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 α-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≤2 and 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≤2. 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.
Related
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- A Parameter-Free Adaptive Reflected Gradient Method for Monotone Variational Inequalities
- Extragradient and Past Extragradient Methods for (L0,L1)-Lipschitz Variational Inequalities
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BibTeX
@misc{chu2026parameterfree,
title = {{Parameter-Free Adaptive Extragradient Methods for Generalized-Smooth Variational Inequalities}},
author = {Zhaoyang Chu and Hongwei Liu and Yakui Huang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14839},
eprint = {2610.14839},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14839},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Parameter-Free Adaptive Extragradient Methods for Generalized-Smooth Variational Inequalities
AU - Zhaoyang Chu
AU - Hongwei Liu
AU - Yakui Huang
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14839
UR - https://arxiv.org/abs/2610.14839
AB - Variational inequalities (VIs) provide a unified framework for optimization, saddle-point, equilibrium, and learning problems, where operators need not be globally Lipschitz. The α-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≤2 and 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≤2. 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.
ER -