Adaptive extra-gradient methods for min-max optimization and games

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 methods automatically detect the problem's smoothness properties, without requiring any prior tuning by the optimizer. As a result, the algorithm simultaneously achieves the optimal smooth/non-smooth min-max 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 divergent loss functions (such as log-likelihood learning and the like). This "off-the-shelf" 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 family of methods at hand.