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

TL;DR

This paper introduces a polynomial-time algorithm for sampling from the Gibbs measure of mixed p-spin spherical spin glasses with high accuracy, using stochastic localization and improved estimators, applicable near the shattering phase transition.

Contribution

The paper develops a new algorithm based on stochastic localization and refined estimators to sample efficiently from spherical spin glasses in a broader parameter regime.

Findings

Achieves sampling up to vanishing total variation error in polynomial time.

Extends applicability to models near the shattering phase transition.

Improves estimator accuracy for mean estimation of tilted measures.

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.