The Harder Path: Last Iterate Convergence for Uncoupled Learning in Zero-Sum Games with Bandit Feedback

Côme Fiegel, Pierre Menard, Tadashi Kozuno, Michal Valko, Vianney Perchet
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:17131-17152, 2025.

Abstract

We study the problem of learning in zero-sum matrix games with repeated play and bandit feedback. Specifically, we focus on developing uncoupled algorithms that guarantee, without communication between players, convergence of the last-iterate to a Nash equilibrium. Although the non-bandit case has been studied extensively, this setting has only been explored recently, with a bound of $\mathcal{O}(T^{-1/8})$ on the exploitability gap. We show that, for uncoupled algorithms, guaranteeing convergence of the policy profiles to a Nash equilibrium is detrimental to the performances, with the best attainable rate being $\mathcal{O}(T^{-1/4})$ in contrast to the usual $\mathcal{O}(T^{-1/2})$ rate for convergence of the average iterates. We then propose two algorithms that achieve this optimal rate. The first algorithm leverages a straightforward tradeoff between exploration and exploitation, while the second employs a regularization technique based on a two-step mirror descent approach.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-fiegel25a, title = {The Harder Path: Last Iterate Convergence for Uncoupled Learning in Zero-Sum Games with Bandit Feedback}, author = {Fiegel, C\^{o}me and Menard, Pierre and Kozuno, Tadashi and Valko, Michal and Perchet, Vianney}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {17131--17152}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/fiegel25a/fiegel25a.pdf}, url = {https://proceedings.mlr.press/v267/fiegel25a.html}, abstract = {We study the problem of learning in zero-sum matrix games with repeated play and bandit feedback. Specifically, we focus on developing uncoupled algorithms that guarantee, without communication between players, convergence of the last-iterate to a Nash equilibrium. Although the non-bandit case has been studied extensively, this setting has only been explored recently, with a bound of $\mathcal{O}(T^{-1/8})$ on the exploitability gap. We show that, for uncoupled algorithms, guaranteeing convergence of the policy profiles to a Nash equilibrium is detrimental to the performances, with the best attainable rate being $\mathcal{O}(T^{-1/4})$ in contrast to the usual $\mathcal{O}(T^{-1/2})$ rate for convergence of the average iterates. We then propose two algorithms that achieve this optimal rate. The first algorithm leverages a straightforward tradeoff between exploration and exploitation, while the second employs a regularization technique based on a two-step mirror descent approach.} }
Endnote
%0 Conference Paper %T The Harder Path: Last Iterate Convergence for Uncoupled Learning in Zero-Sum Games with Bandit Feedback %A Côme Fiegel %A Pierre Menard %A Tadashi Kozuno %A Michal Valko %A Vianney Perchet %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-fiegel25a %I PMLR %P 17131--17152 %U https://proceedings.mlr.press/v267/fiegel25a.html %V 267 %X We study the problem of learning in zero-sum matrix games with repeated play and bandit feedback. Specifically, we focus on developing uncoupled algorithms that guarantee, without communication between players, convergence of the last-iterate to a Nash equilibrium. Although the non-bandit case has been studied extensively, this setting has only been explored recently, with a bound of $\mathcal{O}(T^{-1/8})$ on the exploitability gap. We show that, for uncoupled algorithms, guaranteeing convergence of the policy profiles to a Nash equilibrium is detrimental to the performances, with the best attainable rate being $\mathcal{O}(T^{-1/4})$ in contrast to the usual $\mathcal{O}(T^{-1/2})$ rate for convergence of the average iterates. We then propose two algorithms that achieve this optimal rate. The first algorithm leverages a straightforward tradeoff between exploration and exploitation, while the second employs a regularization technique based on a two-step mirror descent approach.
APA
Fiegel, C., Menard, P., Kozuno, T., Valko, M. & Perchet, V.. (2025). The Harder Path: Last Iterate Convergence for Uncoupled Learning in Zero-Sum Games with Bandit Feedback. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:17131-17152 Available from https://proceedings.mlr.press/v267/fiegel25a.html.

Related Material