  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.
