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(logn/loglogn) 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 A. Blum and R. Kannan (1993). The algorithm is simple and is based on random sampling.