## Adaptive Extra-Gradient Methods for Min-Max Optimization and Games

28 Sep 2020 (modified: 25 Jan 2021)ICLR 2021 PosterReaders: Everyone
• Keywords: min-max optimization, games, mirror-prox, adaptive methods, regime agnostic methods
• Abstract: We present a new family of min-max optimization algorithms that automatically exploit the geometry of the gradient data observed at earlier iterations to perform more informative extra-gradient steps in later ones. Thanks to this adaptation mechanism, the proposed method automatically detects whether the problem is smooth or not, without requiring any prior tuning by the optimizer. As a result, the algorithm simultaneously achieves order-optimal convergence rates, i.e., it converges to an $\varepsilon$-optimal solution within $\mathcal{O}(1/\varepsilon)$ iterations in smooth problems, and within $\mathcal{O}(1/\varepsilon^2)$ iterations in non-smooth ones. Importantly, these guarantees do not require any of the standard boundedness or Lipschitz continuity conditions that are typically assumed in the literature; in particular, they apply even to problems with singularities (such as resource allocation problems and the like). This adaptation is achieved through the use of a geometric apparatus based on Finsler metrics and a suitably chosen mirror-prox template that allows us to derive sharp convergence rates for the methods at hand.
• One-sentence Summary: We develop an adaptive mirror-prox method for min-max problems and games that achieves order-optimal rates in both smooth and non-smooth problems.
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