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Sampling from Spherical Spin Glasses in Total Variation via Algorithmic Stochastic Localization

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arxiv 2404.15651 v1 pith:QWD7IPAB submitted 2024-04-24 math.PR cond-mat.dis-nnmath-phmath.MP

Brice Huang , Andrea Montanari , Huy Tuan Pham This is my paper · ORCID
classification math.PRcond-mat.dis-nnmath-phmath.MP
keywords algorithmspinmeasuretotalvariationalgorithmicapproacherror
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abstract

We consider the problem of algorithmically sampling from the Gibbs measure of a mixed $p$-spin spherical spin glass. We give a polynomial-time algorithm that samples from the Gibbs measure up to vanishing total variation error, for any model whose mixture satisfies $$\xi''(s) < \frac{1}{(1-s)^2}, \qquad \forall s\in [0,1).$$ This includes the pure $p$-spin glasses above a critical temperature that is within an absolute ($p$-independent) constant of the so-called shattering phase transition. Our algorithm follows the algorithmic stochastic localization approach introduced in (Alaoui, Montanari, Sellke, 20022). A key step of this approach is to estimate the mean of a sequence of tilted measures. We produce an improved estimator for this task by identifying a suitable correction to the TAP fixed point selected by approximate message passing (AMP). As a consequence, we improve the algorithm's guarantee over previous work, from normalized Wasserstein to total variation error. In particular, the new algorithm and analysis opens the way to perform inference about one-dimensional projections of the measure.

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Cited by 5 Pith papers

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  1. Polynomial-time sampling despite disorder chaos

    cs.CC 2025-08 conditional novelty 8.0 of 10

    Disorder chaos does not prevent polynomial-time Wasserstein sampling: Glauber dynamics samples the hardcore model on G(n,1/2) in O(n) time even though tiny graph perturbations radically change the target distribution.

  2. Sequential Dynamics in Ising Spin Glasses

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    Block-sequential updates on the SK model are exactly characterized by a system of integro-difference equations, conjectured to coincide with systematic scan dynamics as the block size vanishes.

  3. Weak Poincar\'e Inequalities via Approximate Stochastic Localization: Application to Sampling the Sherrington-Kirkpatrick Model

    math.PR 2026-07 conditional novelty 7.0 of 10

    Approximate stochastic localization plus conductance transfers yield a weak Poincaré inequality for the SK model at β < 1/2, enabling efficient Glauber sampling from a warm start.

  4. Markov Chains Approximate Message Passing

    cs.DS 2025-12 conditional novelty 6.0 of 10

    For spiked Wigner inference, Glauber dynamics and AMP reach the same correlation fixed point, with a phase transition at βλ=1 conditional on SK mixing.

  5. Sampling from Binary Quadratic Distributions via Stochastic Localization

    math.ST 2025-05 conditional novelty 6.0 of 10

    Stochastic localization drives binary quadratic posteriors into a strong-field regime where Glauber, Metropolis-Hastings, and DULA-type samplers satisfy Poincaré inequalities with polynomial mixing time.

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