Regret lower bounds and extended Upper Confidence Bounds policies in stochastic multi-armed bandit problem

Antoine Salomon 1, * Jean-Yves Audibert 1, 2, 3 Issam El Alaoui 1
* Corresponding author
1 IMAGINE [Marne-la-Vallée]
LIGM - Laboratoire d'Informatique Gaspard-Monge, CSTB - Centre Scientifique et Technique du Bâtiment, ENPC - École des Ponts ParisTech
2 SIERRA - Statistical Machine Learning and Parsimony
DI-ENS - Département d'informatique de l'École normale supérieure, ENS Paris - École normale supérieure - Paris, Inria Paris-Rocquencourt, CNRS - Centre National de la Recherche Scientifique : UMR8548
Abstract : This paper is devoted to regret lower bounds in the classical model of stochastic multi-armed bandit. A well-known result of Lai and Robbins, which has then been extended by Burnetas and Katehakis, has established the presence of a logarithmic bound for all consistent policies. We relax the notion of consistence, and exhibit a generalisation of the logarithmic bound. We also show the non existence of logarithmic bound in the general case of Hannan consistency. To get these results, we study variants of popular Upper Confidence Bounds (ucb) policies. As a by-product, we prove that it is impossible to design an adaptive policy that would select the best of two algorithms by taking advantage of the properties of the environment.
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Submitted on : Friday, December 16, 2011 - 2:48:03 PM
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  • HAL Id : hal-00652865, version 1
  • ARXIV : 1112.3827

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Antoine Salomon, Jean-Yves Audibert, Issam El Alaoui. Regret lower bounds and extended Upper Confidence Bounds policies in stochastic multi-armed bandit problem. 2011. ⟨hal-00652865⟩

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