Finite Depth and Width Corrections to the Neural Tangent Kernel

by   Boris Hanin, et al.

We prove the precise scaling, at finite depth and width, for the mean and variance of the neural tangent kernel (NTK) in a randomly initialized ReLU network. The standard deviation is exponential in the ratio of network depth to width. Thus, even in the limit of infinite overparameterization, the NTK is not deterministic if depth and width simultaneously tend to infinity. Moreover, we prove that for such deep and wide networks, the NTK has a non-trivial evolution during training by showing that the mean of its first SGD update is also exponential in the ratio of network depth to width. This is sharp contrast to the regime where depth is fixed and network width is very large. Our results suggest that, unlike relatively shallow and wide networks, deep and wide ReLU networks are capable of learning data-dependent features even in the so-called lazy training regime.


page 1

page 2

page 3

page 4


Neural Tangent Kernel Beyond the Infinite-Width Limit: Effects of Depth and Initialization

Neural Tangent Kernel (NTK) is widely used to analyze overparametrized n...

Robustness in deep learning: The good (width), the bad (depth), and the ugly (initialization)

We study the average robustness notion in deep neural networks in (selec...

Duality of Width and Depth of Neural Networks

Here, we report that the depth and the width of a neural network are dua...

Variance-Preserving Initialization Schemes Improve Deep Network Training: But Which Variance is Preserved?

Before training a neural net, a classic rule of thumb is to randomly ini...

Residual Tangent Kernels

A recent body of work has focused on the theoretical study of neural net...

Is deeper better? It depends on locality of relevant features

It has been recognized that a heavily overparameterized artificial neura...