Convergence of Mean-Field Langevin Stochastic Descent-Ascent for Distributional Minimax Optimization

Zhangyi Liu, Feng Liu, Rui Gao, Shuang Li
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:38869-38893, 2025.

Abstract

We study convergence properties of the discrete-time Mean-Field Langevin Stochastic Gradient Descent-Ascent (MFL-SGDA) algorithm for solving distributional minimax optimization. These problems arise in various applications, such as zero-sum games, generative adversarial networks and distributionally robust learning. Despite the significance of MFL-SGDA in these contexts, the discrete-time convergence rate remains underexplored. To address this gap, we establish a last-iterate convergence rate of $O(\frac{1}{\epsilon}\log\frac{1}{\epsilon})$ for MFL-SGDA. This rate is nearly optimal when compared to the complexity lower bound of its Euclidean counterpart. This rate also matches the complexity of mean-field Langevin stochastic gradient descent for distributional minimization and the outer-loop iteration complexity of an existing double-loop algorithm for distributional minimax problems. By leveraging an elementary analysis framework that avoids PDE-based techniques, we overcome previous limitations and achieve a faster convergence rate.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-liu25al, title = {Convergence of Mean-Field {L}angevin Stochastic Descent-Ascent for Distributional Minimax Optimization}, author = {Liu, Zhangyi and Liu, Feng and Gao, Rui and Li, Shuang}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {38869--38893}, 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/liu25al/liu25al.pdf}, url = {https://proceedings.mlr.press/v267/liu25al.html}, abstract = {We study convergence properties of the discrete-time Mean-Field Langevin Stochastic Gradient Descent-Ascent (MFL-SGDA) algorithm for solving distributional minimax optimization. These problems arise in various applications, such as zero-sum games, generative adversarial networks and distributionally robust learning. Despite the significance of MFL-SGDA in these contexts, the discrete-time convergence rate remains underexplored. To address this gap, we establish a last-iterate convergence rate of $O(\frac{1}{\epsilon}\log\frac{1}{\epsilon})$ for MFL-SGDA. This rate is nearly optimal when compared to the complexity lower bound of its Euclidean counterpart. This rate also matches the complexity of mean-field Langevin stochastic gradient descent for distributional minimization and the outer-loop iteration complexity of an existing double-loop algorithm for distributional minimax problems. By leveraging an elementary analysis framework that avoids PDE-based techniques, we overcome previous limitations and achieve a faster convergence rate.} }
Endnote
%0 Conference Paper %T Convergence of Mean-Field Langevin Stochastic Descent-Ascent for Distributional Minimax Optimization %A Zhangyi Liu %A Feng Liu %A Rui Gao %A Shuang Li %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-liu25al %I PMLR %P 38869--38893 %U https://proceedings.mlr.press/v267/liu25al.html %V 267 %X We study convergence properties of the discrete-time Mean-Field Langevin Stochastic Gradient Descent-Ascent (MFL-SGDA) algorithm for solving distributional minimax optimization. These problems arise in various applications, such as zero-sum games, generative adversarial networks and distributionally robust learning. Despite the significance of MFL-SGDA in these contexts, the discrete-time convergence rate remains underexplored. To address this gap, we establish a last-iterate convergence rate of $O(\frac{1}{\epsilon}\log\frac{1}{\epsilon})$ for MFL-SGDA. This rate is nearly optimal when compared to the complexity lower bound of its Euclidean counterpart. This rate also matches the complexity of mean-field Langevin stochastic gradient descent for distributional minimization and the outer-loop iteration complexity of an existing double-loop algorithm for distributional minimax problems. By leveraging an elementary analysis framework that avoids PDE-based techniques, we overcome previous limitations and achieve a faster convergence rate.
APA
Liu, Z., Liu, F., Gao, R. & Li, S.. (2025). Convergence of Mean-Field Langevin Stochastic Descent-Ascent for Distributional Minimax Optimization. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:38869-38893 Available from https://proceedings.mlr.press/v267/liu25al.html.

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