Entropic mean-field min-max problems via Best Response flow
Razvan-Andrei Lascu, Mateusz B. Majka, Łukasz Szpruch
TL;DR
This paper develops a continuous-time, entropy-regularized mean-field framework for solving convex-concave min-max games and proves exponential convergence to the unique mixed Nash equilibrium via a KL-based Lyapunov functional. By constructing Gibbs-type best-response maps $\Psi_{\sigma}$ and $\Phi_{\sigma}$, the authors show that the MF-BR flow on the space of absolutely continuous measures with the Total Variation metric converges at rate $e^{-\alpha t}$, with NI error decaying as $\mathcal{O}(\sigma^2 e^{-\alpha t})$, and a bilinear payoffs case where NI convergence is $e^{-\alpha t}$ independent of $\sigma$. They also connect MF-BR to fictitious play through a time-rescaling argument, contrasting their results with Wasserstein/Fisher-Rao gradient-flow analyses that require additional assumptions. The contributions establish existence, uniqueness, and explicit convergence rates under standard convex-concave and bounded-derivative conditions on $F$, providing a principled tool for learning mixed Nash equilibria in distributed, entropy-regularized min-max games with potential applications to GANs and adversarial learning.
Abstract
We investigate the convergence properties of a continuous-time optimization method, the \textit{Mean-Field Best Response} flow, for solving convex-concave min-max games with entropy regularization. We introduce suitable Lyapunov functions to establish exponential convergence to the unique mixed Nash equilibrium. Additionally, we demonstrate the convergence of the fictitious play flow as a by-product of our analysis.