Nonasymptotic Bounds on the L 2 Error of Neural Network Regression Estimates

The estimation of multivariate regression functions from bounded i.i.d. data is considered. The L ₂ error with integration with respect to the design measure is used as an error criterion. The distribution of the design is assumed to be concentrated on a finite set. Neural network estimates are defined by minimizing the empirical L ₂ risk over various sets of feedforward neural networks. Nonasymptotic bounds on the L ₂ error of these estimates are presented. The results imply that neural networks are able to adapt to additive regression functions and to regression functions which are a sum of ridge functions, and hence are able to circumvent the curse of dimensionality in these cases.