Sampling from the Sherrington-Kirkpatrick Gibbs measure via algorithmic stochastic localization
Ahmed El Alaoui, Andrea Montanari, Mark Sellke
TL;DR
The paper develops an algorithmic stochastic localization framework to efficiently sample from the Sherrington-Kirkpatrick Gibbs measure μ_A at high temperature, achieving a polynomial-time (O(n^2)) sampler μ_A^{alg} that is close to μ_A in normalized Wasserstein distance for β<1/2. The approach combines stochastic localization with an AMP-based mean estimator and a TAP-guided natural gradient refinement, and crucially leverages a planted-model reduction and state-evolution analysis to establish performance guarantees. A complementary hardness result shows that no stable algorithm can approximately sample for β>1 due to disorder chaos, illustrating a sharp algorithmic barrier in the replica symmetry-broken phase. Collectively, the work advances the frontier of efficient sampling in spin glasses, providing both constructive methods and fundamental limits grounded in the Parisi framework and disorder-chaos phenomena.
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 $μ$ in polynomial time. We prove that, for any inverse temperature $β<1/2$, there exists an algorithm with complexity $O(n^2)$ that samples from a distribution $μ^{alg}$ which is close in normalized Wasserstein distance to $μ$. Namely, there exists a coupling of $μ$ and $μ^{alg}$ 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 $β<1/4$. 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 $β>1$, even under the normalized Wasserstein metric. Our sampling method is based on an algorithmic implementation of stochastic localization, which progressively tilts the measure $μ$ towards a single configuration, together with an approximate message passing algorithm that is used to approximate the mean of the tilted measure.