# Non-Asymptotic Sequential Tests for Overlapping Hypotheses and application to near optimal arm identification in bandit models

2 MC2 - Modèles de calcul, Complexité, Combinatoire
LIP - Laboratoire de l'Informatique du Parallélisme
3 SEQUEL - Sequential Learning
Inria Lille - Nord Europe, CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL) - UMR 9189
Abstract : In this paper, 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 $≥ ε− or ≤ε; if µ ∈ (−ε,ε)$, both answers are considered to be correct. Then, we consider PAC-best arm identification in a bandit model: given K probability distributions on R with means $µ_1,. .. , µ_K$ , we derive the asymptotic complexity of identifying, with risk at most $δ$, an index $I ∈ {1,. .. , K}$ such that $µ_I ≥ max_i µ_i −ε$. We provide non asymptotic bounds on the error of a parallel General Likelihood Ratio Test, which can also be used for more general testing problems. We further propose lower bound on the number of observation 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.
Keywords :
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-02123833
Contributor : Aurélien Garivier <>
Submitted on : Thursday, May 9, 2019 - 10:44:48 AM
Last modification on : Friday, May 10, 2019 - 1:27:56 AM

### Files

GK_AOS19.pdf
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### Identifiers

• HAL Id : hal-02123833, version 1
• ARXIV : 1905.03495

### Citation

Aurélien Garivier, Emilie Kaufmann. Non-Asymptotic Sequential Tests for Overlapping Hypotheses and application to near optimal arm identification in bandit models. 2019. ⟨hal-02123833⟩

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