Jamming in multilayer supervised learning models

TL;DR

This paper explores the universality of jamming transitions in multilayer neural networks, revealing conditions under which they exhibit sphere-like universality classes and proposing a dimensional reduction mechanism for finite-dimensional cases.

Contribution

It introduces a mean-field framework for multilayer neural networks in jamming, identifying a dimensional reduction that links their behavior to infinite-dimensional spheres.

Findings

Jamming in multilayer networks can recover sphere universality when isostatic.

Exact mean-field equations reveal a dimensional reduction mechanism.

The mechanism may explain universality in finite-dimensional systems.

Abstract

Critical jamming transitions are characterized by an astonishing degree of universality. Analytic and numerical evidence points to the existence of a large universality class that encompasses finite and infinite dimensional spheres and continuous constraint satisfaction problems (CCSP) such as the non-convex perceptron and related models. In this paper we investigate multilayer neural networks (MLNN) learning random associations as models for CCSP which could potentially define different jamming universality classes. As opposed to simple perceptrons and infinite dimensional spheres, which are described by a single effective field in terms of which the constraints appear to be one-dimensional, the description of MLNN, involves multiple fields, and the constraints acquire a multidimensional character. We first study the models numerically and show that similarly to the perceptron, whenever jamming is isostatic, the sphere universality class is recovered, we then write the exact mean-field equations for the models and identify a dimensional reduction mechanism that leads to a scaling regime identical to one of the infinite dimensional spheres. We suggest that this mechanism could be general enough to explain finite dimensional universality.