Q-learning with Logarithmic Regret

Kunhe Yang, Lin Yang, Simon Du
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, PMLR 130:1576-1584, 2021.

Abstract

This paper presents the first non-asymptotic result showing a model-free algorithm can achieve logarithmic cumulative regret for episodic tabular reinforcement learning if there exists a strictly positive sub-optimality gap. We prove that the optimistic Q-learning studied in [Jin et al. 2018] enjoys a ${\mathcal{O}}\!\left(\frac{SA\cdot \mathrm{poly}\left(H\right)}{\Delta_{\min}}\log\left(SAT\right)\right)$ cumulative regret bound where $S$ is the number of states, $A$ is the number of actions, $H$ is the planning horizon, $T$ is the total number of steps, and $\Delta_{\min}$ is the minimum sub-optimality gap of the optimal Q-function. This bound matches the information theoretical lower bound in terms of $S,A,T$ up to a $\log\left(SA\right)$ factor. We further extend our analysis to the discounted setting and obtain a similar logarithmic cumulative regret bound.

Cite this Paper

BibTeX
@InProceedings{pmlr-v130-yang21b, title = { Q-learning with Logarithmic Regret }, author = {Yang, Kunhe and Yang, Lin and Du, Simon}, booktitle = {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics}, pages = {1576--1584}, 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/yang21b/yang21b.pdf}, url = {https://proceedings.mlr.press/v130/yang21b.html}, abstract = { This paper presents the first non-asymptotic result showing a model-free algorithm can achieve logarithmic cumulative regret for episodic tabular reinforcement learning if there exists a strictly positive sub-optimality gap. We prove that the optimistic Q-learning studied in [Jin et al. 2018] enjoys a ${\mathcal{O}}\!\left(\frac{SA\cdot \mathrm{poly}\left(H\right)}{\Delta_{\min}}\log\left(SAT\right)\right)$ cumulative regret bound where $S$ is the number of states, $A$ is the number of actions, $H$ is the planning horizon, $T$ is the total number of steps, and $\Delta_{\min}$ is the minimum sub-optimality gap of the optimal Q-function. This bound matches the information theoretical lower bound in terms of $S,A,T$ up to a $\log\left(SA\right)$ factor. We further extend our analysis to the discounted setting and obtain a similar logarithmic cumulative regret bound. } }
Endnote
%0 Conference Paper %T Q-learning with Logarithmic Regret %A Kunhe Yang %A Lin Yang %A Simon Du %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-yang21b %I PMLR %P 1576--1584 %U https://proceedings.mlr.press/v130/yang21b.html %V 130 %X This paper presents the first non-asymptotic result showing a model-free algorithm can achieve logarithmic cumulative regret for episodic tabular reinforcement learning if there exists a strictly positive sub-optimality gap. We prove that the optimistic Q-learning studied in [Jin et al. 2018] enjoys a ${\mathcal{O}}\!\left(\frac{SA\cdot \mathrm{poly}\left(H\right)}{\Delta_{\min}}\log\left(SAT\right)\right)$ cumulative regret bound where $S$ is the number of states, $A$ is the number of actions, $H$ is the planning horizon, $T$ is the total number of steps, and $\Delta_{\min}$ is the minimum sub-optimality gap of the optimal Q-function. This bound matches the information theoretical lower bound in terms of $S,A,T$ up to a $\log\left(SA\right)$ factor. We further extend our analysis to the discounted setting and obtain a similar logarithmic cumulative regret bound.
APA
Yang, K., Yang, L. & Du, S.. (2021). Q-learning with Logarithmic Regret . Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 130:1576-1584 Available from https://proceedings.mlr.press/v130/yang21b.html.

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