发表机构
National University of Singapore（新加坡国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究LinUCB在小间隔机制下的采样分配，证明其经验采样分布收敛到D-最优设计，并据此细化遗憾分析、建立估计器的中心极限定理。
AI 中文摘要
我们研究了LinUCB在小间隔机制下的采样分配问题，其中奖励间隔在决策时间范围$n$内至多为$n^{-1/2}$量级。这种缩放捕捉了支撑最坏情况遗憾下界困难实例的特性，对于这些实例，LinUCB已知在$n$的对数因子范围内接近最优。利用平均场视角，我们通过经验采样分布来表征这种分配，这是一个宏观对象，它平均了时间范围内自适应决策的影响，并确定了其当$n\to\infty$时的极限。我们证明在该机制下，由LinUCB诱导的经验采样分布收敛到D-最优设计集合。这一核心结果揭示，在小间隔机制下，LinUCB不仅实现了接近最优的极小极大遗憾，而且以一种对学习奖励参数渐近有效的方式分配样本，从而将遗憾驱动的在线学习与信息高效的实验设计联系起来。基于最优设计极限，我们获得了两个有用的推论。首先，我们通过刻画其极限中的前导阶常数，细化了LinUCB在小间隔机制下的渐近遗憾分析。其次，我们表明，尽管LinUCB采用自适应采样策略，正则化最小二乘估计器在小间隔机制下满足中心极限型定理，从而能够对奖励参数进行有效的统计推断。
英文摘要
We study the sampling allocation of LinUCB in the small-gap regime, where the reward gaps are of order at most $n^{-1/2}$ over the decision horizon $n$. This scaling captures the hard instances underlying worst-case regret lower bounds, for which LinUCB is known to be near optimal up to logarithmic factors in $n$. Using a mean-field perspective, we characterize this allocation through the empirical sampling distribution, a macroscopic object that averages the effect of adaptive decisions over the horizon, and identify its limit as $n\to\infty$. We establish that in this regime, the empirical sampling distribution induced by LinUCB converges to the set of D-optimal designs. This central result reveals that, in the small-gap regime, LinUCB not only achieves near optimal minimax regret but also allocates samples in a way that is asymptotically efficient for learning the reward parameter, thereby connecting regret-driven online learning with information-efficient experimental design. Building on the optimal design limit, we obtain two useful consequences. First, we refine the asymptotic regret analysis of LinUCB in the small-gap regime by characterizing its leading-order constant in the limit. Second, we show that, despite LinUCB's adaptive sampling strategy, the regularized least-squares estimator satisfies a central-limit-type theorem in the small-gap regime, thereby enabling valid statistical inference for the reward parameter.
