Closing the Gap on the Sample Complexity of 1-Identification

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.