Learning functions of k relevant variables

We consider a fundamental problem in computational learning theory: learning an arbitrary Boolean function that depends on an unknown set of k out of n Boolean variables. We give an algorithm for learning such functions from uniform random examples that runs in time roughly ðnÞ o oþ1; where oo2:376 is the matrix multiplication exponent. We thus obtain the first-polynomial factor improvement on the naive n time bound which can be achieved via exhaustive search. Our algorithm and analysis exploit new structural properties of Boolean functions. r 2003 Elsevier Inc. All rights reserved.