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.

翻译：我们考虑高温无外场条件下自旋玻璃的Sherrington-Kirkpatrick模型，研究在多项式时间内从吉布斯分布$\mu$中采样的问题。我们证明：对于任意逆温度$\beta<1/2$，存在一个复杂度为$O(n^2)$的算法，可从与$\mu$在归一化Wasserstein距离上接近的分布$\mu^{alg}$中采样。具体而言，存在$\mu$与$\mu^{alg}$的一个耦合，使得若从该耦合中抽取配对$(x,x^{alg})\in\{-1,+1\}^n\times \{-1,+1\}^n$，则有$n^{-1}\mathbb E\{||x-x^{alg}||_2^2\}=o_n(1)$。此前Bauerschmidt与Bodineau以及Eldan、Koehler与Zeitouni的最佳结果，仅能在更强度量下为$\beta<1/4$情况提供近似采样的高效算法。我们通过引入适用于采样算法的特定"稳定性"性质（该性质被许多标准技术所满足），用负面结果补充了这一结论：我们证明即使采用归一化Wasserstein度量，对于$\beta>1$的情况，任何稳定算法都无法实现近似采样。我们的采样方法基于随机局部化的算法实现——逐步将测度$\mu$倾斜至单一构型，同时结合近似消息传递算法来近似倾斜测度的均值。