Sampling from the Sherrington-Kirkpatrick Gibbs measure via algorithmic stochastic localization

TL;DR

This paper presents a polynomial-time algorithm for sampling from the high-temperature Sherrington-Kirkpatrick spin glass model's Gibbs measure, extending previous results to higher temperatures and establishing limitations for stable algorithms at low temperatures.

Contribution

It introduces an efficient sampling algorithm for the SK model at inverse temperature below 1/2 using stochastic localization, and proves stability-based limitations for sampling at higher temperatures.

Findings

Efficient $O(n^2)$ sampling algorithm for $eta<1/2$

Approximate sampling close in Wasserstein distance

No stable algorithm can sample for $eta>1$

Abstract

We consider the Sherrington-Kirkpatrick model of spin glasses at high-temperature and no external field, and study the problem of sampling from the Gibbs distribution $\mu$ in polynomial time. We prove that, for any inverse temperature $\beta<1/2$, there exists an algorithm with complexity $O(n^2)$ that samples from a distribution $\mu^{alg}$ which is close in normalized Wasserstein distance to $\mu$. Namely, there exists a coupling of $\mu$ and $\mu^{alg}$ such that if $(x,x^{alg})\in\{-1,+1\}^n\times \{-1,+1\}^n$ is a pair drawn from this coupling, then $n^{-1}\mathbb E\{||x-x^{alg}||_2^2\}=o_n(1)$. The best previous results, by Bauerschmidt and Bodineau and by Eldan, Koehler, and Zeitouni, implied efficient algorithms to approximately sample (under a stronger metric) for $\beta<1/4$. We complement this result with a negative one, by introducing a suitable "stability" property for sampling algorithms, which is verified by many standard techniques. We prove that no stable algorithm can approximately sample for $\beta>1$, even under the normalized Wasserstein metric. Our sampling method is based on an algorithmic implementation of stochastic localization, which progressively tilts the measure $\mu$ towards a single configuration, together with an approximate message passing algorithm that is used to approximate the mean of the tilted measure.