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Consider the multivariate nonparametric regression model. It is shown thatestimators based on sparsely connected deep neural networks with ReLUactivation function and properly chosen network architecture achieve theminimax rates of convergence (up to $\log n$-factors) under a generalcomposition assumption on the regression function. The framework includes manywell-studied structural constraints such as (generalized) additive models.While there is a lot of flexibility in the network architecture, the tuningparameter is the sparsity of the network. Specifically, we consider largenetworks with number of potential network parameters exceeding the sample size.The analysis gives some insights into why multilayer feedforward neuralnetworks perform well in practice. Interestingly, for ReLU activation functionthe depth (number of layers) of the neural network architectures plays animportant role and our theory suggests that for nonparametric regression,scaling the network depth with the sample size is natural. It is also shownthat under the composition assumption wavelet estimators can only achievesuboptimal rates.

Nonparametric regression using deep neural networks with ReLU activation function