Regret Guarantees for Model-Based Reinforcement Learning with Long-Term Average Constraints

Mridul Agarwal, Qinbo Bai, Vaneet Aggarwal

Keywords: constrained Markov Decision Processes, constrained reinforcement learning

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.

TL;DR: Posterior samling algorithm to bound constraint violations. Analysis using Bellman error of the optimal transition probabilities for the optimal policy of the sampled MDP.