A Fine-Grained Spectral Perspective on Neural Networks

Greg Yang, Hadi Salman

Keywords: Neural Tangent Kernel, Neural Network Gaussian Process, Spectral theory, Eigenvalues, Harmonic analysis

TL;DR: Eigenvalues of Conjugate (aka NNGP) and Neural Tangent Kernel can be computed in closed form over the Boolean cube and reveal the effects of hyperparameters on neural network inductive bias, training, and generalization.

Abstract:

Are neural networks biased toward simple functions?
Does depth always help learn more complex features?
Is training the last layer of a network as good as training all layers?
These questions seem unrelated at face value, but in this work we give all of them a common treatment from the spectral perspective.
We will study the spectra of the *Conjugate Kernel, CK,* (also called the *Neural Network-Gaussian Process Kernel*), and the *Neural Tangent Kernel, NTK*.
Roughly, the CK and the NTK tell us respectively ``"what a network looks like at initialization" and "``what a network looks like during and after training."
Their spectra then encode valuable information about the initial distribution and the training and generalization properties of neural networks.
By analyzing the eigenvalues, we lend novel insights into the questions put forth at the beginning, and we verify these insights by extensive experiments of neural networks.
We believe the computational tools we develop here for analyzing the spectra of CK and NTK serve as a solid foundation for future studies of deep neural networks.
We have open-sourced the code for it and for generating the plots in this paper at github.com/jxVmnLgedVwv6mNcGCBy/NNspectra.

Code: https://github.com/jxVmnLgedVwv6mNcGCBy/NNspectra

Community Implementations: [ 2 code implementations](https://www.catalyzex.com/paper/arxiv:1907.10599/code)