Parameter-Free Adaptive Extragradient Methods for Generalized-Smooth Variational Inequalities

Abstract

Variational inequalities (VIs) provide a unified framework for optimization, saddle-point, equilibrium, and learning problems, where operators need not be globally Lipschitz. The $\alpha$-symmetric generalized-smoothness condition allows local operator variation to grow with the operator norm, but existing extragradient methods often require unknown problem-specific constants. To address this issue, we propose PF-AEG, an explicit, line-search-free extragradient method whose stepsizes use quantities generated during the iterations, without prior knowledge of these constants. Under $p$-quasi-sharpness, we prove full-sequence convergence and last-iterate squared-distance rates: linear convergence for $0<p\le2$ and $\mathcal{O}(k^{-2/(p-2)})$ for $p>2$. The latter matches existing parameter-dependent rates under comparable assumptions, while the linear guarantee extends the previous $p=2$ result to $0<p\le2$. Building on the idea of PF-AEG, we develop AEG-VR, a variance-reduced adaptive extragradient method for finite-sum VIs. We prove almost-sure convergence and, under mild conditions, obtain analogous expected last-iterate rates, while retaining independence from the unknown problem-specific constants in stepsize selection. Experiments on synthetic generalized-smooth VIs and several learning problems show that the proposed methods are competitive with, and in several tested regimes outperform, existing baselines.