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Convolutional neural networks learned by minimizing the cross-entropy lossare nowadays the standard for image classification. Till now, the statisticaltheory behind those networks is lacking. We analyze the rate of convergence ofthe misclassification risk of the estimates towards the optimalmisclassification risk. Under suitable assumptions on the smoothness andstructure of the aposteriori probability it is shown that these estimatesachieve a rate of convergence which is independent of the dimension of theimage. The study shed light on the good performance of CNNs learned bycross-entropy loss and partly explains their success in practical applications.

Statistical theory for image classification using deep convolutional neural networks with cross-entropy loss