Efficient active learning of sparse halfspaces

We study the problem of efficient PAC active learning of homogeneous linear classifiers (halfspaces) in ${\mathbb{R}}^d$, where the goal is to learn a halfspace with low error using as few label queries as possible. Under the extra assumption that there is a $t$-sparse halfspace that performs well on the data ($t\ll d$), we would like our active learning algorithm to be {\em attribute efficient}, i.e. to have label requirements sublinear in $d$. In this paper, we provide a computationally efficient algorithm that achieves this goal. Under certain distributional assumptions on the data, our algorithm achieves a label complexity of $O(t \cdot $\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{l}\mathrm{o}\mathrm{g}$(d, \frac 1 \epsilon))$. In contrast, existing algorithms in this setting are either computationally inefficient, or subject to label requirements polynomial in $d$ or $\frac{1}{ϵ}$.