  Min-max optimization problems arise in several key machine learning setups,
including adversarial learning and generative modeling. In their general form,
in absence of convexity/concavity assumptions, finding pure equilibria of the
underlying two-player zero-sum game is computationally hard [Daskalakis et al.,
2021]. In this work we focus instead in finding mixed equilibria, and consider
the associated lifted problem in the space of probability measures. By adding
entropic regularization, our main result establishes global convergence towards
the global equilibrium by using simultaneous gradient ascent-descent with
respect to the Wasserstein metric -- a dynamics that admits efficient particle
discretization in high-dimensions, as opposed to entropic mirror descent. We
complement this positive result with a related entropy-regularized loss which
is not bilinear but still convex-concave in the Wasserstein geometry, and for
which simultaneous dynamics do not converge yet timescale separation does.
Taken together, these results showcase the benign geometry of bilinear games in
the space of measures, enabling particle dynamics with global qualitative
convergence guarantees.
