A Provably Efficient Algorithm for Linear Markov Decision Process with Low Switching Cost

Abstract: Many real-world applications, such as those in medical domains, recommendation systems, etc, can be formulated as large state space reinforcement learning problems with only a small budget of the number of policy changes, i.e., low switching cost. This paper focuses on the linear Markov Decision Process (MDP) recently studied in [Yang et al 2019, Jin et al 2020] where the linear function approximation is used for generalization on the large state space. We present the first algorithm for linear MDP with a low switching cost. Our algorithm achieves an $\widetilde{O}\left(\sqrt{d<sup>3H<sup>4K}\right)$ regret bound with a near-optimal $O\left(d H\log K\right)$ global switching cost where $d$ is the feature dimension, $H$ is the planning horizon and $K$ is the number of episodes the agent plays. Our regret bound matches the best existing polynomial algorithm by [Jin et al 2020] and our switching cost is exponentially smaller than theirs. When specialized to tabular MDP, our switching cost bound improves those in [Bai et al 2019, Zhang et al 20020]. We complement our positive result with an $\Omega\left(dH/\log d\right)$ global switching cost lower bound for any no-regret algorithm.