November 2021 Optimization of mean-field spin glasses
Ahmed El Alaoui, Andrea Montanari, Mark Sellke
Author Affiliations +
Ann. Probab. 49(6): 2922-2960 (November 2021). DOI: 10.1214/21-AOP1519


Mean-field spin glasses are families of random energy functions (Hamiltonians) on high-dimensional product spaces. In this paper, we consider the case of Ising mixed p-spin models,; namely, Hamiltonians HN:ΣNR on the Hamming hypercube ΣN={±1}N, which are defined by the property that {HN(σ)}σΣN is a centered Gaussian process with covariance E{HN(σ1)HN(σ2)} depending only on the scalar product σ1,σ2.

The asymptotic value of the optimum maxσΣNHN(σ) was characterized in terms of a variational principle known as the Parisi formula, first proved by Talagrand and, in a more general setting, by Panchenko. The structure of superlevel sets is extremely rich and has been studied by a number of authors. Here, we ask whether a near optimal configuration σ can be computed in polynomial time.

We develop a message passing algorithm whose complexity per-iteration is of the same order as the complexity of evaluating the gradient of HN, and characterize the typical energy value it achieves. When the p-spin model HN satisfies a certain no-overlap gap assumption, for any ε>0, the algorithm outputs σΣN such that HN(σ)(1ε)maxσHN(σ), with high probability. The number of iterations is bounded in N and depends uniquely on ε. More generally, regardless of whether the no-overlap gap assumption holds, the energy achieved is given by an extended variational principle, which generalizes the Parisi formula.

Funding Statement

This work was partially supported by NSF Grants CCF-1714305, IIS-1741162 and ONR N00014-18-1-2729.


Download Citation

Ahmed El Alaoui. Andrea Montanari. Mark Sellke. "Optimization of mean-field spin glasses." Ann. Probab. 49 (6) 2922 - 2960, November 2021.


Received: 1 March 2020; Revised: 1 February 2021; Published: November 2021
First available in Project Euclid: 7 December 2021

MathSciNet: MR4348682
zbMATH: 07467487
Digital Object Identifier: 10.1214/21-AOP1519

Primary: 68Q87 , 82C44
Secondary: 60K35

Keywords: finding ground states , optimization algorithms , Spin glasses , the Parisi formula

Rights: Copyright © 2021 Institute of Mathematical Statistics


This article is only available to subscribers.
It is not available for individual sale.

Vol.49 • No. 6 • November 2021
Back to Top