Theory of the Frequency Principle for General Deep Neural Networks

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.