发表机构
School of Data Science, The Chinese University of Hong Kong, Shenzhen; School of Management and Economics, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）数据科学学院; 香港中文大学（深圳）经管学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对UCB1等稳定索引算法，推导后-bandit推理中样本均值偏差和Z统计量的精确表达式，揭示了有效探索率这一偏差起源，发现索引函数选择存在遗憾-偏差权衡，采用新颖经验流体近似开展分析。
AI 中文摘要
多臂赌博机（Bandit）算法为下游推理生成数据，但自适应采样会导致后-bandit样本均值出现偏差。我们针对稳定索引算法（包括UCB1及其推广形式）分析该偏差，推导样本均值偏差和期望Z统计量的精确主阶表达式。该表征通过一个关键的、依赖索引函数的量揭示偏差的算法起源，我们将其命名为有效探索率。例如，在UCB1下，有效探索率为√log T量级，任意非唯一最优臂的标准化偏差以极慢的1/√log T速率衰减。我们还表明索引函数的选择会同时影响遗憾和偏差，揭示了遗憾-偏差权衡：更具探索性的算法可降低偏差但会增加遗憾。我们对偏差的精确表征采用了一种新颖的、基于算法采样动力学的经验流体近似，该近似可能具有独立研究价值。
英文摘要
Bandit algorithms generate data for downstream inference, but adaptive sampling biases post-bandit sample means. We analyze this bias for stable index algorithms, including UCB1 and its generalizations, and derive sharp leading-order expressions for the sample-mean bias and expected $Z$-statistic, in bandit experiments of fixed horizon $T$. Our characterization reveals the algorithmic origin of bias through a key index-function-dependent quantity, which we term effective exploration rate. For example, under UCB1, the effective exploration rate is of order $\sqrt{\log T}$, and the standardized bias of any arm (that is not uniquely optimal) decays at the extremely slow rate $1/\sqrt{\log T}$. We also show how the choice of the index function affects both regret and bias, which reveals a regret-bias trade-off: more exploratory algorithm reduces bias but increases regret. We further show how bias most severely distorts confidence intervals and hypothesis tests when the tested arm is one of the tied-optimal arms. Our sharp characterization for bias uses a novel empirical fluid approximation of the algorithm's sampling dynamics, which may be of independent interest.
