## Achieving Sub-linear Regret in Infinite Horizon Average Reward Constrained MDP with Linear Function Approximation

22 Sept 2022, 12:36 (modified: 17 Nov 2022, 01:42)ICLR 2023 Conference Blind SubmissionReaders: Everyone
Keywords: Reinforcement Learning Theory, Infinite horizon Average Reward, Theory of Constrained Reinforcement Learning, Linear MDP, Model-free RL, Soft-max
TL;DR: We provide the first sub-linear regret and sub-linear constraint violation for constrained MDP for linear function approximation using model-free RL algorithm
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.
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