Extragradient Method: $O(1/K)$ Last-Iterate Convergence for Monotone Variational Inequalities and Connections With Cocoercivity

TL;DR

This paper proves the first last-iterate $O(1/K)$ convergence rate for the extragradient method in monotone variational inequalities without extra assumptions, and explores connections with cocoercivity of related operators.

Contribution

It establishes the first last-iterate convergence rate for EG under minimal assumptions and analyzes cocoercivity properties of related optimization operators.

Findings

First last-iterate $O(1/K)$ convergence rate for EG in monotone VIPs.

EG's update operators exhibit specific (non-)cocoercivity properties.

Connections between EG, Optimistic Gradient, and Hamiltonian Gradient methods are clarified.

Abstract

Extragradient method (EG) (Korpelevich, 1976) is one of the most popular methods for solving saddle point and variational inequalities problems (VIP). Despite its long history and significant attention in the optimization community, there remain important open questions about convergence of EG. In this paper, we resolve one of such questions and derive the first last-iterate $O(1/K)$ convergence rate for EG for monotone and Lipschitz VIP without any additional assumptions on the operator unlike the only known result of this type (Golowich et al., 2020) that relies on the Lipschitzness of the Jacobian of the operator. The rate is given in terms of reducing the squared norm of the operator. Moreover, we establish several results on the (non-)cocoercivity of the update operators of EG, Optimistic Gradient Method, and Hamiltonian Gradient Method, when the original operator is monotone and Lipschitz.