An Upper Limit of Decaying Rate with Respect to Frequency in Deep Neural Network

TL;DR

This paper establishes an upper limit on the decay rate of frequency components in deep neural network training, revealing an intrinsic high-frequency learning difficulty and suggesting pre-conditioning as a potential solution.

Contribution

It provides a theoretical upper bound on the frequency decay rate in DNNs and proves the high-frequency curse as an intrinsic challenge.

Findings

There exists a critical decaying rate in DNN training.

Below the limit, DNNs interpolate with regular functions.

Above the limit, DNNs interpolate with trivial functions.

Abstract

Deep neural network (DNN) usually learns the target function from low to high frequency, which is called frequency principle or spectral bias. This frequency principle sheds light on a high-frequency curse of DNNs -- difficult to learn high-frequency information. Inspired by the frequency principle, a series of works are devoted to develop algorithms for overcoming the high-frequency curse. A natural question arises: what is the upper limit of the decaying rate w.r.t. frequency when one trains a DNN? In this work, our theory, confirmed by numerical experiments, suggests that there is a critical decaying rate w.r.t. frequency in DNN training. Below the upper limit of the decaying rate, the DNN interpolates the training data by a function with a certain regularity. However, above the upper limit, the DNN interpolates the training data by a trivial function, i.e., a function is only non-zero at training data points. Our results indicate a better way to overcome the high-frequency curse is to design a proper pre-condition approach to shift high-frequency information to low-frequency one, which coincides with several previous developed algorithms for fast learning high-frequency information. More importantly, this work rigorously proves that the high-frequency curse is an intrinsic difficulty of DNNs.