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# Learning Functions of Halfspaces Using Prefix Covers

*Proceedings of the 25th Annual Conference on Learning Theory*, PMLR 23:15.1-15.10, 2012.

#### Abstract

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 ε in time \emphO((nk/ε)^k+1) under any product distribution on 0, 1^\emphn using read-once branching programs as a hypothesis. This gives the first \emphpoly(n, 1/ε) algorithm for learning even the intersection of 2 halfspaces under the uniform distribution on 0, 1^\emphn previously known algorithms had an exponential dependence either on the accuracy parameter ε or the dimension \emphn. 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.