Regret guarantees for model-based reinforcement learning with long-term average constraints
Mridul Agarwal, Qinbo Bai, Vaneet Aggarwal
Abstract
We consider the problem of constrained Markov Decision Process (CMDP) where an agent interacts with an ergodic Markov Decision Process. At every interaction, the agent obtains a reward and incurs $K$ costs. The agent aims to maximize the long-term average reward while simultaneously keeping the $K$ long-term average costs lower than a certain threshold. In this paper, we propose \NAM, a posterior sampling based algorithm using which the agent can learn optimal policies to interact with the CMDP. We show that with the assumption of slackness, characterized by $\kappa$, the optimization problem is feasible for the sampled MDPs. Further, for MDP with $S$ states, $A$ actions, and mixing time $T_M$, we prove that following \NAM{} algorithm, the agent can bound the regret of not accumulating rewards from an optimal policy by $\Tilde{O}(T_MS\sqrt{AT})$. Further, we show that the violations for any of the $K$ constraints is also bounded by $\Tilde{O}(T_MS\sqrt{AT})$. To the best of our knowledge, this is the first work that obtains a $\Tilde{O}(\sqrt{T})$ regret bounds for ergodic MDPs with long-term average constraints using a posterior sampling method.
BibTeX
@InProceedings{pmlr-v180-agarwal22b,
title = {Regret guarantees for model-based reinforcement learning with long-term average constraints},
author = {Agarwal, Mridul and Bai, Qinbo and Aggarwal, Vaneet},
booktitle = {Proceedings of the Thirty-Eighth Conference on Uncertainty in Artificial Intelligence},
pages = {22--31},
year = {2022},
editor = {Cussens, James and Zhang, Kun},
volume = {180},
series = {Proceedings of Machine Learning Research},
month = {01--05 Aug},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v180/agarwal22b/agarwal22b.pdf},
url = {https://proceedings.mlr.press/v180/agarwal22b.html},
abstract = {We consider the problem of constrained Markov Decision Process (CMDP) where an agent interacts with an ergodic Markov Decision Process. At every interaction, the agent obtains a reward and incurs $K$ costs. The agent aims to maximize the long-term average reward while simultaneously keeping the $K$ long-term average costs lower than a certain threshold. In this paper, we propose \NAM, a posterior sampling based algorithm using which the agent can learn optimal policies to interact with the CMDP. We show that with the assumption of slackness, characterized by $\kappa$, the optimization problem is feasible for the sampled MDPs. Further, for MDP with $S$ states, $A$ actions, and mixing time $T_M$, we prove that following \NAM{} algorithm, the agent can bound the regret of not accumulating rewards from an optimal policy by $\Tilde{O}(T_MS\sqrt{AT})$. Further, we show that the violations for any of the $K$ constraints is also bounded by $\Tilde{O}(T_MS\sqrt{AT})$. To the best of our knowledge, this is the first work that obtains a $\Tilde{O}(\sqrt{T})$ regret bounds for ergodic MDPs with long-term average constraints using a posterior sampling method.}
}