  Finding the mixed Nash equilibria (MNE) of a two-player zero sum continuous
game is an important and challenging problem in machine learning. A canonical
algorithm to finding the MNE is the noisy gradient descent ascent method which
in the infinite particle limit gives rise to the {\em Mean-Field Gradient
Descent Ascent} (GDA) dynamics on the space of probability measures. In this
paper, we first study the convergence of a two-scale Mean-Field GDA dynamics
for finding the MNE of the entropy-regularized objective. More precisely we
show that for each finite temperature (or regularization parameter), the
two-scale Mean-Field GDA with a suitable {\em finite} scale ratio converges
exponentially to the unique MNE without assuming the convexity or concavity of
the interaction potential. The key ingredient of our proof lies in the
construction of new Lyapunov functions that dissipate exponentially along the
Mean-Field GDA. We further study the simulated annealing of the Mean-Field GDA
dynamics. We show that with a temperature schedule that decays logarithmically
in time the annealed Mean-Field GDA converges to the MNE of the original
unregularized objective.
