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January 2021 TAP free energy, spin glasses and variational inference
Zhou Fan, Song Mei, Andrea Montanari
Ann. Probab. 49(1): 1-45 (January 2021). DOI: 10.1214/20-AOP1443


We consider the Sherrington–Kirkpatrick model of spin glasses with ferromagnetically biased couplings. For a specific choice of the couplings mean, the resulting Gibbs measure is equivalent to the Bayesian posterior for a high-dimensional estimation problem known as “${\mathbb{Z}}_{2}$ synchronization.” Statistical physics suggests to compute the expectation with respect to this Gibbs measure (the posterior mean in the synchronization problem), by minimizing the so-called Thouless–Anderson–Palmer (TAP) free energy, instead of the mean field (MF) free energy. We prove that this identification is correct, provided the ferromagnetic bias is larger than a constant (i.e., the noise level is small enough in synchronization). Namely, we prove that the scaled $\ell _{2}$ distance between any low energy local minimizers of the TAP free energy and the mean of the Gibbs measure vanishes in the large size limit. Our proof technique is based on upper bounding the expected number of critical points of the TAP free energy using the Kac–Rice formula.


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Zhou Fan. Song Mei. Andrea Montanari. "TAP free energy, spin glasses and variational inference." Ann. Probab. 49 (1) 1 - 45, January 2021.


Received: 1 June 2019; Revised: 1 March 2020; Published: January 2021
First available in Project Euclid: 22 January 2021

Digital Object Identifier: 10.1214/20-AOP1443

Primary: 60F10

Keywords: Bayesian inference , Free probability , Kac–Rice formula , Sherrington–Kirkpatrick model , TAP complexity

Rights: Copyright © 2021 Institute of Mathematical Statistics


Vol.49 • No. 1 • January 2021
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