Gradient descent dynamics and the jamming transition in infinite dimensions

TL;DR

This paper investigates the dynamics of gradient descent in high-dimensional soft particle systems, revealing a critical jamming transition and deriving self-consistent equations in the infinite-dimensional limit, with implications for learning theory.

Contribution

It provides a mean field theoretical analysis of gradient descent dynamics and the jamming transition in infinite dimensions, including equations and partial solutions, and compares with finite-dimensional systems.

Findings

Identification of a critical jamming transition at a specific density.

Derivation of self-consistent dynamical equations in the infinite-dimensional limit.

Estimation of the jamming transition point in dimensions 2 to 22.

Abstract

Gradient descent dynamics in complex energy landscapes, i.e. featuring multiple minima, finds application in many different problems, from soft matter to machine learning. Here, we analyze one of the simplest examples, namely that of soft repulsive particles in the limit of infinite spatial dimension $d$. The gradient descent dynamics then displays a jamming transition: at low density, it reaches zero-energy states in which particles' overlaps are fully eliminated, while at high density the energy remains finite and overlaps persist. At the transition, the dynamics becomes critical. In the $d\to \infty$ limit, a set of self-consistent dynamical equations can be derived via mean field theory. We analyze these equations and we present some partial progress towards their solution. We also study the Random Lorentz Gas in a range of $d=2\ldots 22$, and obtain a robust estimate for the jamming transition in $d\to\infty$. The jamming transition is analogous to the capacity transition in supervised learning, and in the appendix we discuss this analogy in the case of a simple one-layer fully-connected perceptron.