How many samples are needed to train a deep neural network?

Even though neural networks have become standard tools in many areas, many important statistical questions remain open. This paper studies the question of how much data are needed to train a ReLU feed-forward neural network. Our theoretical and empirical results suggest that the generalization error of ReLU feed-forward neural networks scales at the rate $1/\sqrt{n}$ in the sample size $n$-rather than the "parametric rate" $1/n$, which might be suggested by traditional statistical theories. Thus, broadly speaking, our results underpin the common belief that neural networks need "many" training samples. Along the way, we also establish new technical insights, such as the first lower bounds of the entropy of ReLU feed-forward networks.