Tensor Programs I: Wide Feedforward or Recurrent Neural Networks of Any Architecture are Gaussian Processes

TL;DR

This paper demonstrates that wide neural networks, including modern architectures like RNNs, CNNs, and transformers, are equivalent to Gaussian processes, extending previous results and providing a unifying framework.

Contribution

It generalizes the Gaussian process correspondence to all modern neural network architectures using tensor programs, offering a comprehensive theoretical framework.

Findings

Wide neural networks are Gaussian processes.

The Gaussian process correspondence extends to RNNs, CNNs, transformers, and more.

Open-source kernels for various architectures are provided.

Abstract

Wide neural networks with random weights and biases are Gaussian processes, as originally observed by Neal (1995) and more recently by Lee et al. (2018) and Matthews et al. (2018) for deep fully-connected networks, as well as by Novak et al. (2019) and Garriga-Alonso et al. (2019) for deep convolutional networks. We show that this Neural Network-Gaussian Process correspondence surprisingly extends to all modern feedforward or recurrent neural networks composed of multilayer perceptron, RNNs (e.g. LSTMs, GRUs), (nD or graph) convolution, pooling, skip connection, attention, batch normalization, and/or layer normalization. More generally, we introduce a language for expressing neural network computations, and our result encompasses all such expressible neural networks. This work serves as a tutorial on the *tensor programs* technique formulated in Yang (2019) and elucidates the Gaussian Process results obtained there. We provide open-source implementations of the Gaussian Process kernels of simple RNN, GRU, transformer, and batchnorm+ReLU network at github.com/thegregyang/GP4A.