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

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.