Extragradient Method: $O(1/K)$ Last-Iterate Convergence for Monotone Variational Inequalities and Connections With Cocoercivity
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 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.
Keywords
Cite
@article{arxiv.2110.04261,
title = {Extragradient Method: $O(1/K)$ Last-Iterate Convergence for Monotone Variational Inequalities and Connections With Cocoercivity},
author = {Eduard Gorbunov and Nicolas Loizou and Gauthier Gidel},
journal= {arXiv preprint arXiv:2110.04261},
year = {2022}
}
Comments
AISTATS 2022; 37 pages, 4 figures. Changes in v2: structure was changed, minor typos are fixed, several additional clarifications were added. Code: https://github.com/eduardgorbunov/extragradient_last_iterate_AISTATS_2022