Phase Transitions in the Fluctuations of Functionals of Random Neural Networks
Provides a theoretical foundation for understanding fluctuations in deep random neural networks, relevant to researchers in statistical learning and random matrix theory.
The paper establishes central and non-central limit theorems for functionals of Gaussian outputs from infinitely-wide random neural networks, showing that asymptotic behavior depends on fixed points of the covariance function, leading to three distinct limiting regimes.
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.