Last-Iterate Convergence of Regularized Gradient Methods for Stochastic Monotone Variational Inequalities

Shinji Ito, Taira Tsuchiya, Kaito Ariu, Kenshi Abe
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:50047-50071, 2026.

Abstract

We study last-iterate convergence for stochastic smooth and monotone variational inequalities (VIs), a framework that captures convex-concave saddle points and Nash equilibrium computation in monotone games with noisy payoff feedback. In contrast to the well-understood average-iterate guarantees, anytime last-iterate guarantees in stochastic settings remain limited, despite their relevance for uncoupled learning dynamics that output a single current strategy. We analyze two single-call regularized methods, the regularized gradient (RG) and the regularized optimistic gradient (ROG) methods, and establish anytime last-iterate convergence rates in terms of the squared gap function. For monotone VIs, RG attains $O(t^{-2/5})$ while ROG achieves the variance-adaptive rate $O(\sigma^{4/5} t^{-2/5} + t^{-1})$, where $\sigma^2$ is the noise variance. For $\lambda$-strongly monotone VIs, ROG yields $O(\sigma^2 / (\lambda^2 t) + t^{-c})$ for any constant $c \ge 2$. These results give anytime last-iterate guarantees without knowing the horizon and show that optimism improves convergence in the low-noise regime.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-ito26a, title = {Last-Iterate Convergence of Regularized Gradient Methods for Stochastic Monotone Variational Inequalities}, author = {Ito, Shinji and Tsuchiya, Taira and Ariu, Kaito and Abe, Kenshi}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {50047--50071}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/ito26a/ito26a.pdf}, url = {https://proceedings.mlr.press/v306/ito26a.html}, abstract = {We study last-iterate convergence for stochastic smooth and monotone variational inequalities (VIs), a framework that captures convex-concave saddle points and Nash equilibrium computation in monotone games with noisy payoff feedback. In contrast to the well-understood average-iterate guarantees, anytime last-iterate guarantees in stochastic settings remain limited, despite their relevance for uncoupled learning dynamics that output a single current strategy. We analyze two single-call regularized methods, the regularized gradient (RG) and the regularized optimistic gradient (ROG) methods, and establish anytime last-iterate convergence rates in terms of the squared gap function. For monotone VIs, RG attains $O(t^{-2/5})$ while ROG achieves the variance-adaptive rate $O(\sigma^{4/5} t^{-2/5} + t^{-1})$, where $\sigma^2$ is the noise variance. For $\lambda$-strongly monotone VIs, ROG yields $O(\sigma^2 / (\lambda^2 t) + t^{-c})$ for any constant $c \ge 2$. These results give anytime last-iterate guarantees without knowing the horizon and show that optimism improves convergence in the low-noise regime.} }
Endnote
%0 Conference Paper %T Last-Iterate Convergence of Regularized Gradient Methods for Stochastic Monotone Variational Inequalities %A Shinji Ito %A Taira Tsuchiya %A Kaito Ariu %A Kenshi Abe %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-ito26a %I PMLR %P 50047--50071 %U https://proceedings.mlr.press/v306/ito26a.html %V 306 %X We study last-iterate convergence for stochastic smooth and monotone variational inequalities (VIs), a framework that captures convex-concave saddle points and Nash equilibrium computation in monotone games with noisy payoff feedback. In contrast to the well-understood average-iterate guarantees, anytime last-iterate guarantees in stochastic settings remain limited, despite their relevance for uncoupled learning dynamics that output a single current strategy. We analyze two single-call regularized methods, the regularized gradient (RG) and the regularized optimistic gradient (ROG) methods, and establish anytime last-iterate convergence rates in terms of the squared gap function. For monotone VIs, RG attains $O(t^{-2/5})$ while ROG achieves the variance-adaptive rate $O(\sigma^{4/5} t^{-2/5} + t^{-1})$, where $\sigma^2$ is the noise variance. For $\lambda$-strongly monotone VIs, ROG yields $O(\sigma^2 / (\lambda^2 t) + t^{-c})$ for any constant $c \ge 2$. These results give anytime last-iterate guarantees without knowing the horizon and show that optimism improves convergence in the low-noise regime.
APA
Ito, S., Tsuchiya, T., Ariu, K. & Abe, K.. (2026). Last-Iterate Convergence of Regularized Gradient Methods for Stochastic Monotone Variational Inequalities. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:50047-50071 Available from https://proceedings.mlr.press/v306/ito26a.html.

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