Tensor Programs II: Neural Tangent Kernel for Any Architecture
This work addresses the theoretical understanding of neural network training dynamics for researchers, by generalizing NTK results to any architecture and clarifying the gradient independence assumption.
The authors proved that the Neural Tangent Kernel (NTK) converges to a deterministic limit for any neural network architecture as widths go to infinity, and provided a method to calculate this limit, including reference implementations for architectures like transformers and RNNs.
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.