Optimal Best-Arm Identification under Fixed Confidence with Multiple Optima

Abstract: We study the problem of best-arm identification in stochastic multi-armed bandits under the fixed-confidence setting, with a particular focus on instances that admit multiple optimal arms. While the Track-and-Stop algorithm of Garivier and Kaufmann (2016) is widely conjectured to be instance-optimal, its performance in the presence of multiple optima has remained insufficiently understood. In this work, we revisit the Track-and-Stop strategy and propose a modified stopping rule that ensures instance-optimality even when the set of optimal arms is not a singleton. Our analysis introduces a new information-theoretic lower bound that explicitly accounts for multiple optimal arms, and we demonstrate that our stopping rule tightly matches this bound.