Noise-Tolerant Distribution-Free Learning of General Geometric Concepts

We present an efficient algorithm for PAC-learning a very general class of geometric concepts over Rd for fixed d. More specifically, let T be any set of s halfspaces. Let x = (x1,...,xd) be an arbitrary point in Rd. With each t Є T we associate a boolean indicator function It(x) which is 1 if and only if x is in the halfspace t. The concept class Cds that we study consists of all concepts formed by any boolean function over It1, ...Its for ti Є T. This class is much more general than any geometric concept class known to be PAC-learnable. Our results can be extended easily to learn efficiently any boolean combination of a polynomial number of concepts selected from any concept class C over Rd given that the VC-dimension of C has dependence only on d and there is a polynomial time algorithm to determine if there is a concept from C consistent with a given set of labeled examples. We also present a statistical query version of our algorithm that can tolerate random classification noise. Finally we present a generalization of the standard ε-net result of Haussler and Welzl [1987] and apply it to give an alternative noise-tolerant algorithm for d = 2 based on geometric subdivisions