Mirror Descent-Ascent for mean-field min-max problems
Razvan-Andrei Lascu Mateusz B. Majka Łukasz Szpruch
We study two variants of the mirror descent-ascent algorithm for solving
min-max problems on the space of measures: simultaneous and sequential. We work
under assumptions of convexity-concavity and relative smoothness of the payoff
function with respect to a suitable Bregman divergence, defined on the space of
measures via flat derivatives. We show that the convergence rates to mixed Nash
equilibria, measured in the Nikaid\`o-Isoda error, are of order
$\mathcal{O}\left(N^{-1/2}\right)$ and $\mathcal{O}\left(N^{-2/3}\right)$ for
the simultaneous and sequential schemes, respectively, which is in line with
the state-of-the-art results for related finite-dimensional algorithms.