We develop several new algorithms for learning Markov Decision Processes in an infinite-horizon average-reward setting with linear function approximation. Using the optimism principle and assuming that the MDP has a linear structure, we first propose a computationally inefficient algorithm with optimal $widetilde{O}(sqrt{T})$ regret and another computationally efficient variant with $widetilde{O}(T^{3/4})$ regret, where $T$ is the number of interactions. Next, taking inspiration from adversarial linear bandits, we develop yet another efficient algorithm with $widetilde{O}(sqrt{T})$ regret under a different set of assumptions, improving the best existing result by Hao et al. (2020) with $widetilde{O}(T^{2/3})$ regret. Moreover, we draw a connection between this algorithm and the Natural Policy Gradient algorithm proposed by Kakade (2002), and show that our analysis improves the sample complexity bound recently given by Agarwal et al. (2020).
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