Adaptive and Universal Algorithms for Variational Inequalities with Optimal Convergence

TL;DR

This paper introduces adaptive, universal algorithms for variational inequalities that automatically adjust to unknown parameters, achieve optimal convergence rates across various settings, and work on unbounded domains, improving efficiency and applicability.

Contribution

The authors develop new adaptive algorithms for variational inequalities that are universal, optimal, and applicable to unbounded domains, surpassing previous methods in efficiency and scope.

Findings

Achieve optimal convergence rates in multiple settings

Remove bounded domain restriction from previous algorithms

Require only one or two operator evaluations per iteration

Abstract

We develop new adaptive algorithms for variational inequalities with monotone operators, which capture many problems of interest, notably convex optimization and convex-concave saddle point problems. Our algorithms automatically adapt to unknown problem parameters such as the smoothness and the norm of the operator, and the variance of the stochastic evaluation oracle. We show that our algorithms are universal and simultaneously achieve the optimal convergence rates in the non-smooth, smooth, and stochastic settings. The convergence guarantees of our algorithms improve over existing adaptive methods by a $\Omega(\sqrt{\ln T})$ factor, matching the optimal non-adaptive algorithms. Additionally, prior works require that the optimization domain is bounded. In this work, we remove this restriction and give algorithms for unbounded domains that are adaptive and universal. Our general proof techniques can be used for many variants of the algorithm using one or two operator evaluations per iteration. The classical methods based on the ExtraGradient/MirrorProx algorithm require two operator evaluations per iteration, which is the dominant factor in the running time in many settings.