Open Problem: Best Arm Identification: Almost Instance-Wise Optimality and the Gap Entropy Conjecture

TL;DR

This paper investigates the optimal sample complexity for the best arm identification problem in stochastic bandits, proposing a conjecture that introduces the gap entropy as a fundamental instance-wise lower bound, aiming to resolve a longstanding open problem.

Contribution

The authors introduce the gap entropy as a new measure and conjecture it as the instance-wise lower bound for BEST-1-ARM, advancing understanding of optimal sample complexity.

Findings

Proposes the gap entropy as a new complexity measure.

Conjectures the gap entropy as the fundamental instance-wise lower bound.

Highlights the gap between existing upper and lower bounds for the problem.

Abstract

The best arm identification problem (BEST-1-ARM) is the most basic pure exploration problem in stochastic multi-armed bandits. The problem has a long history and attracted significant attention for the last decade. However, we do not yet have a complete understanding of the optimal sample complexity of the problem: The state-of-the-art algorithms achieve a sample complexity of $O(\sum_{i=2}^{n} \Delta_{i}^{-2}(\ln\delta^{-1} + \ln\ln\Delta_i^{-1}))$ ($\Delta_{i}$ is the difference between the largest mean and the $i^{th}$ mean), while the best known lower bound is $\Omega(\sum_{i=2}^{n} \Delta_{i}^{-2}\ln\delta^{-1})$ for general instances and $\Omega(\Delta^{-2} \ln\ln \Delta^{-1})$ for the two-arm instances. We propose to study the instance-wise optimality for the BEST-1-ARM problem. Previous work has proved that it is impossible to have an instance optimal algorithm for the 2-arm problem. However, we conjecture that modulo the additive term $\Omega(\Delta_2^{-2} \ln\ln \Delta_2^{-1})$ (which is an upper bound and worst case lower bound for the 2-arm problem), there is an instance optimal algorithm for BEST-1-ARM. Moreover, we introduce a new quantity, called the gap entropy for a best-arm problem instance, and conjecture that it is the instance-wise lower bound. Hence, resolving this conjecture would provide a final answer to the old and basic problem.