On the exact computation of linear frequency principle dynamics and its generalization

TL;DR

This paper derives an exact differential equation describing how neural networks learn different frequency components during training, revealing the dynamics and generalization properties related to frequency evolution.

Contribution

It introduces the Linear Frequency-Principle (LFP) model for infinite-width two-layer neural networks, providing exact analysis of frequency dynamics during training.

Findings

Higher frequencies evolve slower than lower frequencies depending on activation smoothness.

LFP model minimizes a Frequency-Principle norm that penalizes higher frequencies.

Generalization error is bounded by the FP-norm of the target function.

Abstract

Recent works show an intriguing phenomenon of Frequency Principle (F-Principle) that deep neural networks (DNNs) fit the target function from low to high frequency during the training, which provides insight into the training and generalization behavior of DNNs in complex tasks. In this paper, through analysis of an infinite-width two-layer NN in the neural tangent kernel (NTK) regime, we derive the exact differential equation, namely Linear Frequency-Principle (LFP) model, governing the evolution of NN output function in the frequency domain during the training. Our exact computation applies for general activation functions with no assumption on size and distribution of training data. This LFP model unravels that higher frequencies evolve polynomially or exponentially slower than lower frequencies depending on the smoothness/regularity of the activation function. We further bridge the gap between training dynamics and generalization by proving that LFP model implicitly minimizes a Frequency-Principle norm (FP-norm) of the learned function, by which higher frequencies are more severely penalized depending on the inverse of their evolution rate. Finally, we derive an \textit{a priori} generalization error bound controlled by the FP-norm of the target function, which provides a theoretical justification for the empirical results that DNNs often generalize well for low frequency functions.