  We study episodic reinforcement learning in non-stationary linear (a.k.a.
low-rank) Markov Decision Processes (MDPs), i.e, both the reward and transition
kernel are linear with respect to a given feature map and are allowed to evolve
either slowly or abruptly over time. For this problem setting, we propose
OPT-WLSVI an optimistic model-free algorithm based on weighted least squares
value iteration which uses exponential weights to smoothly forget data that are
far in the past. We show that our algorithm, when competing against the best
policy at each time, achieves a regret that is upper bounded by
$\widetilde{\mathcal{O}}(d^{5/4}H^2 \Delta^{1/4} K^{3/4})$ where $d$ is the
dimension of the feature space, $H$ is the planning horizon, $K$ is the number
of episodes and $\Delta$ is a suitable measure of non-stationarity of the MDP.
Moreover, we point out technical gaps in the study of forgetting strategies in
non-stationary linear bandits setting made by previous works and we propose a
fix to their regret analysis.
