Minimax and Bayes Optimal Best-Arm Identification

TL;DR

This paper develops an adaptive sampling strategy for best-arm identification that is proven to be both asymptotically minimax and Bayes optimal, optimizing simple regret in fixed-budget experiments.

Contribution

It introduces a novel two-stage adaptive procedure combining a Gaussian minimax game with an optimal sampling ratio, achieving theoretical optimality in simple regret.

Findings

The proposed strategy is asymptotically minimax optimal.

The strategy is also Bayes optimal for simple regret.

Upper bounds match the lower bounds exactly, including constants.

Abstract

This study investigates minimax and Bayes optimal strategies for fixed-budget best-arm identification. We consider an adaptive procedure consisting of a sampling phase followed by a recommendation phase, and we design an adaptive experiment within this framework to efficiently identify the best arm, defined as the one with the highest expected outcome. In our proposed strategy, the sampling phase consists of two stages. The first stage is a pilot phase, in which we allocate samples uniformly across arms to eliminate clearly suboptimal arms and to estimate outcome variances. Before entering the second stage, we solve a Gaussian minimax game, which yields a sampling ratio and a decision rule. In the second stage, samples are allocated according to this sampling ratio. After the sampling phase, the procedure enters the recommendation phase, where we select an arm using the decision rule. We prove that this single strategy is simultaneously asymptotically minimax and Bayes optimal for the simple regret, and we establish upper bounds that coincide exactly with our lower bounds, including the constant terms.