Critical feature learning in deep neural networks

TL;DR

This paper develops a theoretical framework for understanding feature learning in deep neural networks by analyzing finite-width effects, kernel evolution, and the role of fluctuations in the Bayesian prior.

Contribution

It introduces a systematic theory of network kernels in finite-width deep networks, linking feature learning to criticality and prior fluctuations.

Findings

Kernel distribution depends inversely on network width N.

Backward propagation aligns kernels with target features.

Finite-width fluctuations enable kernel adaptation to data.

Abstract

A key property of neural networks driving their success is their ability to learn features from data. Understanding feature learning from a theoretical viewpoint is an emerging field with many open questions. In this work we capture finite-width effects with a systematic theory of network kernels in deep non-linear neural networks. We show that the Bayesian prior of the network can be written in closed form as a superposition of Gaussian processes, whose kernels are distributed with a variance that depends inversely on the network width N . A large deviation approach, which is exact in the proportional limit for the number of data points $P = \alpha N \rightarrow \infty$, yields a pair of forward-backward equations for the maximum a posteriori kernels in all layers at once. We study their solutions perturbatively to demonstrate how the backward propagation across layers aligns kernels with the target. An alternative field-theoretic formulation shows that kernel adaptation of the Bayesian posterior at finite-width results from fluctuations in the prior: larger fluctuations correspond to a more flexible network prior and thus enable stronger adaptation to data. We thus find a bridge between the classical edge-of-chaos NNGP theory and feature learning, exposing an intricate interplay between criticality, response functions, and feature scale.