Аннотация
We provide improved convergence rates for constrained convex-concave min-max
problems and monotone variational inequalities with higher-order smoothness. In
min-max settings where the $p^th$-order derivatives are Lipschitz continuous,
we give an algorithm HigherOrderMirrorProx that achieves an iteration
complexity of $O(1/T^p+12)$ when given access to an oracle for
finding a fixed point of a $p^th$-order equation. We give analogous rates for
the weak monotone variational inequality problem. For $p>2$, our results
improve upon the iteration complexity of the first-order Mirror Prox method of
Nemirovski 2004 and the second-order method of Monteiro and Svaiter 2012.
We further instantiate our entire algorithm in the unconstrained $p=2$ case.
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