On the Complexity of Best Arm Identification in Multi-Armed Bandit Models

TL;DR

This paper investigates the complexity of identifying the top m arms in multi-armed bandit models, providing new lower bounds and algorithms for fixed-confidence and fixed-budget settings, with implications for understanding bandit problem difficulty.

Contribution

It introduces the first distribution-dependent lower bounds for m > 1 in fixed-confidence settings and refines bounds for two-armed bandits, along with improved stopping rules.

Findings

Fixed-budget complexity can be lower than fixed-confidence complexity.

Derived matching algorithms for Gaussian and Bernoulli bandits.

Provided new technical lemmas for deviation and change of measure.

Abstract

The stochastic multi-armed bandit model is a simple abstraction that has proven useful in many different contexts in statistics and machine learning. Whereas the achievable limit in terms of regret minimization is now well known, our aim is to contribute to a better understanding of the performance in terms of identifying the m best arms. We introduce generic notions of complexity for the two dominant frameworks considered in the literature: fixed-budget and fixed-confidence settings. In the fixed-confidence setting, we provide the first known distribution-dependent lower bound on the complexity that involves information-theoretic quantities and holds when m is larger than 1 under general assumptions. In the specific case of two armed-bandits, we derive refined lower bounds in both the fixed-confidence and fixed-budget settings, along with matching algorithms for Gaussian and Bernoulli bandit models. These results show in particular that the complexity of the fixed-budget setting may be smaller than the complexity of the fixed-confidence setting, contradicting the familiar behavior observed when testing fully specified alternatives. In addition, we also provide improved sequential stopping rules that have guaranteed error probabilities and shorter average running times. The proofs rely on two technical results that are of independent interest : a deviation lemma for self-normalized sums (Lemma 19) and a novel change of measure inequality for bandit models (Lemma 1).