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 R-d for fixed d. More specifically, let T be any set of s halfspaces. Let x = (x(1),...,x(d)) be an arbitrary point in R-d. With each t is an element of T we associate a boolean indicator function I-t(x) which is 1 if and only if x is in the halfspace t. The concept class, C-s(d), that we study consists of all concepts formed by any Boolean function over I-t1, ..., I-ts for t(i) is an element of 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 R-d 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 epsilon-net result of Haussler and Welzl [1987] and apply it to give an alternative noise-tolerant algorithm for d = 2 based on geometric subdivisions.