Near-Optimal Single-Loop Predictor--Corrector Extragradient Method for Strongly Convex--Strongly Concave Minimax Optimization
Abstract
We study smooth strongly convex--strongly concave minimax optimization in the deterministic unconstrained setting, without assuming a bilinear or separable structure. Although existing multi-loop methods attain near-optimal condition-number dependence, standard single-loop methods generally exhibit a substantial complexity gap. To close this gap, we propose the Single-Loop Predictor--Corrector Extragradient Method with Damped Momentum (PCE-DM), which combines an extragradient prediction--correction scheme with a novel auxiliary feedback recursion for the weaker-curvature variable. PCE-DM uses fixed parameters and two new full-gradient evaluations per iteration after one initialization query, while requiring no inner solves, accuracy schedules, or staged restarts. We develop a Lyapunov analysis that controls the predictor--corrector mismatch through corrected-gradient increments and establish last-iterate linear convergence. Specifically, PCE-DM computes an ε-accurate relative solution, measured by the squared Euclidean distance to the saddle point, within 𝒪 (√(κxκy) log(2κxκy/ε)) full-gradient queries. This result closes the condition-number complexity gap between standard single-loop methods and near-optimal multi-loop methods, matching the known lower-bound order up to logarithmic factors while retaining fixed, explicit single-loop updates. Numerical experiments on regularized linear regression and AUC maximization demonstrate the computational efficiency of PCE-DM.
Related
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BibTeX
@misc{zhang2026nearoptimal,
title = {{Near-Optimal Single-Loop Predictor--Corrector Extragradient Method for Strongly Convex--Strongly Concave Minimax Optimization}},
author = {Minhao Zhang and Zi Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20327},
eprint = {2609.20327},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20327},
}
RIS
TY - UNPB
TI - Near-Optimal Single-Loop Predictor--Corrector Extragradient Method for Strongly Convex--Strongly Concave Minimax Optimization
AU - Minhao Zhang
AU - Zi Xu
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.20327
UR - https://arxiv.org/abs/2609.20327
AB - We study smooth strongly convex--strongly concave minimax optimization in the deterministic unconstrained setting, without assuming a bilinear or separable structure. Although existing multi-loop methods attain near-optimal condition-number dependence, standard single-loop methods generally exhibit a substantial complexity gap. To close this gap, we propose the Single-Loop Predictor--Corrector Extragradient Method with Damped Momentum (PCE-DM), which combines an extragradient prediction--correction scheme with a novel auxiliary feedback recursion for the weaker-curvature variable. PCE-DM uses fixed parameters and two new full-gradient evaluations per iteration after one initialization query, while requiring no inner solves, accuracy schedules, or staged restarts. We develop a Lyapunov analysis that controls the predictor--corrector mismatch through corrected-gradient increments and establish last-iterate linear convergence. Specifically, PCE-DM computes an ε-accurate relative solution, measured by the squared Euclidean distance to the saddle point, within 𝒪 (√(κ_xκ_y) log(2κ_xκ_y/ε)) full-gradient queries. This result closes the condition-number complexity gap between standard single-loop methods and near-optimal multi-loop methods, matching the known lower-bound order up to logarithmic factors while retaining fixed, explicit single-loop updates. Numerical experiments on regularized linear regression and AUC maximization demonstrate the computational efficiency of PCE-DM.
ER -