On convergence of the cavity and Bolthausen's TAP iterations to the local magnetization

TL;DR

This paper proves that a new iterative scheme, related to the cavity and Bolthausen's TAP iterations, converges to the local magnetization in the Sherrington-Kirkpatrick model under certain conditions, linking physics and algorithmic approaches.

Contribution

Introduces a new iterative scheme motivated by cavity equations and proves its convergence to local magnetization, aligning with Bolthausen's iteration and AMP algorithms.

Findings

New iterative scheme converges to local magnetization.

Establishes connection between cavity equations and AMP algorithms.

Proves convergence under local overlap concentration conditions.

Abstract

The cavity and TAP equations are high-dimensional systems of nonlinear equations of the local magnetization in the Sherrington-Kirkpatrick model. In the seminal work [Comm. Math. Phys., 325(1):333-366, 2014], Bolthausen introduced an iterative scheme that produces an asymptotic solution to the TAP equations if the model lies inside the Almeida-Thouless transition line. However, it was unclear if this asymptotic solution coincides with the local magnetization. In this work, motivated by the cavity equations, we introduce a new iterative scheme and establish a weak law of large numbers. We show that our new scheme is asymptotically the same as the so-called Approximate Message Passing algorithm, a generalization of Bolthausen's iteration, that has been popularly adapted in compressed sensing, Bayesian inferences, etc. Based on this, we confirm that our cavity iteration and Bolthausen's scheme both converge to the local magnetization as long as the overlap is locally uniformly concentrated.