Convergence of Proximal Point and Extragradient-Based Methods Beyond Monotonicity: the Case of Negative Comonotonicity

TL;DR

This paper extends the analysis of Proximal Point, Extragradient, and Optimistic Gradient methods to negative comonotonicity, a weaker condition than monotonicity, providing tight complexity bounds for non-monotone min-max problems.

Contribution

It offers the first tight complexity analyses of these methods under negative comonotonicity, broadening their applicability beyond traditional monotonic settings.

Findings

Provided tight complexity bounds for Proximal Point, Extragradient, and Optimistic Gradient methods.

Extended theoretical guarantees of these algorithms to negative comonotonicity.

Closed gaps in understanding their performance beyond monotonic assumptions.

Abstract

Algorithms for min-max optimization and variational inequalities are often studied under monotonicity assumptions. Motivated by non-monotone machine learning applications, we follow the line of works [Diakonikolas et al., 2021, Lee and Kim, 2021, Pethick et al., 2022, B\"ohm, 2022] aiming at going beyond monotonicity by considering the weaker negative comonotonicity assumption. In particular, we provide tight complexity analyses for the Proximal Point, Extragradient, and Optimistic Gradient methods in this setup, closing some questions on their working guarantees beyond monotonicity.