Tight Bounds on the Smallest Eigenvalue of the Neural Tangent Kernel for Deep ReLU Networks

TL;DR

This paper derives tight bounds on the smallest eigenvalue of the Neural Tangent Kernel for deep ReLU networks, applicable to both infinite and finite widths, advancing understanding of neural network training and generalization.

Contribution

It provides the first tight bounds on NTK eigenvalues for deep ReLU networks in finite-width settings, generalizing previous two-layer results.

Findings

Tight bounds on NTK smallest eigenvalue for deep ReLU networks.

Analysis of singular values of hidden feature matrices.

Bounds on Lipschitz constants of feature maps.

Abstract

A recent line of work has analyzed the theoretical properties of deep neural networks via the Neural Tangent Kernel (NTK). In particular, the smallest eigenvalue of the NTK has been related to the memorization capacity, the global convergence of gradient descent algorithms and the generalization of deep nets. However, existing results either provide bounds in the two-layer setting or assume that the spectrum of the NTK matrices is bounded away from 0 for multi-layer networks. In this paper, we provide tight bounds on the smallest eigenvalue of NTK matrices for deep ReLU nets, both in the limiting case of infinite widths and for finite widths. In the finite-width setting, the network architectures we consider are fairly general: we require the existence of a wide layer with roughly order of $N$ neurons, $N$ being the number of data samples; and the scaling of the remaining layer widths is arbitrary (up to logarithmic factors). To obtain our results, we analyze various quantities of independent interest: we give lower bounds on the smallest singular value of hidden feature matrices, and upper bounds on the Lipschitz constant of input-output feature maps.