A Theoretical Analysis of Deep Q-Learning

Jianqing Fan, Zhaoran Wang, Yuchen Xie, Zhuoran Yang
Proceedings of the 2nd Conference on Learning for Dynamics and Control, PMLR 120:486-489, 2020.

Abstract

Despite the great empirical success of deep reinforcement learning, its theoretical foundation is less well understood. In this work, we make the first attempt to theoretically understand the deep Q-network (DQN) algorithm (Mnih et al., 2015) from both algorithmic and statistical perspectives. In specific, we focus on the fitted Q iteration (FQI) algorithm with deep neural networks, which is a slight simplification of DQN that captures the tricks of experience replay and target network used in DQN. Under mild assumptions, we establish the algorithmic and statistical rates of convergence for the action-value functions of the iterative policy sequence obtained by FQI. In particular, the statistical error characterizes the bias and variance that arise from approximating the action-value function using deep neural network, while the algorithmic error converges to zero at a geometric rate. As a byproduct, our analysis provides justifications for the techniques of experience replay and target network, which are crucial to the empirical success of DQN. Furthermore, as a simple extension of DQN, we propose the Minimax-DQN algorithm for zero-sum Markov game with two players, which is deferred to the appendix due to space limitations.

Cite this Paper

BibTeX
@InProceedings{pmlr-v120-yang20a, title = {A Theoretical Analysis of Deep Q-Learning}, author = {Fan, Jianqing and Wang, Zhaoran and Xie, Yuchen and Yang, Zhuoran}, booktitle = {Proceedings of the 2nd Conference on Learning for Dynamics and Control}, pages = {486--489}, year = {2020}, editor = {Bayen, Alexandre M. and Jadbabaie, Ali and Pappas, George and Parrilo, Pablo A. and Recht, Benjamin and Tomlin, Claire and Zeilinger, Melanie}, volume = {120}, series = {Proceedings of Machine Learning Research}, month = {10--11 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v120/yang20a/yang20a.pdf}, url = {https://proceedings.mlr.press/v120/yang20a.html}, abstract = {Despite the great empirical success of deep reinforcement learning, its theoretical foundation is less well understood. In this work, we make the first attempt to theoretically understand the deep Q-network (DQN) algorithm (Mnih et al., 2015) from both algorithmic and statistical perspectives. In specific, we focus on the fitted Q iteration (FQI) algorithm with deep neural networks, which is a slight simplification of DQN that captures the tricks of experience replay and target network used in DQN. Under mild assumptions, we establish the algorithmic and statistical rates of convergence for the action-value functions of the iterative policy sequence obtained by FQI. In particular, the statistical error characterizes the bias and variance that arise from approximating the action-value function using deep neural network, while the algorithmic error converges to zero at a geometric rate. As a byproduct, our analysis provides justifications for the techniques of experience replay and target network, which are crucial to the empirical success of DQN. Furthermore, as a simple extension of DQN, we propose the Minimax-DQN algorithm for zero-sum Markov game with two players, which is deferred to the appendix due to space limitations.} }
Endnote
%0 Conference Paper %T A Theoretical Analysis of Deep Q-Learning %A Jianqing Fan %A Zhaoran Wang %A Yuchen Xie %A Zhuoran Yang %B Proceedings of the 2nd Conference on Learning for Dynamics and Control %C Proceedings of Machine Learning Research %D 2020 %E Alexandre M. Bayen %E Ali Jadbabaie %E George Pappas %E Pablo A. Parrilo %E Benjamin Recht %E Claire Tomlin %E Melanie Zeilinger %F pmlr-v120-yang20a %I PMLR %P 486--489 %U https://proceedings.mlr.press/v120/yang20a.html %V 120 %X Despite the great empirical success of deep reinforcement learning, its theoretical foundation is less well understood. In this work, we make the first attempt to theoretically understand the deep Q-network (DQN) algorithm (Mnih et al., 2015) from both algorithmic and statistical perspectives. In specific, we focus on the fitted Q iteration (FQI) algorithm with deep neural networks, which is a slight simplification of DQN that captures the tricks of experience replay and target network used in DQN. Under mild assumptions, we establish the algorithmic and statistical rates of convergence for the action-value functions of the iterative policy sequence obtained by FQI. In particular, the statistical error characterizes the bias and variance that arise from approximating the action-value function using deep neural network, while the algorithmic error converges to zero at a geometric rate. As a byproduct, our analysis provides justifications for the techniques of experience replay and target network, which are crucial to the empirical success of DQN. Furthermore, as a simple extension of DQN, we propose the Minimax-DQN algorithm for zero-sum Markov game with two players, which is deferred to the appendix due to space limitations.
APA
Fan, J., Wang, Z., Xie, Y. & Yang, Z.. (2020). A Theoretical Analysis of Deep Q-Learning. Proceedings of the 2nd Conference on Learning for Dynamics and Control, in Proceedings of Machine Learning Research 120:486-489 Available from https://proceedings.mlr.press/v120/yang20a.html.

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