Nearly Minimax Optimal Reinforcement Learning for Linear Markov Decision Processes

Jiafan He, Heyang Zhao, Dongruo Zhou, Quanquan Gu
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:12790-12822, 2023.

Abstract

We study reinforcement learning (RL) with linear function approximation. For episodic time-inhomogeneous linear Markov decision processes (linear MDPs) whose transition probability can be parameterized as a linear function of a given feature mapping, we propose the first computationally efficient algorithm that achieves the nearly minimax optimal regret $\tilde O(d\sqrt{H^3K})$, where $d$ is the dimension of the feature mapping, $H$ is the planning horizon, and $K$ is the number of episodes. Our algorithm is based on a weighted linear regression scheme with a carefully designed weight, which depends on a new variance estimator that (1) directly estimates the variance of the optimal value function, (2) monotonically decreases with respect to the number of episodes to ensure a better estimation accuracy, and (3) uses a rare-switching policy to update the value function estimator to control the complexity of the estimated value function class. Our work provides a complete answer to optimal RL with linear MDPs, and the developed algorithm and theoretical tools may be of independent interest.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-he23d, title = {Nearly Minimax Optimal Reinforcement Learning for Linear {M}arkov Decision Processes}, author = {He, Jiafan and Zhao, Heyang and Zhou, Dongruo and Gu, Quanquan}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {12790--12822}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/he23d/he23d.pdf}, url = {https://proceedings.mlr.press/v202/he23d.html}, abstract = {We study reinforcement learning (RL) with linear function approximation. For episodic time-inhomogeneous linear Markov decision processes (linear MDPs) whose transition probability can be parameterized as a linear function of a given feature mapping, we propose the first computationally efficient algorithm that achieves the nearly minimax optimal regret $\tilde O(d\sqrt{H^3K})$, where $d$ is the dimension of the feature mapping, $H$ is the planning horizon, and $K$ is the number of episodes. Our algorithm is based on a weighted linear regression scheme with a carefully designed weight, which depends on a new variance estimator that (1) directly estimates the variance of the optimal value function, (2) monotonically decreases with respect to the number of episodes to ensure a better estimation accuracy, and (3) uses a rare-switching policy to update the value function estimator to control the complexity of the estimated value function class. Our work provides a complete answer to optimal RL with linear MDPs, and the developed algorithm and theoretical tools may be of independent interest.} }
Endnote
%0 Conference Paper %T Nearly Minimax Optimal Reinforcement Learning for Linear Markov Decision Processes %A Jiafan He %A Heyang Zhao %A Dongruo Zhou %A Quanquan Gu %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-he23d %I PMLR %P 12790--12822 %U https://proceedings.mlr.press/v202/he23d.html %V 202 %X We study reinforcement learning (RL) with linear function approximation. For episodic time-inhomogeneous linear Markov decision processes (linear MDPs) whose transition probability can be parameterized as a linear function of a given feature mapping, we propose the first computationally efficient algorithm that achieves the nearly minimax optimal regret $\tilde O(d\sqrt{H^3K})$, where $d$ is the dimension of the feature mapping, $H$ is the planning horizon, and $K$ is the number of episodes. Our algorithm is based on a weighted linear regression scheme with a carefully designed weight, which depends on a new variance estimator that (1) directly estimates the variance of the optimal value function, (2) monotonically decreases with respect to the number of episodes to ensure a better estimation accuracy, and (3) uses a rare-switching policy to update the value function estimator to control the complexity of the estimated value function class. Our work provides a complete answer to optimal RL with linear MDPs, and the developed algorithm and theoretical tools may be of independent interest.
APA
He, J., Zhao, H., Zhou, D. & Gu, Q.. (2023). Nearly Minimax Optimal Reinforcement Learning for Linear Markov Decision Processes. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:12790-12822 Available from https://proceedings.mlr.press/v202/he23d.html.

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