Analysis of convolutional neural network image classifiers in a hierarchical max-pooling model with additional local pooling

Deep learning, i.e., estimation of a functional relationship by a deep neural network, belongs nowadays to the most successful and most widely used methods in machine learning, see, e.g., Schmidhuber (2015) and the literature cited therein. In many applications the most successful networks are deep convolutional networks. E.g., since 2012, the ImageNet Large Scale Visual Recognition Challenge (ILSVRC) was won each year by deep convolutional neural networks (cf., Russakovsky et al. (2015)). Deep convolutional neural networks can be considered as a special case of deep feedforward neural networks, where symmetry constraints are imposed on the weights of the networks. For deep feedforward neural networks, recently a large number of impressive rate of convergence results have been shown. E.g., in Kohler and Krzyżak (2017), Bauer and Kohler (2019), Schmidt-Hieber (2020), Kohler and Langer (2021) and Suzuki and Nitanda (2019) it was shown that these networks achieve in nonparametric regression a