The recent success of neural networks in pattern recognition and classification problems suggests that they possess qualities distinct from other more classical classifiers, such as SVMs or boosting classifiers. This paper studies the performance of plug-in classifiers based on neural networks in a binary classification setting as measured by their excess risks. Compared to the typical settings imposed in the literature, we consider a more general scenario that resembles actual practice in two respects: first, the function class to be approximated includes the Barron functions as a proper subset, and second, the neural network classifier constructed is the minimizer of a surrogate loss for which the gradient descent-based numerical optimizations can be easily applied. While the class of distributions we consider is quite large that optimal rates cannot be faster thann–1/3, it is a regime in which dimension-free rates are possible, and approximation power of neural networks can be taken advantage of. In particular, we analyze the estimation and approximation properties of neural networks to obtain a dimension-free, uniform rate of convergence for the excess risk. Finally, we show that the rate obtained is, in fact, minimax optimal up to a logarithmic factor, and the minimax lower bound shows the effect of the margin assumption in this regime.

In this paper, we prove the universal consistency of wide and deep ReLU neural network classifiers. We also give sufficient conditions for a class of probability measures for which classifiers based on neural networks achieve minimax optimal rates of convergence. The result applies to a wide range of known function classes. In particular, while most previous works impose explicit smoothness assumptions on the regression function, our framework encompasses more general settings. The proposed neural networks are either the minimizers of the $0$-$1$ loss that exhibit a benign overfitting behavior.