Near-Minimax Optimal Estimation With Shallow ReLU Neural Networks

We study the problem of estimating an unknown function from noisy data using shallow (single-hidden layer) ReLU neural networks. The estimators we study minimize the sum of squared data-fitting errors plus a regularization term proportional to the Euclidean norm of the network weights. This minimization corresponds to the common approach of training a neural network with weight decay. We quantify the performance (mean-squared error) of these neural network estimators when the data-generating function belongs to the space of functions of second-order bounded variation in the Radon domain. This space of functions was recently proposed as the natural function space associated with shallow ReLU neural networks. We derive a minimax lower bound for the estimation problem for this function space and show that the neural network estimators are minimax optimal up to logarithmic factors. We also show that this is a "mixed variation" function space that contains classical multivariate function spaces including certain Sobolev spaces and certain spectral Barron spaces. Finally, we use these results to quantify a gap between neural networks and linear methods (which include kernel methods). This paper sheds light on the phenomenon that neural networks seem to break the curse of dimensionality.