Nonparametric regression using deep neural networks with ReLU activation function

Consider the multivariate nonparametric regression model. It is shown that estimators based on sparsely connected deep neural networks with ReLU activation function and properly chosen network architecture achieve the minimax rates of convergence (up to log n-factors) under a general composition assumption on the regression function. The framework includes many well-studied structural constraints such as (generalized) additive models. While there is a lot of flexibility in the network architecture, the tuning parameter is the sparsity of the network. Specifically, we consider large networks with number of potential parameters being much bigger than the sample size. The analysis gives some insights why multilayer feedforward neural networks perform well in practice. Interestingly, the depth (number of layers) of the neural network architectures plays an important role and our theory suggests that scaling the network depth with the logarithm of the sample size is natural.