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# Online Optimization in X-Armed Bandits

## Author

Listed:
• Sébastien Bubeck

() (SEQUEL - Sequential Learning - LIFL - Laboratoire d'Informatique Fondamentale de Lille - Université de Lille, Sciences et Technologies - Inria - Institut National de Recherche en Informatique et en Automatique - Université de Lille, Sciences Humaines et Sociales - CNRS - Centre National de la Recherche Scientifique - LAGIS - Laboratoire d'Automatique, Génie Informatique et Signal - Université de Lille, Sciences et Technologies - Ecole Centrale de Lille - CNRS - Centre National de la Recherche Scientifique - Inria Lille - Nord Europe - Inria - Institut National de Recherche en Informatique et en Automatique)

• Rémi Munos

() (SEQUEL - Sequential Learning - LIFL - Laboratoire d'Informatique Fondamentale de Lille - Université de Lille, Sciences et Technologies - Inria - Institut National de Recherche en Informatique et en Automatique - Université de Lille, Sciences Humaines et Sociales - CNRS - Centre National de la Recherche Scientifique - LAGIS - Laboratoire d'Automatique, Génie Informatique et Signal - Université de Lille, Sciences et Technologies - Ecole Centrale de Lille - CNRS - Centre National de la Recherche Scientifique - Inria Lille - Nord Europe - Inria - Institut National de Recherche en Informatique et en Automatique)

• Gilles Stoltz

() (DMA - Département de Mathématiques et Applications - ENS Paris - ENS Paris - École normale supérieure - Paris - CNRS - Centre National de la Recherche Scientifique, GREGH - Groupement de Recherche et d'Etudes en Gestion à HEC - HEC Paris - Ecole des Hautes Etudes Commerciales - CNRS - Centre National de la Recherche Scientifique)

• Csaba Szepesvari

() (Department of Computing Science [Edmonton] - University of Alberta [Edmonton])

## Abstract

We consider a generalization of stochastic bandit problems where the set of arms, X, is allowed to be a generic topological space. We constraint the mean-payoff function with a dissimilarity function over X in a way that is more general than Lipschitz. We construct an arm selection policy whose regret improves upon previous result for a large class of problems. In particular, our results imply that if X is the unit hypercube in a Euclidean space and the mean-payoff function has a finite number of global maxima around which the behavior of the function is locally Holder with a known exponent, then the expected regret is bounded up to a logarithmic factor by $\sqrt{n}$, i.e., the rate of the growth of the regret is independent of the dimension of the space. Moreover, we prove the minimax optimality of our algorithm for the class of mean-payoff functions we consider.

## Suggested Citation

• Sébastien Bubeck & Rémi Munos & Gilles Stoltz & Csaba Szepesvari, 2008. "Online Optimization in X-Armed Bandits," Post-Print inria-00329797, HAL.
• Handle: RePEc:hal:journl:inria-00329797
Note: View the original document on HAL open archive server: https://hal.inria.fr/inria-00329797
as

File URL: https://hal.inria.fr/inria-00329797/document

## Citations

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Cited by:

1. Sébastien Bubeck & Rémi Munos & Gilles Stoltz & Csaba Szepesvari, 2011. "X-Armed Bandits," Post-Print hal-00450235, HAL.

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