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. Given 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 $O(t \mathrm{polylog}(d, \frac 1 \epsilon))$. In contrast, existing algorithms in this setting are either computationally inefficient, or subject to label requirements polynomial in $d$ or $\frac 1 \epsilon$.