Even simple neural nets cannot be trained reliably with a polynomial number of examples

A variation of L.G. Valiant´s ´PAC´ model of learnability (Commun. ACM, vol.27, no.11, p.1134-42, 1984; Proc. 9th Int. Joint Conf. Artif. Intell., Aug. 1985) is used to investigate the learning power of artificial neural nets with threshold nodes. It is shown that there are cases where simple nets require an exponential number of training examples for reliably determining their sets of parameters. Polynomially many training examples may not be enough to determine the set of parameters even for a net of three threshold nodes, if it has to perform reliably in two different environments.<>