Thompson Sampling for Linear-Quadratic Control Problems

Marc Abeille, Alessandro Lazaric
Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, PMLR 54:1246-1254, 2017.

Abstract

We consider the exploration-exploitation tradeoff in linear quadratic (LQ) control problems, where the state dynamics is linear and the cost function is quadratic in states and controls. We analyze the regret of Thompson sampling (TS) (a.k.a. posterior-sampling for reinforcement learning) in the frequentist setting, i.e., when the parameters characterizing the LQ dynamics are fixed. Despite the empirical and theoretical success in a wide range of problems from multi-armed bandit to linear bandit, we show that when studying the frequentist regret TS in control problems, we need to trade-off the frequency of sampling optimistic parameters and the frequency of switches in the control policy. This results in an overall regret of $O(T^2/3)$, which is significantly worse than the regret $O(\sqrtT)$ achieved by the optimism-in-face-of-uncertainty algorithm in LQ control problems.

Cite this Paper

BibTeX
@InProceedings{pmlr-v54-abeille17b, title = {{Thompson Sampling for Linear-Quadratic Control Problems}}, author = {Abeille, Marc and Lazaric, Alessandro}, booktitle = {Proceedings of the 20th International Conference on Artificial Intelligence and Statistics}, pages = {1246--1254}, year = {2017}, editor = {Singh, Aarti and Zhu, Jerry}, volume = {54}, series = {Proceedings of Machine Learning Research}, month = {20--22 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v54/abeille17b/abeille17b.pdf}, url = {https://proceedings.mlr.press/v54/abeille17b.html}, abstract = {We consider the exploration-exploitation tradeoff in linear quadratic (LQ) control problems, where the state dynamics is linear and the cost function is quadratic in states and controls. We analyze the regret of Thompson sampling (TS) (a.k.a. posterior-sampling for reinforcement learning) in the frequentist setting, i.e., when the parameters characterizing the LQ dynamics are fixed. Despite the empirical and theoretical success in a wide range of problems from multi-armed bandit to linear bandit, we show that when studying the frequentist regret TS in control problems, we need to trade-off the frequency of sampling optimistic parameters and the frequency of switches in the control policy. This results in an overall regret of $O(T^2/3)$, which is significantly worse than the regret $O(\sqrtT)$ achieved by the optimism-in-face-of-uncertainty algorithm in LQ control problems.} }
Endnote
%0 Conference Paper %T Thompson Sampling for Linear-Quadratic Control Problems %A Marc Abeille %A Alessandro Lazaric %B Proceedings of the 20th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2017 %E Aarti Singh %E Jerry Zhu %F pmlr-v54-abeille17b %I PMLR %P 1246--1254 %U https://proceedings.mlr.press/v54/abeille17b.html %V 54 %X We consider the exploration-exploitation tradeoff in linear quadratic (LQ) control problems, where the state dynamics is linear and the cost function is quadratic in states and controls. We analyze the regret of Thompson sampling (TS) (a.k.a. posterior-sampling for reinforcement learning) in the frequentist setting, i.e., when the parameters characterizing the LQ dynamics are fixed. Despite the empirical and theoretical success in a wide range of problems from multi-armed bandit to linear bandit, we show that when studying the frequentist regret TS in control problems, we need to trade-off the frequency of sampling optimistic parameters and the frequency of switches in the control policy. This results in an overall regret of $O(T^2/3)$, which is significantly worse than the regret $O(\sqrtT)$ achieved by the optimism-in-face-of-uncertainty algorithm in LQ control problems.
APA
Abeille, M. & Lazaric, A.. (2017). Thompson Sampling for Linear-Quadratic Control Problems. Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 54:1246-1254 Available from https://proceedings.mlr.press/v54/abeille17b.html.

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