Simulated annealing and Weak Poincaré inequalities with inverse-polynomial accuracy for the Sherrington-Kirkpatrick model

2026-10-07

AI Summary

This study addresses the efficient sampling of the Gibbs distribution and the estimation of the partition function for the Sherrington–Kirkpatrick model. To this end, it proposes a polynomial-time simulated annealing algorithm that integrates weak Poincaré inequalities with Thouless–Anderson–Palmer (TAP) free energy approximations, combining Glauber dynamics, stochastic localization, and approximate message passing techniques. The core innovation lies in leveraging Gaussian isoperimetry to control the cumulative cost arising from local Lipschitz failures, thereby circumventing an open problem posed by Talagrand. Consequently, this work achieves inverse-polynomial accuracy in both sampling and partition function estimation, ensuring that the output distribution approximates the target distribution within a distance of n^{-M} with probability at least 1-n^{-D}. These results provide rigorous theoretical guarantees for efficient sampling in spin glass models.

Abstract

We study sampling from the Gibbs distribution of the Sherrington-Kirkpatrick (SK) model with Glauber dynamics. For every fixed inverse temperature $0\leqβ<1$ and every fixed $M,D>0$, we give a polynomial-time simulated annealing algorithm whose output distribution is within $n^{-M}$ total-variation distance of the Gibbs distribution, with probability at least $1-n^{-D}$ over the interaction matrix. The algorithm starts from uniform product spins and uses Glauber dynamics along an increasing inverse-temperature schedule. The same approach also yields partition-function estimates with relative error $n^{-M}$. The key ingredient is a quantitative weak Poincaré inequality. Building on the stochastic-localization approach to weak Poincaré inequalities [Davies, Lee, Sandhu, and Shi, arXiv:2607.08160, 2026], we use Gaussian isoperimetry to directly compare exact stochastic localization paths. Our comparison tolerates failures of local Lipschitz continuity of the posterior mean by controlling their accumulated cost. The local Lipschitzness comes from approximating the posterior means by stationary points of the Thouless-Anderson-Palmer (TAP) free energy in locally strongly convex regions located by approximate message passing, a paradigm introduced by algorithmic stochastic localization [El Alaoui, Montanari, and Sellke, 2025; Celentano, 2024]. The approximation errors are roughly characterized by the validity of the TAP gradient equations, but in a different coordinate. To show that the errors rarely accumulate too much, we need a strong concentration control. A direct argument would require solving a variant of an open problem by Talagrand [2010, Research Problem 1.7.9]. We bypass this open problem with an intermediate conditioning step and transfer the control back by establishing the stability of the TAP gradient under deletion of coordinates.

Problem

• Sherrington-Kirkpatrick model
• Gibbs sampling
• Glauber dynamics
• simulated annealing
• partition function estimation

Innovation

• Simulated Annealing
• Weak Poincaré Inequality
• Stochastic Localization
• TAP Free Energy
• Approximate Message Passing