Learning intersections and thresholds of halfspaces

• DOI: 10.1016/j.jcss.2003.11.002
• Publication Date: 2004-01-16T10:15:36Z
• Authors: Adam R. Klivans; Ryan O'Donnell; Rocco A. Servedio
• Keywords: FOS: Computer and information sciences; Computer Networks and Communications; Applied Mathematics; Computational learning theory; Halfspaces; Noise sensitivity; 0102 computer and information sciences; 01 natural sciences; Fourier analysis; Theoretical Computer Science; Computational Theory and Mathematics; 89999 Information and Computing Sciences not elsewhere classified; Polynomial threshold functions

We give the first polynomial time algorithm to learn any function of a constant number of halfspaces under the uniform distribution on the Boolean hypercube to within any constant error parameter. We also give the first quasipolynomial time algorithm for learning any Boolean function of a polylog number of polynomial-weight halfspaces under any distribution on the Boolean hypercube. As special cases of these results we obtain algorithms for learning intersections and thresholds of halfspaces. Our uniform distribution learning algorithms involve a novel non-geometric approach to learning halfspaces; we use Fourier techniques together with a careful analysis of the noise sensitivity of functions of halfspaces. Our algorithms for learning under any distribution use techniques from real approximation theory to construct low-degree polynomial threshold functions. Finally, we also observe that any function of a constant number of polynomial-weight halfspaces can be learned in polynomial time in the model of exact learning from membership and equivalence queries.