Mirror Descent-Ascent for mean-field min-max problems
This work addresses optimization in mean-field games for researchers, but it is incremental as it extends finite-dimensional results to measure spaces.
The paper tackles min-max problems on the space of measures by proposing mirror descent-ascent variants, achieving convergence rates of O(N^{-1/2}) for simultaneous and O(N^{-2/3}) for sequential schemes to mixed Nash equilibria.
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 Nikaidò-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.