Generalized Neyman Allocation for Locally Minimax Optimal Best-Arm Identification

TL;DR

This paper introduces the Generalized Neyman Allocation algorithm, achieving asymptotic minimax optimality for best-arm identification in small-gap bandit problems, matching theoretical lower bounds.

Contribution

It generalizes Neyman allocation to multi-armed bandits and provides tight bounds, addressing a key open problem in fixed-budget best-arm identification.

Findings

GNA algorithm is asymptotically locally minimax optimal.

Worst-case bounds match the theoretical lower bounds.

The approach refines existing algorithms and addresses open problems.

Abstract

This study investigates an asymptotically locally minimax optimal algorithm for fixed-budget best-arm identification (BAI). We propose the Generalized Neyman Allocation (GNA) algorithm and demonstrate that its worst-case upper bound on the probability of misidentifying the best arm aligns with the worst-case lower bound under the small-gap regime, where the gap between the expected outcomes of the best and suboptimal arms is small. Our lower and upper bounds are tight, matching exactly including constant terms within the small-gap regime. The GNA algorithm generalizes the Neyman allocation for two-armed bandits (Neyman, 1934; Kaufmann et al., 2016) and refines existing BAI algorithms, such as those proposed by Glynn & Juneja (2004). By proposing an asymptotically minimax optimal algorithm, we address the longstanding open issue in BAI (Kaufmann, 2020) and treatment choice (Kasy & Sautmann, 202) by restricting a class of distributions to the small-gap regimes.