Learning an Intersection of a Constant Number of Halfspaces over a Uniform Distribution

We present a polynomial-time algorithm to learn an intersection of a constant number of halfspaces inndimensions, over the uniform distribution on ann-dimensional ball. The algorithm we present in fact can learn an intersection of an arbitrary (polynomial) number of halfspaces over this distribution, if the subspace spanned by the normal vectors to the bounding hyperplanes has constant dimension. This generalizes previous results for this distribution, in particular a result of Baum who showed how to learn an intersection of two halfspaces defined by hyperplanes that pass through the origin. (His results also held for a variety of symmetric distributions.) Our algorithm uses estimates of second moments to find vectors in a low-dimensional “relevant subspace.” We believe that the algorithmic techniques studied here may be useful in other geometric learning applications. Our algorithm succeeds even in the presence of random noise, since the only use we make of the examples is to calculate the expectation of certain simple quantities.