  Convolutional neural networks (CNNs) trained with cross-entropy loss have
proven to be extremely successful in classifying images. In recent years, much
work has been done to also improve the theoretical understanding of neural
networks. Nevertheless, it seems limited when these networks are trained with
cross-entropy loss, mainly because of the unboundedness of the target function.
In this paper, we aim to fill this gap by analyzing the rate of the excess risk
of a CNN classifier trained by cross-entropy loss. Under suitable assumptions
on the smoothness and structure of the a posteriori probability, it is shown
that these classifiers achieve a rate of convergence which is independent of
the dimension of the image. These rates are in line with the practical
observations about CNNs.
