This paper studies the question of lower bounds on the number of neurons and examples necessary to program a given task into feed forward neural networks. We introduce the notion of information complexity of a network to complement that of neural complexity. Neural complexity deals with lower bounds for neural resources (numbers of neurons) needed by a network to perform a given task within a given tolerance. Information complexity measures lower bounds for the information (i.e. number of examples) needed about the desired input-output function. We study the interaction of the two complexities, and so lower bounds for the complexity of building and then programming feed-forward nets for given tasks. We show something unexpected a priori--the interaction of the two can be simply bounded, so that they can be studied essentially independently. We construct radial basis function (RBF) algorithms of order n3 that are information-optimal, and give example applications.