Local exponential stability of mean-field Langevin descent-ascent and associated particle system

Geuntaek Seo; Minseop Shin; Pierre Monmarché; Beomjun Choi

cs.LGFeb 2, 2026

Local exponential stability of mean-field Langevin descent-ascent and associated particle system

Authors: Geuntaek Seo, Minseop Shin, Pierre Monmarché, Beomjun Choi

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, together with its associated interacting particle system. For general nonconvex-nonconcave payoffs, Wang and Chizat (COLT 2024) asked whether the original single-timescale MFL-DA converges to the mixed Nash equilibrium and, if so, at what rate. We prove a local affirmative answer in Wasserstein space: if the initial datum is sufficiently close to the mixed Nash equilibrium, then the mean-field dynamics converges to it exponentially fast at a quantitative rate. We further show that the finite- $N$ particle system inherits this stability up to times exponential in $N$ , with an $N$ -independent exponential rate modulo a finite-particle error floor. Combined with the recent counterexample of Mourrat and Pillaud-Vivien for MFL-DA, which shows that global convergence cannot hold in general, our theorem completes the positive local counterpart of the Wang-Chizat question: the mixed Nash equilibrium has a robust basin of attraction, stable under both the mean-field flow and its finite-particle approximation.

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