Learning intersections and thresholds of halfspaces

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Abstract

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.

Keywords

Computational learning theory
Halfspaces
Fourier analysis
Noise sensitivity
Polynomial threshold functions

Cited by (0)

1

Supported in part by NSF Grant CCR-97-01304. This work was performed while at the Department of Mathematics, MIT.

2

Supported by NSF Grant CCR-99-12342. This work was performed while at the Department of Mathematics, MIT.

3

Supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship and by NSF Grant CCR-98-77049. This work was performed while at the Division of Engineering and Applied Sciences, Harvard University.