Optimal Best-Arm Identification under Fixed Confidence with Multiple Optima

TL;DR

This paper establishes a new lower bound and an asymptotically optimal algorithm for best-arm identification in multi-armed bandits with multiple known optimal arms, improving efficiency in fixed-confidence settings.

Contribution

It introduces a tighter information-theoretic lower bound and modifies the Track-and-Stop algorithm to achieve asymptotic optimality when the number of optimal arms is known.

Findings

Derived a new lower bound on sample complexity for multi-optimal arms

Proposed a tie-aware stopping rule for Track-and-Stop

Proved asymptotic optimality of the modified algorithm

Abstract

We study best-arm identification in stochastic multi-armed bandits under the fixed-confidence setting, focusing on instances with multiple optimal arms. Unlike prior work that addresses the unknown-number-of-optimal-arms case, we consider the setting where the number of optimal arms is known in advance. We derive a new information-theoretic lower bound on the expected sample complexity that leverages this structural knowledge and is strictly tighter than previous bounds. Building on the Track-and-Stop algorithm, we propose a modified, tie-aware stopping rule and prove that it achieves asymptotic instance-optimality, matching the new lower bound. Our results provide the first formal guarantee of optimality for Track-and-Stop in multi-optimal settings with known cardinality, offering both theoretical insights and practical guidance for efficiently identifying any optimal arm.