Learning Functions of Halfspaces

We give an algorithm that learns arbitrary Boolean functions of $k$ arbitrary halfspaces over ${\mathbb{R}}^n$, in the challenging distribution-free Probably Approximately Correct (PAC) learning model, running in time $2^{\sqrt{n}\cdot (\log n{)}^{O(k)}}$. This is the first algorithm that can PAC learn even intersections of two halfspaces in time $2^{o(n)}\mathrm{.}$