  The recent success of neural networks in pattern recognition and
classification problems suggests that neural networks 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 instead of the $0$-$1$ loss so that gradient descent-based
numerical optimizations can be easily applied. While the class of functions we
consider is quite large that optimal rates cannot be faster than
$n^{-\frac{1}{3}}$, 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.
