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Markov Decision Processes with Long-Term Average Constraints

We consider the problem of constrained Markov Decision Process (CMDP) where an agent interacts with a unichain Markov Decision Process. At every interaction, the agent obtains a reward. Further, there are $K$ cost functions. 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 CMDP-PSRL, a posterior sampling based algorithm using which the agent can learn optimal policies to interact with the CMDP. Further, for MDP with $S$ states, $A$ actions, and diameter $D$, we prove that following CMDP-PSRL algorithm, the agent can bound the regret of not accumulating rewards from optimal policy by $\Tilde{O}(poly(DSA)\sqrt{T})$. Further, we show that the violations for any of the $K$ constraints is also bounded by $\Tilde{O}(poly(DSA)\sqrt{T})$. To the best of our knowledge, this is the first work which obtains a $\Tilde{O}(\sqrt{T})$ regret bounds for ergodic MDPs with long-term average constraints.