Nonstationary Reinforcement Learning with Linear Function Approximation

We consider reinforcement learning (RL) in episodic Markov decision processes (MDPs) with linear function approximation under drifting environment. Specifically, both the reward and state transition functions can evolve over time but their total variations do not exceed a $\textit{variation budget}$. We first develop $\texttt{LSVI-UCB-Restart}$ algorithm, an optimistic modification of least-squares value iteration with periodic restart, and bound its dynamic regret when variation budgets are known. Then we propose a parameter-free algorithm $\texttt{Ada-LSVI-UCB-Restart}$ that extends to unknown variation budgets. We also derive the first minimax dynamic regret lower bound for nonstationary linear MDPs and as a byproduct establish a minimax regret lower bound for linear MDPs unsolved by Jin et al. (2020). Finally, we provide numerical experiments to demonstrate the effectiveness of our proposed algorithms.