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arXiv · 2404.15651

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

Brice Huang·Andrea Montanari·Huy Tuan Pham
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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 $$ξ''(s) < \frac{1}{(1-s)^2}, \qquad \forall s\in [0,1).$$ 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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Brice Huang, Andrea Montanari, Huy Tuan Pham. 2024-04-24. Sampling from Spherical Spin Glasses in Total Variation via Algorithmic Stochastic Localization. https://arxiv.org/abs/2404.15651

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