Improved Algorithms for Misspecified Linear Markov Decision Processes

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 Kmax{εmis,εtol}, where εmis is the degree of misspecification and εtol is a user-specified error tolerance. (P2) Its space and per-episode time complexities remain bounded as K→∞. (P3) It does not require εmis as input. To our knowledge, this is the first algorithm satisfying all three properties. For concrete choices of εtol, 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) in the contextual bandit setting. We also provide an intuitive interpretation of their result, which informs the design of our algorithm.