Neyman allocation is minimax optimal for best arm identification with two arms

TL;DR

This paper proves that Neyman allocation is the minimax optimal sampling strategy for best arm identification in two-treatment scenarios, allocating samples proportionally to outcome standard deviations without adaptation.

Contribution

It establishes the minimax optimality of Neyman allocation for two-arm best treatment identification under the local asymptotic regret criterion.

Findings

Neyman allocation is optimal for two-treatment best arm identification.

Optimal allocation is proportional to treatment outcome standard deviations.

When variances are equal, the optimal ratio is 1/2.

Abstract

This note describes the optimal policy rule, according to the local asymptotic minimax regret criterion, for best arm identification when there are only two treatments. It is shown that the optimal sampling rule is the Neyman allocation, which allocates a constant fraction of units to each treatment in a manner that is proportional to the standard deviation of the treatment outcomes. When the variances are equal, the optimal ratio is one-half. This policy is independent of the data, so there is no adaptation to previous outcomes. At the end of the experiment, the policy maker adopts the treatment with higher average outcomes.