Optimal Learning Rates of Deep Convolutional Neural Networks: Additive Ridge Functions

Convolutional neural networks have shown extraordinary abilities in many applications, especially those related to the classification tasks. However, for the regression problem, the abilities of convolutional structures have not been fully understood, and further investigation is needed. In this paper, we consider the mean squared error analysis for deep convolutional neural networks. We show that, for additive ridge functions, convolutional neural networks followed by one fully connected layer with ReLU activation functions can reach optimal mini-max rates (up to a log factor). The convergence rates are dimension independent. This work shows the statistical optimality of convolutional neural networks and may shed light on why convolutional neural networks are able to behave well for high dimensional input.