Sampling from the Sherrington-Kirkpatrick Gibbs measure via algorithmic stochastic localization
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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)∈{-1,+1}^n×{-1,+1}^n is a pair drawn from this coupling, then n^-1𝔼{||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.
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