Extragradient Method for $(L_0, L_1)$-Lipschitz Root-finding Problems

2025-10-25

AI Summary

Existing extragradient (EG) methods for minimax optimization, root-finding, and variational inequalities rely on the restrictive strong $L$-Lipschitz assumption, failing to capture complex operator structures arising in modern machine learning. Method: We propose the first adaptive step-size strategy based on dynamic operator-norm estimation, operating under the significantly milder $alpha$-symmetric $(L_0, L_1)$-Lipschitz condition. Contribution/Results: We establish sublinear convergence for monotone operators and linear convergence for strongly monotone operators. Moreover, we provide the first local convergence guarantee under the weak Minty condition—a substantially weaker requirement than standard monotonicity. Experiments demonstrate that our method achieves robust, practical convergence across diverse non-standard operator settings, markedly broadening the applicability of extragradient methods beyond classical assumptions.

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: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Introduced by Korpelevich in 1976, the extragradient method (EG) has become a cornerstone technique for solving min-max optimization, root-finding problems, and variational inequalities (VIs). Despite its longstanding presence and significant attention within the optimization community, most works focusing on understanding its convergence guarantees assume the strong L-Lipschitz condition. In this work, building on the proposed assumptions by Zhang et al. [2024b] for minimization and Vankov et al.[2024] for VIs, we focus on the more relaxed $α$-symmetric $(L_0, L_1)$-Lipschitz condition. This condition generalizes the standard Lipschitz assumption by allowing the Lipschitz constant to scale with the operator norm, providing a more refined characterization of problem structures in modern machine learning. Under the $α$-symmetric $(L_0, L_1)$-Lipschitz condition, we propose a novel step size strategy for EG to solve root-finding problems and establish sublinear convergence rates for monotone operators and linear convergence rates for strongly monotone operators. Additionally, we prove local convergence guarantees for weak Minty operators. We supplement our analysis with experiments validating our theory and demonstrating the effectiveness and robustness of the proposed step sizes for EG.

Problem

• Extends extragradient method to relaxed Lipschitz root-finding problems
• Proposes novel step size strategy for generalized Lipschitz conditions
• Establishes convergence rates for monotone and strongly monotone operators

Innovation

• Extragradient method with novel step size strategy
• Relaxed alpha-symmetric Lipschitz condition application
• Convergence guarantees for monotone and strongly monotone operators