标题： 精确渐近性和方差感知UCB的精化遗憾 显示英文标题

标题： Precise Asymptotics and Refined Regret of Variance-Aware UCB

摘要： 本文研究了多臂老虎机（Multi-Armed Bandit, MAB）问题中上置信界-方差（Upper Confidence Bound-Variance, UCB-V）算法的行为，这是经典上置信界（UCB）算法的一个变种，其决策过程纳入了方差估计。更具体地说，我们给出了UCB-V的拉臂速率的渐近刻画，推广了Kalvit和Zeevi（2021）以及Khamaru和Zhang（2024）最近关于经典UCB的结果。 与经典UCB形成有趣的对比的是，我们的分析揭示了UCB-V的行为可能会表现出不稳定性，这意味着拉臂速率可能并不总是渐近确定的。 除了渐近刻画之外，我们还在高概率情形下提供了拉臂速率的非渐近界，为遗憾分析提供了见解。 作为这一高概率结果的应用，我们证明了UCB-V可以实现一个更为精细的遗憾界，即使对于更复杂、更先进的考虑方差意识的在线决策算法而言，这一遗憾界此前也是未知的。 显示英文摘要

摘要： In this paper, we study the behavior of the Upper Confidence Bound-Variance (UCB-V) algorithm for the Multi-Armed Bandit (MAB) problems, a variant of the canonical Upper Confidence Bound (UCB) algorithm that incorporates variance estimates into its decision-making process. More precisely, we provide an asymptotic characterization of the arm-pulling rates for UCB-V, extending recent results for the canonical UCB in Kalvit and Zeevi (2021) and Khamaru and Zhang (2024). In an interesting contrast to the canonical UCB, our analysis reveals that the behavior of UCB-V can exhibit instability, meaning that the arm-pulling rates may not always be asymptotically deterministic. Besides the asymptotic characterization, we also provide non-asymptotic bounds for the arm-pulling rates in the high probability regime, offering insights into the regret analysis. As an application of this high probability result, we establish that UCB-V can achieve a more refined regret bound, previously unknown even for more complicate and advanced variance-aware online decision-making algorithms.