A Random Sampling Based Algorithm for Learning the Intersection of Half-spaces

We present an algorithm for learning the intersection of half-spaces in n dimensions. Over nearly-uniform distributions, it runs in polynomial time for up to O(log n= log log n) half-spaces or, more generally , for any number of half-spaces whose normal vectors lie in an O(log n= log log n) dimensional subspace. Over less restricted \non-concentrated" distributions it runs in polynomial time for a constant number of half-spaces. This generalizes an earlier result of Blum and Kannan 4]. The algorithm is simple and is based on random sampling.