Mirror Descent-Ascent for mean-field min-max problems
Razvan-Andrei Lascu, Mateusz B. Majka, Łukasz Szpruch
TL;DR
The work advances mean-field min–max optimization by casting training tasks like GANs into convex-concave games over probability measures and applying infinite-dimensional mirror descent-ascent. By distinguishing simultaneous and sequential playing, the authors prove time-averaged convergence to mixed Nash equilibria with NI errors decaying as $O(N^{-1/2})$ and $O(N^{-2/3})$, respectively, under appropriate relative smoothness and regularity assumptions. A key novelty is the duality-based analysis for the sequential scheme, which reveals faster convergence through control of dual Bregman divergences and commutators. The results generalize finite-dimensional rates to the mean-field setting and align with practical training practices that alternately optimize generator and discriminator components in GANs. Overall, the paper provides rigorous rates, a duality toolkit, and a principled justification for alternating updates in mean-field min–max problems.
Abstract
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.