We study regret minimization for infinite-horizon average-reward Markov Decision Processes (MDPs) under cost constraints.We start by designing a policy optimization algorithm with carefully designed action-value estimator and bonus term,and show that for ergodic MDPs, our algorithm ensures $O(\sqrt{T})$ regret and constant constraint violation, where $T$ is the total number of time steps.This strictly improves over the algorithm of (Singh et al., 2020), whose regret and constraint violation are both $O(T^{2/3})$.Next, we consider the most general class of weakly communicating MDPs. Through a finite-horizon approximation, we develop another algorithm with $O(T^{2/3})$ regret and constraint violation, which can be further improved to $O(\sqrt{T})$ via a simple modification,albeit making the algorithm computationally inefficient.As far as we know, these are the first set of provable algorithms for weakly communicating MDPs with cost constraints.

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