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Finding Nash equilibria in two-player zero-sum continuous games is a centralproblem in machine learning, e.g. for training both GANs and robust models. Theexistence of pure Nash equilibria requires strong conditions which are nottypically met in practice. Mixed Nash equilibria exist in greater generalityand may be found using mirror descent. Yet this approach does not scale to highdimensions. To address this limitation, we parametrize mixed strategies asmixtures of particles, whose positions and weights are updated using gradientdescent-ascent. We study this dynamics as an interacting gradient flow overmeasure spaces endowed with the Wasserstein-Fisher-Rao metric. We establishglobal convergence to an approximate equilibrium for the related Langevingradient-ascent dynamic. We prove a law of large numbers that relates particledynamics to mean-field dynamics. Our method identifies mixed equilibria in highdimensions and is demonstrably effective for training mixtures of GANs.

A mean-field analysis of two-player zero-sum games