Variance-Aware Confidence Set: Variance-Dependent Bound for Linear Bandits and Horizon-Free Bound for Linear Mixture MDP

• URL: http://arxiv.org/abs/2101.12745v2
• Date: Fri, 19 Feb 2021 15:46:46 GMT
• Title: Variance-Aware Confidence Set: Variance-Dependent Bound for Linear Bandits and Horizon-Free Bound for Linear Mixture MDP
• Authors: Zihan Zhang, Jiaqi Yang, Xiangyang Ji, Simon S. Du
• Abstract summary: We show how to construct variance-aware confidence sets for linear bandits and linear mixture Decision Process (MDP) For linear bandits, we obtain an $widetildeO(mathrmpoly(d)sqrt1 + sum_i=1Ksigma_i2) regret bound, where $d is the feature dimension. For linear mixture MDP, we obtain an $widetildeO(mathrmpoly(d)sqrtK)$ regret bound, where
• Score: 76.94328400919836
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We show how to construct variance-aware confidence sets for linear bandits and linear mixture Markov Decision Process (MDP). Our method yields the following new regret bounds: * For linear bandits, we obtain an $\widetilde{O}(\mathrm{poly}(d)\sqrt{1 + \sum_{i=1}^{K}\sigma_i^2})$ regret bound, where $d$ is the feature dimension, $K$ is the number of rounds, and $\sigma_i^2$ is the (unknown) variance of the reward at the $i$-th round. This is the first regret bound that only scales with the variance and the dimension, with no explicit polynomial dependency on $K$. * For linear mixture MDP, 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 logarithmically with $H$ in the reinforcement learning with linear function approximation setting, thus exponentially improving existing results. Our methods utilize three novel ideas that may be of independent interest: 1) applications of the peeling techniques to the norm of input and the magnitude of variance, 2) a recursion-based approach to estimate the variance, and 3) a convex potential lemma that somewhat generalizes the seminal elliptical potential lemma.

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