Understanding training and generalization in deep learning by Fourier analysis

Background: It is still an open research area to theoretically understand why Deep Neural Networks (DNNs)-- equipped with many more parameters than training data and trained by (stochastic) gradient-based methods-- often achieve remarkably low generalization error. Contribution: We study DNN training by Fourier analysis. Our theoretical framework explains: i) DNN with (stochastic) gradient-based methods endows low-frequency components of the target function with a higher priority during the training; ii) Small initialization leads to good generalization ability of DNN while preserving the DNN's ability of fitting any function. These results are further confirmed by experiments of DNNs fitting the following datasets, i.e., natural images, one-dimensional functions and MNIST dataset.