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

• URL: http://arxiv.org/abs/2111.03289v1
• Date: Fri, 5 Nov 2021 06:47:27 GMT
• Title: Improved Regret Analysis for Variance-Adaptive Linear Bandits and Horizon-Free Linear Mixture MDPs
• Authors: Yeoneung Kim, Insoon Yang, Kwang-Sung Jun
• Abstract summary: In online learning problems, exploiting low variance plays an important role in obtaining tight performance guarantees. We present novel analyses that improve their regret bounds significantly. Our analysis critically relies on a novel elliptical potential count' lemma.
• Score: 12.450760567361531
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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, a 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 $\tilde O(d^{1.5}\sqrt{\sum_{k}^K \sigma_k^2} + d^2)$ where $d$ is the dimension of the features, $K$ is the time horizon, and $\sigma_k^2$ is the noise variance at time step $k$, and $\tilde O$ ignores polylogarithmic dependence, which is a factor of $d^3$ improvement. For linear mixture MDPs, we achieve a horizon-free regret bound of $\tilde O(d^{1.5}\sqrt{K} + d^3)$ where $d$ is the number of base models and $K$ is the number of episodes. This is a factor of $d^3$ improvement in the leading term and $d^6$ in the lower order term. Our analysis critically relies on a novel elliptical potential `count' lemma. This lemma allows a peeling-based regret analysis, which can be of independent interest.

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