Best Arm Identification in Stochastic Bandits: Beyond $β-$optimality

This paper focuses on best arm identification (BAI) in stochastic multi-armed bandits (MABs) in the fixed-confidence, parametric setting. In such pure exploration problems, the accuracy of the sampling strategy critically hinges on the sequential allocation of the sampling resources among the arms. The existing approaches to BAI address the following question: what is an optimal sampling strategy when we spend a $\beta$ fraction of the samples on the best arm? These approaches treat $\beta$ as a tunable parameter and offer efficient algorithms that ensure optimality up to selecting $\beta$, hence $\beta-$optimality. However, the BAI decisions and performance can be highly sensitive to the choice of $\beta$. This paper provides a BAI algorithm that is agnostic to $\beta$, dispensing with the need for tuning $\beta$, and specifies an optimal allocation strategy, including the optimal value of $\beta$. Furthermore, the existing relevant literature focuses on the family of exponential distributions. This paper considers a more general setting of any arbitrary family of distributions parameterized by their mean values (under mild regularity conditions).