Geodesy

Function (biology)

Algorithm

Benchmark (surveying)

Philosophy

Deep neural networks

Biology

Applied mathematics

Computer science

Simple (philosophy)

Evolutionary biology

MNIST database

Mathematics

Geography

Epistemology

Mathematical analysis

Artificial intelligence

Artificial neural network

Perspective (graphical)

Generalization

Fourier transform

Communications in Computational Physics

We study the training process of Deep Neural Networks (DNNs) from the Fourier analysis perspective. We demonstrate a very universal Frequency Principle (F-Principle) --- DNNs often fit target functions from low to high frequencies --- on high-dimensional benchmark datasets such as MNIST/CIFAR10 and deep neural networks such as VGG16. This F-Principle of DNNs is opposite to the behavior of most conventional iterative numerical schemes (e.g., Jacobi method), which exhibit faster convergence for higher frequencies for various scientific computing problems. With a simple theory, we illustrate that this F-Principle results from the regularity of the commonly used activation functions. The F-Principle implies an implicit bias that DNNs tend to fit training data by a low-frequency function. This understanding provides an explanation of good generalization of DNNs on most real datasets and bad generalization of DNNs on parity function or randomized dataset.

Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks