Online Reinforcement Learning in Periodic MDP

We study learning in periodic Markov Decision Process (MDP), a special type of non-stationary MDP where both the state transition probabilities and reward functions vary periodically, under the average reward maximization setting. We formulate the problem as a stationary MDP by augmenting the state space with the period index, and propose a periodic upper confidence bound reinforcement learning-2 (PUCRL2) algorithm. We show that the regret of PUCRL2 varies linearly with the period N and as O (TlogT) with the horizon length T. Utilizing the information about the sparsity of transition matrix of augmented MDP, we propose another algorithm PUCRLB (Periodic Upper Confidence Reinforcement Learning with Bernstein bounds) which enhances upon PUCRL2, both in terms of regret (O (N) dependency on period) and empirical performance. Finally, we propose two other algorithms U-PUCRL2 and U-PUCRLB for extended uncertainty in the environment in which the period is unknown but a set of candidate periods are known. Numerical results demonstrate the efficacy of all the algorithms.