# Online Optimization in X-Armed Bandits

## Author Info

• 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)

Registered author(s):

## 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.

If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.

File URL: http://hal.inria.fr/docs/00/32/97/97/PDF/HOO_non-anonymous.pdf

## Bibliographic Info

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

as
in new window

 Length: Date of creation: 2008 Date of revision: 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/ Contact details of provider: Web page: http://hal.archives-ouvertes.fr/

## References

No references listed on IDEAS
You can help add them by filling out this form.

## Lists

This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

## Corrections

When requesting a correction, please mention this item's handle: RePEc:hal:journl:inria-00329797. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (CCSD)

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

If references are entirely missing, you can add them using this form.

If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.

Please note that corrections may take a couple of weeks to filter through the various RePEc services.

This information is provided to you by IDEAS at the Research Division of the Federal Reserve Bank of St. Louis using RePEc data.