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The authors address the problem of PAC (probably and approximately correct) learning functions f : [0, 1]{sup d} {r_arrow} [{minus}K, K] based on iid (independently and identically distributed) sample generated according to an unknown distribution, by using feedforward sigmoidal networks. They use two basic properties of the neural networks with bounded weights, namely: (a) they form a Euclidean class, and (b) for hidden units of the form tanh ({gamma}z) they are Lipschitz functions. Either property yields sample sizes for PAC function learning under any Lipschitz cost function. The sample size based on the first property is tighter compared to the known bounds based on VC-dimension. The second estimate yields a sample size that can be conveniently adjusted by a single parameter, {gamma}, related to the hidden nodes

On PAC learning of functions with smoothness properties using feedforward sigmoidal networks