Optimal Batched Best Arm Identification

TL;DR

This paper introduces new batched algorithms for best arm identification that optimize sample and batch complexity, achieving asymptotic optimality with minimal batches and providing near-optimal results in finite settings.

Contribution

The paper presents the Tri-BBAI and Opt-BBAI algorithms, the first to achieve asymptotic optimality in batch setting and near-optimality in finite cases, with improved complexity guarantees.

Findings

Tri-BBAI achieves asymptotic optimal sample complexity in 3 batches.

Opt-BBAI attains near-optimal complexity in non-asymptotic settings.

The new procedures do not rely on event-conditioned complexity bounds.

Abstract

We study the batched best arm identification (BBAI) problem, where the learner's goal is to identify the best arm while switching the policy as less as possible. In particular, we aim to find the best arm with probability $1-\delta$ for some small constant $\delta>0$ while minimizing both the sample complexity (total number of arm pulls) and the batch complexity (total number of batches). We propose the three-batch best arm identification (Tri-BBAI) algorithm, which is the first batched algorithm that achieves the optimal sample complexity in the asymptotic setting (i.e., $\delta\rightarrow 0$) and runs in $3$ batches in expectation. Based on Tri-BBAI, we further propose the almost optimal batched best arm identification (Opt-BBAI) algorithm, which is the first algorithm that achieves the near-optimal sample and batch complexity in the non-asymptotic setting (i.e., $\delta$ is finite), while enjoying the same batch and sample complexity as Tri-BBAI when $\delta$ tends to zero. Moreover, in the non-asymptotic setting, the complexity of previous batch algorithms is usually conditioned on the event that the best arm is returned (with a probability of at least $1-\delta$), which is potentially unbounded in cases where a sub-optimal arm is returned. In contrast, the complexity of Opt-BBAI does not rely on such an event. This is achieved through a novel procedure that we design for checking whether the best arm is eliminated, which is of independent interest.