  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 $\mathcal{O}(\sqrt{Tlog T})$ with the
horizon length $T$. Utilizing the information about the sparsity of transition
matrix of augmented MDP, we propose another algorithm PUCRLB which enhances
upon PUCRL2, both in terms of regret ($O(\sqrt{N})$ 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.
