Open Problem: Optimal Best Arm Identification with Fixed Budget

TL;DR

This paper discusses the open problem of determining the optimal asymptotic complexity for best arm identification in the fixed-budget setting of bandit problems, highlighting the gap in current understanding.

Contribution

It identifies and explores open problems and conjectures related to the asymptotic complexity in the fixed-budget setting for best arm identification.

Findings

Highlights the gap in understanding fixed-budget complexity

Discusses open problems and conjectures in the field

Emphasizes the need for further research on instance-dependent complexity

Abstract

Best arm identification or pure exploration problems have received much attention in the COLT community since Bubeck et al. (2009) and Audibert et al. (2010). For any bandit instance with a unique best arm, its asymptotic complexity in the so-called fixed-confidence setting has been completely characterized in Garivier and Kaufmann (2016) and Chernoff (1959), while little is known about the asymptotic complexity in its "dual" setting called fixed-budget setting. This note discusses the open problems and conjectures about the instance-dependent asymptotic complexity in the fixed-budget setting.