Asymptotically Optimal Fixed-Budget Best Arm Identification with Variance-Dependent Bounds

TL;DR

This paper develops an asymptotically optimal strategy for fixed-budget best arm identification that accounts for variance-dependent bounds, ensuring minimax optimality in expected simple regret.

Contribution

It introduces the TS-HIR strategy, which achieves asymptotic minimax optimality by leveraging variance-dependent lower bounds and the HIR estimator.

Findings

The TS-HIR strategy attains asymptotic minimax optimality.

Derived variance-dependent lower bounds for worst-case simple regret.

Validated effectiveness through simulation experiments.

Abstract

We investigate the problem of fixed-budget best arm identification (BAI) for minimizing expected simple regret. In an adaptive experiment, a decision maker draws one of multiple treatment arms based on past observations and observes the outcome of the drawn arm. After the experiment, the decision maker recommends the treatment arm with the highest expected outcome. We evaluate the decision based on the expected simple regret, which is the difference between the expected outcomes of the best arm and the recommended arm. Due to inherent uncertainty, we evaluate the regret using the minimax criterion. First, we derive asymptotic lower bounds for the worst-case expected simple regret, which are characterized by the variances of potential outcomes (leading factor). Based on the lower bounds, we propose the Two-Stage (TS)-Hirano-Imbens-Ridder (HIR) strategy, which utilizes the HIR estimator (Hirano et al., 2003) in recommending the best arm. Our theoretical analysis shows that the TS-HIR strategy is asymptotically minimax optimal, meaning that the leading factor of its worst-case expected simple regret matches our derived worst-case lower bound. Additionally, we consider extensions of our method, such as the asymptotic optimality for the probability of misidentification. Finally, we validate the proposed method's effectiveness through simulations.