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

Brice Huang, Andrea Montanari, Huy Tuan Pham

TL;DR

The paper tackles the problem of efficiently sampling from Gibbs measures of mixed p-spin spherical spin glasses. It develops a polynomial-time algorithm built on algorithmic stochastic localization, augmented by a TAP-based mean estimator and an efficiently computable correction that achieves vanishing total-variation error under a curvature condition on the mixture. Key contributions include a refined estimator for tilted-means, a tractable localization scheme that reduces to strongly log-concave sampling, and a thorough analysis of the TAP landscape and its conditioning, including state-evolution and band-wise magnetization. The results bridge algorithmic sampling with the physics of shattering and replica symmetry breaking, and open ways to infer one-dimensional projections of the measure in high dimensions.

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.

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

TL;DR

The paper tackles the problem of efficiently sampling from Gibbs measures of mixed p-spin spherical spin glasses. It develops a polynomial-time algorithm built on algorithmic stochastic localization, augmented by a TAP-based mean estimator and an efficiently computable correction that achieves vanishing total-variation error under a curvature condition on the mixture. Key contributions include a refined estimator for tilted-means, a tractable localization scheme that reduces to strongly log-concave sampling, and a thorough analysis of the TAP landscape and its conditioning, including state-evolution and band-wise magnetization. The results bridge algorithmic sampling with the physics of shattering and replica symmetry breaking, and open ways to infer one-dimensional projections of the measure in high dimensions.

Abstract

We consider the problem of algorithmically sampling from the Gibbs measure of a mixed -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 This includes the pure -spin glasses above a critical temperature that is within an absolute (-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.
Paper Structure (37 sections, 84 theorems, 540 equations, 2 algorithms)

This paper contains 37 sections, 84 theorems, 540 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and related work
  3. Notations
  4. Main result
  5. Mean estimation of tilted measure
  6. Stochastic localization sampling
  7. The correction ${\boldsymbol \Delta}(\bm)$
  8. Fundamental limits of algorithmic SL, replica symmetry breaking, and shattering
  9. Preliminaries
  10. Stochastic localization
  11. Planted model and contiguity
  12. Basic regularity estimate
  13. Conditioning lemma
  14. Analysis of mean computation algorithm
  15. Analysis of AMP iteration: proof of Proposition \ref{['ppn:amp-performance']}
  16. ...and 22 more sections

Key Result

Theorem 2.1

Suppose $\xi$ satisfies eq:amp-works. There exist constants $K_{\hbox{\tiny\sf AMP}},K_{\mathsf{GD}}^*,\eta,T$ depending on ${\varepsilon}$ and $\xi$ such that running Algorithm alg:main with parameters $K_{\hbox{\tiny\sf AMP}}$, $K_{{\mathsf{GD}}}(N) = K_{\mathsf{GD}}^*\log N$, $\eta$, $T$, the fol Further the complexity of the algorithm is upper bounded by $CN^4\, (N + \chi_{\nabla H}) \log N+ \

Theorems & Definitions (171)

  • Remark 1.1
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 3.1
  • Lemma 3.2: Proved in Section \ref{['sec:partition-fn-fluctuations']}
  • Remark 3.3
  • Proposition 3.4: alaoui2022sampling
  • ...and 161 more