Phase Transitions in the Fluctuations of Functionals of Random Neural Networks

TL;DR

This paper analyzes the asymptotic behavior of functionals of Gaussian outputs in wide random neural networks, revealing three regimes based on covariance fixed points and employing advanced probabilistic tools.

Contribution

It introduces a novel analysis linking the fixed points of the covariance function to the limiting distributions of neural network functionals.

Findings

Identifies three distinct limiting regimes depending on covariance fixed points.

Shows the asymptotic behavior is governed by the stability of the covariance operator.

Employs classical and novel probabilistic techniques to analyze neural network functionals.

Abstract

We establish central and non-central limit theorems for sequences of functionals of the Gaussian output of an infinitely-wide random neural network on the d-dimensional sphere . We show that the asymptotic behaviour of these functionals as the depth of the network increases depends crucially on the fixed points of the covariance function, resulting in three distinct limiting regimes: convergence to the same functional of a limiting Gaussian field, convergence to a Gaussian distribution, convergence to a distribution in the Qth Wiener chaos. Our proofs exploit tools that are now classical (Hermite expansions, Diagram Formula, Stein-Malliavin techniques), but also ideas which have never been used in similar contexts: in particular, the asymptotic behaviour is determined by the fixed-point structure of the iterative operator associated with the covariance, whose nature and stability governs the different limiting regimes.