Spectrum Dependent Learning Curves in Kernel Regression and Wide Neural Networks

TL;DR

This paper provides analytical formulas for the generalization performance of kernel regression and wide neural networks, revealing a spectral principle where models learn successively higher spectral modes as training data increases.

Contribution

It introduces a spectral decomposition approach to understanding learning dynamics in kernel methods and neural networks, linking spectral modes to training set size and data distribution.

Findings

Kernel methods and neural networks learn spectral modes sequentially.

Learning stages depend on data distribution and kernel spectral properties.

Theoretical predictions are validated with simulations on synthetic and real data.

Abstract

We derive analytical expressions for the generalization performance of kernel regression as a function of the number of training samples using theoretical methods from Gaussian processes and statistical physics. Our expressions apply to wide neural networks due to an equivalence between training them and kernel regression with the Neural Tangent Kernel (NTK). By computing the decomposition of the total generalization error due to different spectral components of the kernel, we identify a new spectral principle: as the size of the training set grows, kernel machines and neural networks fit successively higher spectral modes of the target function. When data are sampled from a uniform distribution on a high-dimensional hypersphere, dot product kernels, including NTK, exhibit learning stages where different frequency modes of the target function are learned. We verify our theory with simulations on synthetic data and MNIST dataset.