Direct-Search for a Class of Stochastic Min-Max Problems

Sotirios-Konstantinos Anagnostidis, Aurelien Lucchi, Youssef Diouane
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, PMLR 130:3772-3780, 2021.

Abstract

Recent applications in machine learning have renewed the interest of the community in min-max optimization problems. While gradient-based optimization methods are widely used to solve such problems, there are however many scenarios where these techniques are not well-suited, or even not applicable when the gradient is not accessible. We investigate the use of direct-search methods that belong to a class of derivative-free techniques that only access the objective function through an oracle. In this work, we design a novel algorithm in the context of min-max saddle point games where one sequentially updates the min and the max player. We prove convergence of this algorithm under mild assumptions, where the objective of the max-player satisfies the Polyak-Ł{}ojasiewicz (PL) condition, while the min-player is characterized by a nonconvex objective. Our method only assumes dynamically adjusted accurate estimates of the oracle with a fixed probability. To the best of our knowledge, our analysis is the first one to address the convergence of a direct-search method for min-max objectives in a stochastic setting.

Cite this Paper

BibTeX
@InProceedings{pmlr-v130-anagnostidis21a, title = { Direct-Search for a Class of Stochastic Min-Max Problems }, author = {Anagnostidis, Sotirios-Konstantinos and Lucchi, Aurelien and Diouane, Youssef}, booktitle = {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics}, pages = {3772--3780}, year = {2021}, editor = {Banerjee, Arindam and Fukumizu, Kenji}, volume = {130}, series = {Proceedings of Machine Learning Research}, month = {13--15 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v130/anagnostidis21a/anagnostidis21a.pdf}, url = {https://proceedings.mlr.press/v130/anagnostidis21a.html}, abstract = { Recent applications in machine learning have renewed the interest of the community in min-max optimization problems. While gradient-based optimization methods are widely used to solve such problems, there are however many scenarios where these techniques are not well-suited, or even not applicable when the gradient is not accessible. We investigate the use of direct-search methods that belong to a class of derivative-free techniques that only access the objective function through an oracle. In this work, we design a novel algorithm in the context of min-max saddle point games where one sequentially updates the min and the max player. We prove convergence of this algorithm under mild assumptions, where the objective of the max-player satisfies the Polyak-Ł{}ojasiewicz (PL) condition, while the min-player is characterized by a nonconvex objective. Our method only assumes dynamically adjusted accurate estimates of the oracle with a fixed probability. To the best of our knowledge, our analysis is the first one to address the convergence of a direct-search method for min-max objectives in a stochastic setting. } }
Endnote
%0 Conference Paper %T Direct-Search for a Class of Stochastic Min-Max Problems %A Sotirios-Konstantinos Anagnostidis %A Aurelien Lucchi %A Youssef Diouane %B Proceedings of The 24th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2021 %E Arindam Banerjee %E Kenji Fukumizu %F pmlr-v130-anagnostidis21a %I PMLR %P 3772--3780 %U https://proceedings.mlr.press/v130/anagnostidis21a.html %V 130 %X Recent applications in machine learning have renewed the interest of the community in min-max optimization problems. While gradient-based optimization methods are widely used to solve such problems, there are however many scenarios where these techniques are not well-suited, or even not applicable when the gradient is not accessible. We investigate the use of direct-search methods that belong to a class of derivative-free techniques that only access the objective function through an oracle. In this work, we design a novel algorithm in the context of min-max saddle point games where one sequentially updates the min and the max player. We prove convergence of this algorithm under mild assumptions, where the objective of the max-player satisfies the Polyak-Ł{}ojasiewicz (PL) condition, while the min-player is characterized by a nonconvex objective. Our method only assumes dynamically adjusted accurate estimates of the oracle with a fixed probability. To the best of our knowledge, our analysis is the first one to address the convergence of a direct-search method for min-max objectives in a stochastic setting.
APA
Anagnostidis, S., Lucchi, A. & Diouane, Y.. (2021). Direct-Search for a Class of Stochastic Min-Max Problems . Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 130:3772-3780 Available from https://proceedings.mlr.press/v130/anagnostidis21a.html.

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