  In this paper, we extend mean-field Langevin dynamics to minimax optimization
over probability distributions for the first time with symmetric and provably
convergent updates. We propose mean-field Langevin averaged gradient (MFL-AG),
a single-loop algorithm that implements gradient descent ascent in the
distribution spaces with a novel weighted averaging, and establish
average-iterate convergence to the mixed Nash equilibrium. We also study both
time and particle discretization regimes and prove a new uniform-in-time
propagation of chaos result which accounts for the dependency of the particle
interactions on all previous distributions. Furthermore, we propose mean-field
Langevin anchored best response (MFL-ABR), a symmetric double-loop algorithm
based on best response dynamics with linear last-iterate convergence. Finally,
we study applications to zero-sum Markov games and conduct simulations
demonstrating long-term optimality.
