Nonasymptotic sequential tests for overlapping hypotheses applied to near-optimal arm identification in bandit models

• Published in: Sequential analysis Vol. 40; no. 1; pp. 61 - 96
• Main Authors: Garivier, Aurélien, Kaufmann, Emilie
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 15.01.2021; Taylor & Francis Ltd
• Subjects: Best arm identification; Gaussian distribution; generalized likelihood ratio test; Hypotheses; Information theory; Likelihood ratio; Lower bounds; multi-armed bandits; Normal distribution; sequential testing; Statistical analysis
• ISSN: 0747-4946; 1532-4176
• DOI: 10.1080/07474946.2021.1847965

In this article, we study sequential testing problems with overlapping hypotheses. We first focus on the simple problem of assessing if the mean μ of a Gaussian distribution is smaller or larger than a fixed if both answers are considered to be correct. Then, we consider probably approximately correct best arm identification in a bandit model: given K probability distributions on with means we derive the asymptotic complexity of identifying, with risk at most δ, an index such that We provide nonasymptotic bounds on the error of a parallel general likelihood ratio test, which can also be used for more general testing problems. We further propose a lower bound on the number of observations needed to identify a correct hypothesis. Those lower bounds rely on information-theoretic arguments, and specifically on two versions of a change of measure lemma (a high-level form and a low-level form) whose relative merits are discussed.