Parameter-free non-ergodic extragradient algorithms for solving monotone variational inequalities

2026-04-08

AI Summary

This work addresses the limitations of existing methods for monotone variational inequalities, which typically rely on global smoothness assumptions and lack non-ergodic (last-iterate) convergence guarantees. The authors propose a parameter-free extragradient algorithm that incorporates backtracking line search to adaptively handle operators with only local Lipschitz continuity. Notably, this is the first method to establish non-asymptotic last-iterate convergence rates for constrained monotone variational inequalities without requiring global smoothness. Empirical evaluations demonstrate that the algorithm consistently outperforms state-of-the-art approaches across diverse tasks, including bilinear matrix games, LASSO regression, minimax group fairness optimization, and maximum entropy sampling.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Monotone variational inequalities (VIs) provide a unifying framework for convex minimization, equilibrium computation, and convex-concave saddle-point problems. Extragradient-type methods are among the most effective first-order algorithms for such problems, but their performance hinges critically on stepsize selection. While most existing theory focuses on ergodic averages of the iterates, practical performance is often driven by the significantly stronger behavior of the last iterate. Moreover, available last-iterate guarantees typically rely on fixed stepsizes chosen using problem-specific global smoothness information, which is often difficult to estimate accurately and may not even be applicable. In this paper, we develop parameter-free extragradient methods with non-asymptotic last-iterate guarantees for constrained monotone VIs. For globally Lipschitz operators, our algorithm achieves an $o(1/\sqrt{T})$ last-iterate rate. We then extend the framework to locally Lipschitz operators via backtracking line search and obtain the same rate while preserving parameter-freeness, thereby making parameter-free last-iterate methods applicable to important problem classes for which global smoothness is unrealistic. Our numerical experiments on bilinear matrix games, LASSO, minimax group fairness, and state-of-the-art maximum entropy sampling relaxations demonstrate wide applicability of our results as well as strong last-iterate performance and significant improvements over existing methods.

Problem

• monotone variational inequalities
• extragradient methods
• last-iterate convergence
• parameter-free algorithms
• Lipschitz operators

Innovation

• parameter-free
• last-iterate convergence
• extragradient
• monotone variational inequalities
• backtracking line search