Online Reinforcement Learning in Periodic MDP

We study learning in periodic Markov decision process (MDP), a special type of nonstationary 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
$\mathcal{O}(\sqrt{T \text{log} T})$
with the horizon length
$T$
. Utilizing the information about the sparsity of transition matrix of augmented MDP, we propose another algorithm [periodic upper confidence reinforcement learning with Bernstein bounds (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.