# Bandit Theory meets Compressed Sensing for high dimensional Stochastic Linear Bandit

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Alexandra Carpentier,
Remi Munos
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Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics, PMLR 22:190-198, 2012.

#### Abstract

We consider a linear stochastic bandit problem where the dimension K of the unknown parameter heta is larger than the sampling budget n. Since usual linear bandit algorithms have a regret in O(K\sqrtn), it is in general impossible to obtain a sub-linear regret without further assumption. In this paper we make the assumption that heta is S-sparse, i.e. has at most S-non-zero components, and that the space of arms is the unit ball for the ||.||_2 norm. We combine ideas from Compressed Sensing and Bandit Theory to derive an algorithm with a regret bound in O(S\sqrtn). We detail an application to the problem of optimizing a function that depends on many variables but among which only a small number of them (initially unknown) are relevant.

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