Tensor Programs II: Neural Tangent Kernel for Any Architecture

TL;DR

This paper proves that the Neural Tangent Kernel (NTK) of any randomly initialized neural network converges to a deterministic limit as widths grow, providing a method to compute this limit and analyzing the validity of the gradient independence assumption.

Contribution

It introduces the Simple GIA Check to validate NTK calculations and applies tensor programs to analyze wide neural networks of any architecture.

Findings

NTK converges to a deterministic limit for any architecture

Simple GIA Check determines when GIA-based NTK calculations are correct

Provides implementations for NTKs of RNNs, transformers, and batch normalization

Abstract

We prove that a randomly initialized neural network of *any architecture* has its Tangent Kernel (NTK) converge to a deterministic limit, as the network widths tend to infinity. We demonstrate how to calculate this limit. In prior literature, the heuristic study of neural network gradients often assumes every weight matrix used in forward propagation is independent from its transpose used in backpropagation (Schoenholz et al. 2017). This is known as the *gradient independence assumption (GIA)*. We identify a commonly satisfied condition, which we call *Simple GIA Check*, such that the NTK limit calculation based on GIA is correct. Conversely, when Simple GIA Check fails, we show GIA can result in wrong answers. Our material here presents the NTK results of Yang (2019a) in a friendly manner and showcases the *tensor programs* technique for understanding wide neural networks. We provide reference implementations of infinite-width NTKs of recurrent neural network, transformer, and batch normalization at https://github.com/thegregyang/NTK4A.