  Along with fruitful applications of Deep Neural Networks (DNNs) to realistic
problems, recently, some empirical studies of DNNs reported a universal
phenomenon of Frequency Principle (F-Principle): a DNN tends to learn a target
function from low to high frequencies during the training. The F-Principle has
been very useful in providing both qualitative and quantitative understandings
of DNNs. In this paper, we rigorously investigate the F-Principle for the
training dynamics of a general DNN at three stages: initial stage, intermediate
stage, and final stage. For each stage, a theorem is provided in terms of
proper quantities characterizing the F-Principle. Our results are general in
the sense that they work for multilayer networks with general activation
functions, population densities of data, and a large class of loss functions.
Our work lays a theoretical foundation of the F-Principle for a better
understanding of the training process of DNNs.
