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

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