AMP algorithms and Stein's method: Understanding TAP equations with a new method

TL;DR

This paper introduces a new iterative method for solving TAP equations in the SK model, utilizing Stein's method to analyze convergence and Gaussian limits across all temperatures.

Contribution

It presents a novel iterative construction for TAP solutions that converges up to the AT line, employing Stein's method for mean field analysis.

Findings

Converges to solutions of TAP equations with finite-size Onsager correction.

Weak convergence of effective fields to Gaussian distributions at all temperatures.

Applicable to models with TAP-like equations and Stein operators.

Abstract

We propose a new iterative construction of solutions of the classical TAP equations for the Sherrington-Kirkpatrick model, i.e. with finite-size Onsager correction. The algorithm can be started in an arbitrary point, and converges up to the AT line. The analysis relies on a novel treatment of mean field algorithms through Stein's method. As such, the approach also yields weak convergence of the effective fields at all temperatures towards Gaussians, and can be applied, upon proper alterations, to all models where TAP-like equations and a Stein-operator are available.