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arXiv · 1906.09235

Theory of the Frequency Principle for General Deep Neural Networks

Tao Luo·Zheng Ma·Zhi-Qin John Xu·Yaoyu Zhang
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Abstract

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.

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Tao Luo, Zheng Ma, Zhi-Qin John Xu, Yaoyu Zhang. 2019-06-21. Theory of the Frequency Principle for General Deep Neural Networks. https://doi.org/10.4208/csiam-am.so-2020-0005

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