Closing the Gap on the Sample Complexity of 1-Identification

TL;DR

This paper advances understanding of the sample complexity in 1-identification for multi-armed bandits by deriving tight bounds and proposing an optimal algorithm for instances with qualified arms.

Contribution

It introduces a new lower bound on expected sample complexity and proposes an algorithm that nearly matches this bound across all instances.

Findings

Derived a novel lower bound on expected arm pulls for qualified instances.

Proposed an algorithm with upper bounds matching the lower bound up to polylogarithmic factors.

Complemented existing analysis for multiple qualified arms, addressing an open problem.

Abstract

The 1-identification problem is a fundamental pure-exploration problem in multi-armed bandits. An agent aims to determine whether there exists an arm whose mean reward exceeds a known threshold $\mu_0$, or to output \textsf{None} otherwise. The agent must guarantee correctness with probability at least $1-\delta$, while minimizing the expected number of arm pulls $\mathbb{E}[\tau]$. We study the 1-identification problem and make two main contributions. First, for instances with at least one qualified arm, we derive a new lower bound on $\mathbb{E}[\tau]$ via a novel optimization formulation. Second, we propose a new algorithm and establish upper bounds that match the lower bounds up to polynomial logarithmic factors uniformly over all instances. Our result complements the analysis of $\mathbb{E}\tau$ when there are multiple qualified arms, which is an open problem in the literature.