Relative Lipschitzness in Extragradient Methods and a Direct Recipe for Acceleration

Abstract

We show that standard extragradient methods (i.e. mirror prox and dual extrapolation) recover optimal accelerated rates for first-order minimization of smooth convex functions. To obtain this result we provide a fine-grained characterization of the convergence rates of extragradient methods for solving monotone variational inequalities in terms of a natural condition we call relative Lipschitzness. We further generalize this framework to handle local and randomized notions of relative Lipschitzness and thereby recover rates for box-constrained _ regression based on area convexity and complexity bounds achieved by accelerated (randomized) coordinate descent for smooth convex function minimization.

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