Function Estimation by Feedforward Sigmoidal Networks with Bounded Weights

We address the problem of estimating a function f: [0,1] d ↦ [-L,L] by using feedforward sigmoidal networks with a single hidden layer and bounded weights. The only information about the function is provided by an identically independently distributed sample generated according to an unknown distribution. The quality of the estimate is quantified by the expected cost functional and depends on the sample size. We use Lipschitz properties of the cost functional and of the neural networks to derive the relationship between performance bounds and sample sizes within the framework of Valiant's probably approximately correct learning.