  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
$\widetilde{O}\left(\sqrt{d^3H^4K}\right)$ 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.
