  We consider the problem of computing mixed Nash equilibria of two-player
zero-sum games with continuous sets of pure strategies and with first-order
access to the payoff function. This problem arises for example in
game-theory-inspired machine learning applications, such as
distributionally-robust learning. In those applications, the strategy sets are
high-dimensional and thus methods based on discretisation cannot tractably
return high-accuracy solutions.
  In this paper, we introduce and analyze a particle-based method that enjoys
guaranteed local convergence for this problem. This method consists in
parametrizing the mixed strategies as atomic measures and applying proximal
point updates to both the atoms' weights and positions. It can be interpreted
as a time-implicit discretization of the "interacting" Wasserstein-Fisher-Rao
gradient flow.
  We prove that, under non-degeneracy assumptions, this method converges at an
exponential rate to the exact mixed Nash equilibrium from any initialization
satisfying a natural notion of closeness to optimality. We illustrate our
results with numerical experiments and discuss applications to max-margin and
distributionally-robust classification using two-layer neural networks, where
our method has a natural interpretation as a simultaneous training of the
network's weights and of the adversarial distribution.
