Improved Regret Analysis for Variance-Adaptive Linear Bandits and Horizon-Free Linear Mixture MDPs

Abstract: In online learning problems, exploiting low variance plays an important role in obtaining tight performance guarantees yet is challenging because variances are often not known a priori. Recently, considerable progress has been made by Zhang et al. (2021) where they obtain a variance-adaptive regret bound for linear bandits without knowledge of the variances and a horizon-free regret bound for linear mixture Markov decision processes (MDPs). In this paper, we present novel analyses that improve their regret bounds significantly. For linear bandits, we achieve $$\tilde O(\min{d\sqrt{K}, d^{{1.5}\sqrt{\sum_{k=1}^K} \sigma_k^2}} + d<sup>2)$$ where $d$ is the dimension of the features, $K$ is the time horizon, and $\sigma_k<sup>2$ is the noise variance at time step $k$, and $\tilde O$ ignores polylogarithmic dependence, which is a factor of $d<sup>3$ improvement. For linear mixture MDPs with the assumption of maximum cumulative reward in an episode being in $[0,1]$, we achieve a horizon-free regret bound of $\tilde O(d \sqrt{K} + d<sup>2)$ where $d$ is the number of base models and $K$ is the number of episodes. This is a factor of $d<sup>{3.5}$ improvement in the leading term and $d<sup>7$ in the lower order term. Our analysis critically relies on a novel peeling-based regret analysis that leverages the elliptical potential `count' lemma.