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Gradient flows play a substantial role in addressing many machine learning problems. We examine the convergence in continuous-time of a 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.
Lascu et al. (Fri,) studied this question.