Neural Networks and Quantum Field Theory

TL;DR

This paper develops a theoretical framework connecting neural networks to quantum field theory, using Wilsonian effective field theory to analyze their behavior and likelihoods, and relates overparameterization to model simplicity.

Contribution

It introduces a novel approach linking neural networks with quantum field theory concepts, providing a new perspective on their likelihoods and overparameterization.

Findings

Neural networks in the asymptotic limit behave like Gaussian processes.

Non-Gaussian corrections correspond to particle interactions in field theory.

The formalism applies broadly to architectures approaching Gaussian processes asymptotically.

Abstract

We propose a theoretical understanding of neural networks in terms of Wilsonian effective field theory. The correspondence relies on the fact that many asymptotic neural networks are drawn from Gaussian processes, the analog of non-interacting field theories. Moving away from the asymptotic limit yields a non-Gaussian process and corresponds to turning on particle interactions, allowing for the computation of correlation functions of neural network outputs with Feynman diagrams. Minimal non-Gaussian process likelihoods are determined by the most relevant non-Gaussian terms, according to the flow in their coefficients induced by the Wilsonian renormalization group. This yields a direct connection between overparameterization and simplicity of neural network likelihoods. Whether the coefficients are constants or functions may be understood in terms of GP limit symmetries, as expected from 't Hooft's technical naturalness. General theoretical calculations are matched to neural network experiments in the simplest class of models allowing the correspondence. Our formalism is valid for any of the many architectures that becomes a GP in an asymptotic limit, a property preserved under certain types of training.