  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.
