Pierre Monmarché

Geuntaek Seo, Minseop Shin, Pierre Monmarché et al.

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.

Yulong Lu, Pierre Monmarché

The approximation of mixed Nash equilibria (MNE) for zero-sum games with mean-field interacting players has recently raised much interest in machine learning. In this paper we propose a mean-field gradient descent dynamics for finding the MNE of zero-sum games involving $K$ players with $K\geq 2$. The evolution of the players' strategy distributions follows coupled mean-field gradient descent flows with momentum, incorporating an exponentially discounted time-averaging of gradients. First, in the case of a fixed entropic regularization, we prove an exponential convergence rate for the mean-field dynamics to the mixed Nash equilibrium with respect to the total variation metric. This improves a previous polynomial convergence rate for a similar time-averaged dynamics with different averaging factors. Moreover, unlike previous two-scale approaches for finding the MNE, our approach treats all player types on the same time scale. We also show that with a suitable choice of decreasing temperature, a simulated annealing version of the mean-field dynamics converges to an MNE of the initial unregularized problem.