On the Global Convergence Rates of Softmax Policy Gradient Methods

Jincheng Mei, Chenjun Xiao, Csaba Szepesvari, Dale Schuurmans
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:6820-6829, 2020.

Abstract

We make three contributions toward better understanding policy gradient methods in the tabular setting. First, we show that with the true gradient, policy gradient with a softmax parametrization converges at a $O(1/t)$ rate, with constants depending on the problem and initialization. This result significantly expands the recent asymptotic convergence results. The analysis relies on two findings: that the softmax policy gradient satisfies a Ł{}ojasiewicz inequality, and the minimum probability of an optimal action during optimization can be bounded in terms of its initial value. Second, we analyze entropy regularized policy gradient and show that it enjoys a significantly faster linear convergence rate $O(e^{-t})$ toward softmax optimal policy. This result resolves an open question in the recent literature. Finally, combining the above two results and additional new $\Omega(1/t)$ lower bound results, we explain how entropy regularization improves policy optimization, even with the true gradient, from the perspective of convergence rate. The separation of rates is further explained using the notion of non-uniform Ł{}ojasiewicz degree. These results provide a theoretical understanding of the impact of entropy and corroborate existing empirical studies.

Cite this Paper

BibTeX
@InProceedings{pmlr-v119-mei20b, title = {On the Global Convergence Rates of Softmax Policy Gradient Methods}, author = {Mei, Jincheng and Xiao, Chenjun and Szepesvari, Csaba and Schuurmans, Dale}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {6820--6829}, year = {2020}, editor = {III, Hal Daumé and Singh, Aarti}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/mei20b/mei20b.pdf}, url = {https://proceedings.mlr.press/v119/mei20b.html}, abstract = {We make three contributions toward better understanding policy gradient methods in the tabular setting. First, we show that with the true gradient, policy gradient with a softmax parametrization converges at a $O(1/t)$ rate, with constants depending on the problem and initialization. This result significantly expands the recent asymptotic convergence results. The analysis relies on two findings: that the softmax policy gradient satisfies a Ł{}ojasiewicz inequality, and the minimum probability of an optimal action during optimization can be bounded in terms of its initial value. Second, we analyze entropy regularized policy gradient and show that it enjoys a significantly faster linear convergence rate $O(e^{-t})$ toward softmax optimal policy. This result resolves an open question in the recent literature. Finally, combining the above two results and additional new $\Omega(1/t)$ lower bound results, we explain how entropy regularization improves policy optimization, even with the true gradient, from the perspective of convergence rate. The separation of rates is further explained using the notion of non-uniform Ł{}ojasiewicz degree. These results provide a theoretical understanding of the impact of entropy and corroborate existing empirical studies.} }
Endnote
%0 Conference Paper %T On the Global Convergence Rates of Softmax Policy Gradient Methods %A Jincheng Mei %A Chenjun Xiao %A Csaba Szepesvari %A Dale Schuurmans %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-mei20b %I PMLR %P 6820--6829 %U https://proceedings.mlr.press/v119/mei20b.html %V 119 %X We make three contributions toward better understanding policy gradient methods in the tabular setting. First, we show that with the true gradient, policy gradient with a softmax parametrization converges at a $O(1/t)$ rate, with constants depending on the problem and initialization. This result significantly expands the recent asymptotic convergence results. The analysis relies on two findings: that the softmax policy gradient satisfies a Ł{}ojasiewicz inequality, and the minimum probability of an optimal action during optimization can be bounded in terms of its initial value. Second, we analyze entropy regularized policy gradient and show that it enjoys a significantly faster linear convergence rate $O(e^{-t})$ toward softmax optimal policy. This result resolves an open question in the recent literature. Finally, combining the above two results and additional new $\Omega(1/t)$ lower bound results, we explain how entropy regularization improves policy optimization, even with the true gradient, from the perspective of convergence rate. The separation of rates is further explained using the notion of non-uniform Ł{}ojasiewicz degree. These results provide a theoretical understanding of the impact of entropy and corroborate existing empirical studies.
APA
Mei, J., Xiao, C., Szepesvari, C. & Schuurmans, D.. (2020). On the Global Convergence Rates of Softmax Policy Gradient Methods. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:6820-6829 Available from https://proceedings.mlr.press/v119/mei20b.html.

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