Bandit Theory meets Compressed Sensing for high dimensional Stochastic Linear Bandit

Alexandra Carpentier (1) , Rémi Munos (1)

1 SEQUEL - Sequential Learning

We consider a linear stochastic bandit problem where the dimension $K$ of the unknown parameter $\theta$ 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 $O(K\sqrt{n})$. In this paper we assume that $\theta$ 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 and derive algorithms with regret bounds in $O(S\sqrt{n})$.

Domaines: Statistiques [math.ST]; Théorie [stat.TH]; Apprentissage [cs.LG]

Alexandra Carpentier, Rémi Munos. Bandit Theory meets Compressed Sensing for high dimensional Stochastic Linear Bandit. [Technical Report] 2012. ⟨hal-00659731v2⟩