Title: Achieving Sub-linear Regret in Infinite Horizon Average Reward Constrained MDP with Linear Function Approximation. Abstract: We study the infinite horizon average reward constrained Markov Decision Process (CMDP). In contrast to existing works on model-based, finite state space, we consider the model-free linear CMDP setup.  We first propose a computationally inefficient algorithm and show that $\tilde{\mathcal{O}}(\sqrt{d^3T})$ regret and constraint violation can be achieved, in which $T$ is the number of interactions, and $d$ is the dimension of the feature mapping. We also propose an efficient variant based on the primal-dual adaptation of the LSVI-UCB algorithm and show that $\tilde{\mathcal{O}}((dT)^{3/4})$ regret and constraint violation can be achieved. 
This improves the known regret bound of $\tilde{\mathcal{O}}(T^{5/6})$ for the finite state-space model-free constrained RL which was obtained under a stronger assumption compared to ours.  We also develop an efficient policy-based algorithm via novel adaptation of the MDP-EXP2 algorithm to our primal-dual set up with $\tilde{\mathcal{O}}(\sqrt{T})$ regret and even zero constraint violation bound under a stronger set of assumptions.