Generalization error bounds for two-layer neural networks with Lipschitz loss function

Abstract

We derive generalization error bounds for the training of two-layer neural networks without assuming boundedness of the loss function, using Wasserstein distance estimates on the discrepancy between a probability distribution and its associated empirical measure, together with moment bounds for the associated stochastic gradient method. In the case of independent test data, we obtain a dimension-free rate of order $O(n^{-1/2})$ on the $n$-sample generalization error, whereas without independence assumption, we derive a bound of order $O(n^{-1/(d_{\mathrm{in}}+d_{\mathrm{out}})})$, where $d_{\mathrm{in}}$, $d_{\mathrm{out}}$ denote input and output dimensions. Our bounds and their coefficients can be explicitly computed prior to the training of the model, and are confirmed by numerical simulations.