealing a local displacement convex-concave structure. They provide the first proof that the dynamics exhibit local exponential stability within a Wasserstein neighborhood of the unique mixed Nash equilibrium. This result partially resolves an open problem posed by Wang and Chizat and offers a quantitative characterization of the convergence rate.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Search and Optimization: Non-convex Optimization

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
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

We study the mean-field Langevin descent-ascent (MFL-DA), a coupled optimization dynamics on the space of probability measures for entropically regularized two-player zero-sum games. Although the associated mean-field objective admits a unique mixed Nash equilibrium, the long-time behavior of the original MFL-DA for general nonconvex-nonconcave payoffs has remained largely open. Answering an open question posed by Wang and Chizat (COLT 2024), we provide a partial resolution by proving that this equilibrium is locally exponentially stable: if the initialization is sufficiently close in Wasserstein metric, the dynamics trends to the equilibrium at an exponential rate. The key to our analysis is to establish a coercivity estimate for the entropy near equilibrium via spectral analysis of the linearized operator. We show that this coercivity effectively reveals a local displacement convex-concave structure, thereby driving contraction. This result settles the local stability and quantitative rate questions of Wang and Chizat, leaving global convergence as a remaining open challenge.

Problem

• mean-field Langevin descent-ascent
• local exponential stability
• nonconvex-nonconcave games
• Wasserstein space
• mixed Nash equilibrium

Innovation

• mean-field Langevin dynamics
• local exponential stability
• Wasserstein space
• coercivity estimate
• displacement convex-concave structure