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

arXiv: https://arxiv.org/abs/2608.19903 · HTML: https://mlsanity.com/paper/2608.19903/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 14 references that are in the map.
- First posted on arXiv 20 Aug 2026.

## Builds on (most cited first, 3 of 3)

- [Optimal Best Arm Identification with Fixed Confidence](https://mlsanity.com/paper/1602.04589/) (2016)
- [Mixture Martingales Revisited with Applications to Sequential Tests and Confidence Intervals](https://mlsanity.com/paper/1811.11419/) (2018)
- [An Empirical Process Approach to the Union Bound: Practical Algorithms for Combinatorial and Linear Bandits](https://mlsanity.com/paper/2006.11685/) (2020)

## Abstract (arXiv)

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.