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Convolutional neural networks (CNNs) trained with cross-entropy loss haveproven to be extremely successful in classifying images. In recent years, muchwork has been done to also improve the theoretical understanding of neuralnetworks. Nevertheless, it seems limited when these networks are trained withcross-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 riskof a CNN classifier trained by cross-entropy loss. Under suitable assumptionson the smoothness and structure of the a posteriori probability, it is shownthat these classifiers achieve a rate of convergence which is independent ofthe dimension of the image. These rates are in line with the practicalobservations about CNNs.

Statistical theory for image classification using deep convolutional  neural networks with cross-entropy loss under the hierarchical max-pooling  model