Where Does the Union Bound Go? Best-Arm Identification and Strong FWER Control

2026-08-20

AI Summary

本文探讨了最佳臂识别中联合界的应用问题，通过两种假设方向解释了为何需要Bonferroni校正，并明确了其与强家庭错误率控制之间的关系。

Abstract

In fixed-confidence best-arm identification, proofs often use a union bound across the competing arms. From a multiple-testing point of view this can look puzzling: if the best arm is unique, only one hypothesis of the form ``arm $i$ is best'' can be true. Why then should there be a Bonferroni-type factor of $K-1$? The answer is that there are two natural ways to orient the hypotheses. In one orientation, best-arm identification is literally a strong familywise-error-rate (FWER) problem with $K-1$ true nulls. In the opposite orientation, exactly one null is true, but a pairwise implementation can falsely reject that one null through any of $K-1$ comparisons. Thus the multiplicity has not disappeared; it just pops up in different places. This note makes the equivalence explicit in the terminology of both communities.

Problem

• best-arm identification
• union bound
• strong FWER control
• multiple testing

Innovation

• Best-Arm Identification
• Strong FWER Control
• Union Bound