Min-Max Optimization Made Simple: Approximating the Proximal Point Method via Contraction Maps

TL;DR

This paper introduces a simple first-order method for convex/concave min-max problems that achieves near-optimal convergence rates by approximating the Proximal Point method using contraction maps, with efficient gradient calls.

Contribution

It presents the Clairvoyant Extra Gradient method, which approximates the Proximal Point method efficiently, providing improved convergence analysis for min-max optimization.

Findings

Achieves near-optimal convergence rates for convex/concave min-max problems.

Requires only O(log 1/ε) gradient calls to approximate the Proximal Point method.

Ensures last-iterate convergence in unconstrained settings.

Abstract

In this paper we present a first-order method that admits near-optimal convergence rates for convex/concave min-max problems while requiring a simple and intuitive analysis. Similarly to the seminal work of Nemirovski and the recent approach of Piliouras et al. in normal form games, our work is based on the fact that the update rule of the Proximal Point method (PP) can be approximated up to accuracy $\epsilon$ with only $O(\log 1/\epsilon)$ additional gradient-calls through the iterations of a contraction map. Then combining the analysis of (PP) method with an error-propagation analysis we establish that the resulting first order method, called Clairvoyant Extra Gradient, admits near-optimal time-average convergence for general domains and last-iterate convergence in the unconstrained case.