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

Gradient flows play a substantial role in addressing many machine learning problems. We examine the convergence in continuous-time of a $\mathit{\text{Fisher-Rao}}$ (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.