On the Impacts of the Random Initialization in the Neural Tangent Kernel Theory

TL;DR

This paper investigates how random initialization affects the neural tangent kernel (NTK) theory, revealing that traditional and mirrored initializations lead to different generalization behaviors and that NTK theory alone may not fully explain neural network performance.

Contribution

It demonstrates the convergence of neural network training dynamics to NTK regression and analyzes the function space of Gaussian process limits, highlighting limitations of NTK theory.

Findings

Training dynamics converge to NTK regression with random initialization.

The Gaussian process lies in a specific interpolation space of the NTK RKHS.

Wide neural networks suffer from the curse of dimensionality in generalization.

Abstract

This paper aims to discuss the impact of random initialization of neural networks in the neural tangent kernel (NTK) theory, which is ignored by most recent works in the NTK theory. It is well known that as the network's width tends to infinity, the neural network with random initialization converges to a Gaussian process $f^{\mathrm{GP}}$, which takes values in $L^{2}(\mathcal{X})$, where $\mathcal{X}$ is the domain of the data. In contrast, to adopt the traditional theory of kernel regression, most recent works introduced a special mirrored architecture and a mirrored (random) initialization to ensure the network's output is identically zero at initialization. Therefore, it remains a question whether the conventional setting and mirrored initialization would make wide neural networks exhibit different generalization capabilities. In this paper, we first show that the training dynamics of the gradient flow of neural networks with random initialization converge uniformly to that of the corresponding NTK regression with random initialization $f^{\mathrm{GP}}$. We then show that $\mathbf{P}(f^{\mathrm{GP}} \in [\mathcal{H}^{\mathrm{NT}}]^{s}) = 1$ for any $s < \frac{3}{d+1}$ and $\mathbf{P}(f^{\mathrm{GP}} \in [\mathcal{H}^{\mathrm{NT}}]^{s}) = 0$ for any $s \geq \frac{3}{d+1}$, where $[\mathcal{H}^{\mathrm{NT}}]^{s}$ is the real interpolation space of the RKHS $\mathcal{H}^{\mathrm{NT}}$ associated with the NTK. Consequently, the generalization error of the wide neural network trained by gradient descent is $\Omega(n^{-\frac{3}{d+3}})$, and it still suffers from the curse of dimensionality. On one hand, the result highlights the benefits of mirror initialization. On the other hand, it implies that NTK theory may not fully explain the superior performance of neural networks.