Improved Algorithms for Misspecified Linear Markov Decision Processes
Daniel Vial, Advait Parulekar, Sanjay Shakkottai, R Srikant
Abstract
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{\ensuremath{\varepsilon}mis,\ensuremath{\varepsilon}tol}, where \ensuremath{\varepsilon}mis is the degree of misspecification and \ensuremath{\varepsilon}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 \ensuremath{\varepsilon}mis as input. To our knowledge, this is the first algorithm satisfying all three properties. For concrete choices of \ensuremath{\varepsilon}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) in the contextual bandit setting. We also provide an intuitive interpretation of their result, which informs the design of our algorithm.
BibTeX
@InProceedings{pmlr-v151-vial22a,
title = { Improved Algorithms for Misspecified Linear Markov Decision Processes },
author = {Vial, Daniel and Parulekar, Advait and Shakkottai, Sanjay and Srikant, R},
booktitle = {Proceedings of The 25th International Conference on Artificial Intelligence and Statistics},
pages = {4723--4746},
year = {2022},
editor = {Camps-Valls, Gustau and Ruiz, Francisco J. R. and Valera, Isabel},
volume = {151},
series = {Proceedings of Machine Learning Research},
month = {28--30 Mar},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v151/vial22a/vial22a.pdf},
url = {https://proceedings.mlr.press/v151/vial22a.html},
abstract = { 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{\ensuremath{\varepsilon}mis,\ensuremath{\varepsilon}tol}, where \ensuremath{\varepsilon}mis is the degree of misspecification and \ensuremath{\varepsilon}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 \ensuremath{\varepsilon}mis as input. To our knowledge, this is the first algorithm satisfying all three properties. For concrete choices of \ensuremath{\varepsilon}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) in the contextual bandit setting. We also provide an intuitive interpretation of their result, which informs the design of our algorithm. }
}