Double-descent curves in neural networks: a new perspective using Gaussian processes

TL;DR

This paper offers a new perspective on the double-descent phenomenon in neural networks by linking spectral properties of empirical feature covariance matrices to Gaussian processes using random matrix theory.

Contribution

It introduces an analytical framework connecting neural network spectral distributions with Gaussian process kernels, explaining double-descent behavior.

Findings

Spectral distribution of empirical covariance matrices is a width-dependent perturbation of NNGP kernel spectrum.

Double-descent is explained by the discrepancy between empirical and NNGP kernels.

Analytical expressions for generalization behavior in kernel and GP regression.

Abstract

Double-descent curves in neural networks describe the phenomenon that the generalisation error initially descends with increasing parameters, then grows after reaching an optimal number of parameters which is less than the number of data points, but then descends again in the overparameterized regime. In this paper, we use techniques from random matrix theory to characterize the spectral distribution of the empirical feature covariance matrix as a width-dependent perturbation of the spectrum of the neural network Gaussian process (NNGP) kernel, thus establishing a novel connection between the NNGP literature and the random matrix theory literature in the context of neural networks. Our analytical expression allows us to study the generalisation behavior of the corresponding kernel and GP regression, and provides a new interpretation of the double-descent phenomenon, namely as governed by the discrepancy between the width-dependent empirical kernel and the width-independent NNGP kernel.