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.

使用基于稀疏连接ReLU激活函数的深层神经网络，通过适当选择网络结构实现多变量非参数回归模型的极小极限(最优)收敛速率(最多出现$log n$-因子)，同时为多层前馈神经网络表现良好提供理论解释，并表明在不用结构约束的情况下，调整深度可以使模型的性能更好。

使用具有ReLU激活函数的深度神经网络进行非参数回归