  This paper studies privacy-preserving exploration in Markov Decision
Processes (MDPs) with linear representation. We first consider the setting of
linear-mixture MDPs (Ayoub et al., 2020) (a.k.a.\ model-based setting) and
provide an unified framework for analyzing joint and local differential private
(DP) exploration. Through this framework, we prove a
$\widetilde{O}(K^{3/4}/\sqrt{\epsilon})$ regret bound for
$(\epsilon,\delta)$-local DP exploration and a
$\widetilde{O}(\sqrt{K/\epsilon})$ regret bound for $(\epsilon,\delta)$-joint
DP. We further study privacy-preserving exploration in linear MDPs (Jin et al.,
2020) (a.k.a.\ model-free setting) where we provide a
$\widetilde{O}\left(K^{\frac{3}{5}}/\epsilon^{\frac{2}{5}}\right)$ regret bound
for $(\epsilon,\delta)$-joint DP, with a novel algorithm based on
low-switching. Finally, we provide insights into the issues of designing local
DP algorithms in this model-free setting.
