Deformed semicircle law and concentration of nonlinear random matrices for ultra-wide neural networks

TL;DR

This paper analyzes the spectral distribution of neural network kernels in ultra-wide regimes, revealing a deformed semicircle law and demonstrating concentration results that connect empirical and limiting kernels for improved understanding of neural network behavior.

Contribution

It introduces a deformed semicircle law for spectral distributions in ultra-wide neural networks and provides concentration bounds linking empirical and theoretical kernels.

Findings

Deformed semicircle law describes spectral distribution in ultra-wide neural networks.

Concentration bounds show empirical kernels closely follow their limiting counterparts.

Asymptotic analysis of training and test errors in random feature regression.

Abstract

In this paper, we investigate a two-layer fully connected neural network of the form $f(X)=\frac{1}{\sqrt{d_1}}\boldsymbol{a}^\top \sigma\left(WX\right)$, where $X\in\mathbb{R}^{d_0\times n}$ is a deterministic data matrix, $W\in\mathbb{R}^{d_1\times d_0}$ and $\boldsymbol{a}\in\mathbb{R}^{d_1}$ are random Gaussian weights, and $\sigma$ is a nonlinear activation function. We study the limiting spectral distributions of two empirical kernel matrices associated with $f(X)$: the empirical conjugate kernel (CK) and neural tangent kernel (NTK), beyond the linear-width regime ($d_1\asymp n$). We focus on the $\textit{ultra-wide regime}$, where the width $d_1$ of the first layer is much larger than the sample size $n$. Under appropriate assumptions on $X$ and $\sigma$, a deformed semicircle law emerges as $d_1/n\to\infty$ and $n\to\infty$. We first prove this limiting law for generalized sample covariance matrices with some dependency. To specify it for our neural network model, we provide a nonlinear Hanson-Wright inequality that is suitable for neural networks with random weights and Lipschitz activation functions. We also demonstrate non-asymptotic concentrations of the empirical CK and NTK around their limiting kernels in the spectral norm, along with lower bounds on their smallest eigenvalues. As an application, we show that random feature regression induced by the empirical kernel achieves the same asymptotic performance as its limiting kernel regression under the ultra-wide regime. This allows us to calculate the asymptotic training and test errors for random feature regression using the corresponding kernel regression.