Learning functions of halfspaces using prefix covers

We present a simple query-algorithm for learning arbitrary functions of k halfspaces under any product distribution on the Boolean hypercube. Our algorithms learn any function of k halfspaces to within accuracy eps in time O((nk/eps)^k+1) under any product distribution on 0,1ⁿ using read-once branching programs as a hypothesis.. This gives the fifirst poly(n, 1/eps) algorithm for learning even the intersection of 2 halfspaces under the uniform distribution on 0,1ⁿ; previously known algorithms had an exponential dependence either on the accuracy parameter eps or the dimension n. To prove this result, we identify a new structural property of Boolean functions that yields learnability with queries: that of having a small prefix cover.