  Image classification from independent and identically distributed random
variables is considered. Image classifiers are defined which are based on a
linear combination of deep convolutional networks with max-pooling layer. Here
all the weights are learned by stochastic gradient descent. A general result is
presented which shows that the image classifiers are able to approximate the
best possible deep convolutional network. In case that the a posteriori
probability satisfies a suitable hierarchical composition model it is shown
that the corresponding deep convolutional neural network image classifier
achieves a rate of convergence which is independent of the dimension of the
images.
