TAP free energy, spin glasses, and variational inference

Abstract:
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 “Z2 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 l_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.

This is joint work with Song Mei and Andrea Montanari.

Biography:
Zhou Fan is an Assistant Professor in the Department of Statistics and Data Science at Yale University. His research interests include random matrix theory, high dimensional and multivariate statistics, inference in random graphs and networks, discrete algorithms, and applications in genetics and computational biology. Zhou received his Ph.D. in Statistics at Stanford University, working with Iain M. Johnstone and Andrea Montanari. Prior to this, Zhou developed statistical and software tools for molecular dynamics simulations at D. E. Shaw Research.

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