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Finding Nash equilibria in two-player zero-sum continuous games is a central problem in machine learning, e.g. for training both GANs and robust models. The existence of pure Nash equilibria requires strong conditions which are not typically met in practice. Mixed Nash equilibria exist in greater generality and may be found using mirror descent. Yet this approach does not scale to high dimensions. To address this limitation, we parametrize mixed strategies as mixtures of particles, whose positions and weights are updated using gradient descent-ascent. We study this dynamics as an interacting gradient flow over measure spaces endowed with the Wasserstein-Fisher-Rao metric. We establish global convergence to an approximate equilibrium for the related Langevin gradient-ascent dynamic. We prove a law of large numbers that relates particle dynamics to mean-field dynamics. Our method identifies mixed equilibria in high dimensions and is demonstrably effective for training mixtures of GANs. 

Recent theoretical works have characterized the dynamics of wide shallow neural networks trained via gradient descent in an asymptotic mean-field limit when the width tends towards infinity. At initialization, the random sampling of the parameters leads to deviations from the mean-field limit dictated by the classical Central Limit Theorem (CLT). However, since gradient descent induces correlations among the parameters, it is of interest to analyze how these fluctuations evolve. Here, we use a dynamical CLT to prove that the asymptotic fluctuations around the mean limit remain bounded in mean square throughout training. The upper bound is given by a Monte-Carlo resampling error, with a variance that that depends on the 2-norm of the underlying measure, which also controls the generalization error. This motivates the use of this 2-norm as a regularization term during training. Furthermore, if the mean-field dynamics converges to a measure that interpolates the training data, we prove that the asymptotic deviation eventually vanishes in the CLT scaling. We also complement these results with numerical experiments.