On the Interplay Between Misspecification and Sub-optimality Gap: From Linear Contextual Bandits to Linear MDPs

Weitong Zhang, Jiafan He, Zhiyuan Fan, Quanquan Gu

Abstract: We study linear contextual bandits in the misspecified setting, where the expected reward function can be approximated by a linear function class up to a bounded misspecification level $\zeta>0$. We propose an algorithm based on a novel data selection scheme, which only selects the contextual vectors with large uncertainty for online regression. We show that, when the misspecification level $\zeta$ is dominated by $\tilde O(\Delta / \sqrt{d})$ with $\Delta$ being the minimal sub-optimality gap and $d$ being the dimension of the contextual vectors, our algorithm enjoys the same gap-dependent regret bound $\tilde O ({d^2} /{\Delta})$ as in the well-specified setting up to logarithmic factors. Together with a lower bound adapted from Du et al. (2019); Lattimore et al.(2020), our result suggests an interplay between misspecification level and the sub-optimality gap: (1) the linear contextual bandit model is efficiently learnable when $\zeta \leq \tilde O({\Delta} / \sqrt{d})$; and (2) it is not efficiently learnable when $\zeta \geq \tilde \Omega({\Delta} / {\sqrt{d}})$. We also extend our algorithm to reinforcement learning with linear Markov decision processes (linear MDPs), and obtain a parallel result of gap-dependent regret. Experiments on both synthetic and real-world datasets corroborate our theoretical results.

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