Fluctuation Distributions of Energy Minima in Complex Landscapes

TL;DR

This paper investigates the distribution of energy minima in complex landscapes, revealing non-Gaussian Tracy-Widom-like distributions and proposing a first-passage process model for gradient descent dynamics.

Contribution

It uncovers universal phenomenology of minima energy distributions and introduces a novel first-passage framework linking landscape properties to Tracy-Widom statistics.

Findings

Energy minima distributions are non-Gaussian and Tracy-Widom-like.

Finite-size scaling effects are observed in the distributions.

A first-passage process model explains gradient descent behavior.

Abstract

We discuss the properties of the distributions of energies of minima obtained by gradient descent in complex energy landscapes. We find strikingly similar phenomenology across several prototypical models. We particularly focus on the distribution of energies of minima in the analytically well-understood p-spin-interaction spin glass model. We numerically find non-Gaussian distributions that resemble the Tracy-Widom distributions often found in problems of random correlated variables, and non-trivial finite-size scaling. Based on this, we propose a picture of gradient descent dynamics that highlights the importance of a first-passage process in the eigenvalues of the Hessian. This picture provides a concrete link to problems in which the Tracy-Widom distribution is established. Aspects of this first-passage view of gradient-descent dynamics are generic for non-convex complex landscapes, rationalizing the commonality that we find across models.