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 generalizes an earlier result of Blum and Kannan [4]. The algorithm is simple and is based on random, sampling.