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

Geuntaek Seo, Minseop Shin, Pierre Monmarché, Beomjun Choi

TL;DR

This paper proves that the mean-field Langevin descent-ascent flow for entropically regularized two-player zero-sum games is locally exponentially stable near the unique mixed Nash equilibrium. By performing a spectral analysis of the linearized operator around the equilibrium, the authors show entropy induces a local, displacement-convex-concave structure that yields a positive spectral gap and a contraction in Wasserstein distance. The main result establishes existence of a neighborhood in which the $W_2$ distance to the MNE decays as $e^{-\lambda t}$ and the densities gain $C^{1,\alpha}$ regularity, with the rate governed by the gap $\lambda_{gap}$. Global convergence remains open, but the work provides a rigorous mechanism for local stability and quantitative rates in the mean-field setting, highlighting the utility of entropy regularization in nonconvex-nonconcave minimax dynamics.

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.

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

TL;DR

This paper proves that the mean-field Langevin descent-ascent flow for entropically regularized two-player zero-sum games is locally exponentially stable near the unique mixed Nash equilibrium. By performing a spectral analysis of the linearized operator around the equilibrium, the authors show entropy induces a local, displacement-convex-concave structure that yields a positive spectral gap and a contraction in Wasserstein distance. The main result establishes existence of a neighborhood in which the distance to the MNE decays as and the densities gain regularity, with the rate governed by the gap . Global convergence remains open, but the work provides a rigorous mechanism for local stability and quantitative rates in the mean-field setting, highlighting the utility of entropy regularization in nonconvex-nonconcave minimax dynamics.

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.
Paper Structure (34 sections, 30 theorems, 203 equations)

This paper contains 34 sections, 30 theorems, 203 equations.

Table of Contents

  1. Introduction
  2. Related Works
  3. Motivating the MFL-DA
  4. Timescale variants.
  5. Temperature effects, annealing, and time-averaged gradients.
  6. Particle and time discretizations.
  7. Single-timescale regime.
  8. Main Result
  9. Heuristic Explanation of Key Mechanism
  10. Proof of Main Result
  11. Conclusion
  12. Preliminaries
  13. Notations and Definitions
  14. Sobolev and Hölder Spaces
  15. Compactness and Convergence.
  16. ...and 19 more sections

Key Result

Theorem 1.1

For a given $f\in C^2$, let $(\mu^*,\nu^*)$ be the unique mixed Nash equilibrium of eq-functional. There exist $\delta>0$, $\lambda>0$, and $C<\infty$ depending on $f$ with the following significance: if the initial data satisfies $W_2(\mu_0,\mu^*)+W_2(\nu_0,\nu^*)<\delta$, then the MFL-DA eq-main r

Theorems & Definitions (57)

  • Theorem 1.1: Exponential Stability of Nash Equilibrium
  • Lemma 3.1: Spectral Gap at MNE
  • proof
  • Remark 3.2: Poincaré inequality and an alternative proof
  • Lemma 3.3: Local Convex--Concavity
  • proof
  • Proposition 3.4: Local Contraction
  • proof
  • Lemma 3.5: Stability
  • proof
  • ...and 47 more