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Online Optimization in X-Armed Bandits

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  • Sébastien Bubeck

    ()
    (INRIA Futurs - SEQUEL - INRIA - CNRS : UMR8022 - CNRS : UMR8146 - Université des Sciences et Technologies de Lille - Lille I - Université Charles de Gaulle - Lille III - Ecole Centrale de Lille)

  • Rémi Munos

    ()
    (INRIA Futurs - SEQUEL - INRIA - CNRS : UMR8022 - CNRS : UMR8146 - Université des Sciences et Technologies de Lille - Lille I - Université Charles de Gaulle - Lille III - Ecole Centrale de Lille)

  • Gilles Stoltz

    ()
    (DMA - Département de Mathématiques et Applications - CNRS : UMR8553 - Ecole Normale Supérieure de Paris - ENS Paris, GREGH - Groupement de Recherche et d'Etudes en Gestion à HEC - GROUPE HEC - CNRS : UMR2959)

  • Csaba Szepesvari

    ()
    (Department of Computing Science - Department of Computing Science, University of Alberta)

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    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.

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    Bibliographic Info

    Paper provided by HAL in its series Post-Print with number inria-00329797.

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    Date of creation: 2008
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    Publication status: Published - Presented, Twenty-Second Annual Conference on Neural Information Processing Systems, 2008, Vancouver, Canada
    Handle: RePEc:hal:journl:inria-00329797

    Note: View the original document on HAL open archive server: http://hal.inria.fr/inria-00329797/en/
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    Web page: http://hal.archives-ouvertes.fr/

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