Bandit Theory meets Compressed Sensing for high dimensional Stochastic Linear Bandit

We consider a linear stochastic bandit problem where the dimension K of the unknown parameter is larger than the sampling budget n. In such cases, it is in general impossible to derive sub-linear regret bounds since usual linear bandit algorithms have a regret in O(K p n). In this paper we assume that is S sparse, i.e. has at most S non-zero components, and that the space of arms is the unit ball for the jj:jj2 norm. We combine ideas from Compressed Sensing and Bandit Theory and derive an algorithm with a regret bound in O(S p n). 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.