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 {0,1}^n 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 {0,1}^n; 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.

PDF file

In  COLT'12

Publisher  Journal of Machine Learning Research

Details