  For the misspecified linear Markov decision process (MLMDP) model of Jin et
al. [2020], we propose an algorithm with three desirable properties. (P1) Its
regret after $K$ episodes scales as $K \max \{ \varepsilon_{\text{mis}},
\varepsilon_{\text{tol}} \}$, where $\varepsilon_{\text{mis}}$ is the degree of
misspecification and $\varepsilon_{\text{tol}}$ is a user-specified error
tolerance. (P2) Its space and per-episode time complexities remain bounded as
$K \rightarrow \infty$. (P3) It does not require $\varepsilon_{\text{mis}}$ as
input. To our knowledge, this is the first algorithm satisfying all three
properties. For concrete choices of $\varepsilon_{\text{tol}}$, we also improve
existing regret bounds (up to log factors) while achieving either (P2) or (P3)
(existing algorithms satisfy neither). At a high level, our algorithm
generalizes (to MLMDPs) and refines the Sup-Lin-UCB algorithm, which Takemura
et al. [2021] recently showed satisfies (P3) for contextual bandits. We also
provide an intuitive interpretation of their result, which informs the design
of our algorithm.
