A Fisher-Rao gradient flow for entropic mean-field min-max games

TL;DR

This paper introduces a Fisher-Rao gradient flow approach for convex-concave min-max games with entropy regularization, demonstrating convergence to the unique equilibrium with explicit rates.

Contribution

It proposes a novel Fisher-Rao gradient flow method and provides convergence analysis with explicit rates for entropy-regularized min-max games.

Findings

Convergence to the unique mixed Nash equilibrium is proven.

Explicit convergence rates are derived.

The method effectively addresses entropy-regularized min-max problems.

Abstract

Gradient flows play a substantial role in addressing many machine learning problems. We examine the convergence in continuous-time of a \textit{Fisher-Rao} (Mean-Field Birth-Death) gradient flow in the context of solving convex-concave min-max games with entropy regularization. We propose appropriate Lyapunov functions to demonstrate convergence with explicit rates to the unique mixed Nash equilibrium.