We provide the first useful and rigorous analysis of ensemble sampling for the stochastic linear bandit setting. In particular, we show that, under standard assumptions, for a $d$-dimensional stochastic linear bandit with an interaction horizon $T$, ensemble sampling with an ensemble of size of order $\smash{d \log T}$ incurs regret at most of the order $\smash{(d \log T)^{5/2} \sqrt{T}}$. Ours is the first result in any structured setting not to require the size of the ensemble to scale linearly with $T$ -- which defeats the purpose of ensemble sampling -- while obtaining near $\smash{\sqrt{T}}$ order regret. Ours is also the first result that allows infinite action sets.

翻译：我们首次对随机线性赌博机设置下的集成采样进行了实用且严谨的分析。具体而言，我们证明在标准假设下，对于维度为$d$、交互时域为$T$的随机线性赌博机问题，采用规模为$\smash{d \log T}$量级的集成进行集成采样，其遗憾上界为$\smash{(d \log T)^{5/2} \sqrt{T}}$量级。我们的研究是首个在结构化场景中不要求集成规模与$T$呈线性增长（否则将违背集成采样的初衷）而能获得接近$\smash{\sqrt{T}}$量级遗憾的结果。本研究也是首个支持无限动作集的理论成果。