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## Improved Variance-Aware Confidence Sets for Linear Bandits and Linear Mixture MDP

### Zihan Zhang · Jiaqi Yang · Xiangyang Ji · Simon Du

Keywords: [ Bandits ] [ Theory ] [ Reinforcement Learning and Planning ]

Abstract:
This paper presents new \emph{variance-aware} confidence sets for linear bandits and linear mixture Markov Decision Processes (MDPs).With the new confidence sets, we obtain the follow regret bounds:For linear bandits, we obtain an $\widetilde{O}(\mathrm{poly}(d)\sqrt{1 + \sum_{k=1}^{K}\sigma_k^2})$ data-dependent regret bound, where $d$ is the feature dimension, $K$ is the number of rounds, and $\sigma_k^2$ is the \emph{unknown} variance of the reward at the $k$-th round. This is the first regret bound that only scales with the variance and the dimension but \emph{no explicit polynomial dependency on $K$}.When variances are small, this bound can be significantly smaller than the $\widetilde{\Theta}\left(d\sqrt{K}\right)$ worst-case regret bound.For linear mixture MDPs, we obtain an $\widetilde{O}(\mathrm{poly}(d, \log H)\sqrt{K})$ regret bound, where $d$ is the number of base models, $K$ is the number of episodes, and $H$ is the planning horizon. This is the first regret bound that only scales \emph{logarithmically} with $H$ in the reinforcement learning with linear function approximation setting, thus \emph{exponentially improving} existing results, and resolving an open problem in \citep{zhou2020nearly}.We develop three technical ideas that may be of independent interest:1) applications of the peeling technique to both the input norm and the variance magnitude, 2) a recursion-based estimator for the variance, and 3) a new convex potential lemma that generalizes the seminal elliptical potential lemma.

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