# Lipschitz Bandits without the Lipschitz Constant

* Corresponding author
4 CLASSIC - Computational Learning, Aggregation, Supervised Statistical, Inference, and Classification
DMA - Département de Mathématiques et Applications, ENS Paris - École normale supérieure - Paris, Inria Paris-Rocquencourt
Abstract : We consider the setting of stochastic bandit problems with a continuum of arms. We first point out that the strategies considered so far in the literature only provided theoretical guarantees of the form: given some tuning parameters, the regret is small with respect to a class of environments that depends on these parameters. This is however not the right perspective, as it is the strategy that should adapt to the specific bandit environment at hand, and not the other way round. Put differently, an adaptation issue is raised. We solve it for the special case of environments whose mean-payoff functions are globally Lipschitz. More precisely, we show that the minimax optimal orders of magnitude $L^{d/(d+2)} \, T^{(d+1)/(d+2)}$ of the regret bound against an environment $f$ with Lipschitz constant $L$ over $T$ time instances can be achieved without knowing $L$ or $T$ in advance. This is in contrast to all previously known strategies, which require to some extent the knowledge of $L$ to achieve this performance guarantee.
Document type :
Conference papers
Jyrki Kivinen, Csaba Szepesvári, Esko Ukkonen, Thomas Zeugmann. ALT 2011 - 22nd International Conference on Algorithmic Learning Theory, Oct 2011, Espoo, Finland. Springer, pp.[A venir], 2011, LNCS / Lecture Notes in Artificial Intelligence
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https://hal.archives-ouvertes.fr/hal-00595692
Contributor : Gilles Stoltz <>
Submitted on : Thursday, July 14, 2011 - 1:14:01 PM
Last modification on : Wednesday, January 4, 2017 - 4:25:24 PM
Document(s) archivé(s) le : Saturday, October 15, 2011 - 2:20:29 AM

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• HAL Id : hal-00595692, version 2
• ARXIV : 1105.5041

### Citation

Sébastien Bubeck, Gilles Stoltz, Jia Yuan Yu. Lipschitz Bandits without the Lipschitz Constant. Jyrki Kivinen, Csaba Szepesvári, Esko Ukkonen, Thomas Zeugmann. ALT 2011 - 22nd International Conference on Algorithmic Learning Theory, Oct 2011, Espoo, Finland. Springer, pp.[A venir], 2011, LNCS / Lecture Notes in Artificial Intelligence. <hal-00595692v2>

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