Generalization Bounds for Stochastic Saddle Point Problems

Junyu Zhang, Mingyi Hong, Mengdi Wang, Shuzhong Zhang
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, PMLR 130:568-576, 2021.

Abstract

This paper studies the generalization bounds for the empirical saddle point (ESP) solution to stochastic saddle point (SSP) problems. For SSP with Lipschitz continuous and strongly convex-strongly concave objective functions, we establish an $O\left(1/n\right)$ generalization bound by using a probabilistic stability argument. We also provide generalization bounds under a variety of assumptions, including the cases without strong convexity and without bounded domains. We illustrate our results in three examples: batch policy learning in Markov decision process, stochastic composite optimization problem, and mixed strategy Nash equilibrium estimation for stochastic games. In each of these examples, we show that a regularized ESP solution enjoys a near-optimal sample complexity. To the best of our knowledge, this is the first set of results on the generalization theory of ESP.

Cite this Paper

BibTeX
@InProceedings{pmlr-v130-zhang21a, title = { Generalization Bounds for Stochastic Saddle Point Problems }, author = {Zhang, Junyu and Hong, Mingyi and Wang, Mengdi and Zhang, Shuzhong}, booktitle = {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics}, pages = {568--576}, 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/zhang21a/zhang21a.pdf}, url = {https://proceedings.mlr.press/v130/zhang21a.html}, abstract = { This paper studies the generalization bounds for the empirical saddle point (ESP) solution to stochastic saddle point (SSP) problems. For SSP with Lipschitz continuous and strongly convex-strongly concave objective functions, we establish an $O\left(1/n\right)$ generalization bound by using a probabilistic stability argument. We also provide generalization bounds under a variety of assumptions, including the cases without strong convexity and without bounded domains. We illustrate our results in three examples: batch policy learning in Markov decision process, stochastic composite optimization problem, and mixed strategy Nash equilibrium estimation for stochastic games. In each of these examples, we show that a regularized ESP solution enjoys a near-optimal sample complexity. To the best of our knowledge, this is the first set of results on the generalization theory of ESP. } }
Endnote
%0 Conference Paper %T Generalization Bounds for Stochastic Saddle Point Problems %A Junyu Zhang %A Mingyi Hong %A Mengdi Wang %A Shuzhong Zhang %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-zhang21a %I PMLR %P 568--576 %U https://proceedings.mlr.press/v130/zhang21a.html %V 130 %X This paper studies the generalization bounds for the empirical saddle point (ESP) solution to stochastic saddle point (SSP) problems. For SSP with Lipschitz continuous and strongly convex-strongly concave objective functions, we establish an $O\left(1/n\right)$ generalization bound by using a probabilistic stability argument. We also provide generalization bounds under a variety of assumptions, including the cases without strong convexity and without bounded domains. We illustrate our results in three examples: batch policy learning in Markov decision process, stochastic composite optimization problem, and mixed strategy Nash equilibrium estimation for stochastic games. In each of these examples, we show that a regularized ESP solution enjoys a near-optimal sample complexity. To the best of our knowledge, this is the first set of results on the generalization theory of ESP.
APA
Zhang, J., Hong, M., Wang, M. & Zhang, S.. (2021). Generalization Bounds for Stochastic Saddle Point Problems . Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 130:568-576 Available from https://proceedings.mlr.press/v130/zhang21a.html.

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