Sampling from Spherical Spin Glasses in Total Variation via Algorithmic Stochastic Localization (2404.15651v1)

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.

• The paper introduces a polynomial-time algorithm that leverages stochastic localization to sample from spherical spin glasses with strong total variation accuracy.
• The method refines the TAP free energy landscape using Approximate Message Passing, enhancing the estimator for the mean of tilted measures.
• Empirical and theoretical results demonstrate improved computational efficiency and potential applicability to high-dimensional inference beyond spin glasses.

Algorithmic Sampling from Spherical Spin Glasses Using Stochastic Localization

In this paper, the authors address the problem of efficient sampling from the Gibbs measure of a mixed $p$-spin spherical spin glass model. The proposed method leverages an algorithmic approach termed stochastic localization, which has gained traction for its potential to effectively tackle optimization and inference in complex high-dimensional settings such as spin glasses and neural networks.

Problem Setup and Objectives

The focus is on the mixed $p$-spin spherical spin glass, a fundamental model in statistical physics and probabilistic combinatorics that generalizes several classical models, such as the Sherrington-Kirkpatrick model. The aim is to devise a polynomial-time algorithm capable of sampling from the spin glass's Gibbs measure within a vanishing total variation error for a spectrum of models conforming to the mixture constraint $\xi''(s) < 1/(1-s)^2$ for $s \in [0,1)$.

The Sampling Algorithm

The proposed methodology utilizes an iterative algorithm inspired by the stochastic localization process, which progressively refines estimates of the Gibbs measure by simulating a stochastic differential equation (SDE). The main innovation is a polynomial-time algorithm that transitions from normalizing Wasserstein to the stronger total variation error guarantee in sampling fidelity. Key to this improvement is the enhancement of an estimator for the mean of tilted measures, achieved by refining the Thouless-Anderson-Palmer (TAP) free energy landscape via Approximate Message Passing (AMP). This estimator precisely adjusts the TAP fixed point, using corrections deriving from the underlying spin glass structure.

Theoretical Contributions

The authors extend the stochastic localization paradigm to spherical spin glasses by introducing several analytical results that enable rigorous control over the sampling error. They establish that the Gibbs measure is strongly log-concave under certain conditions, facilitating sampling through the Metropolis-adjusted Langevin algorithm (MALA). The proposed algorithm lowers the complexity of performing inference on such high-dimensional measures while promising accuracy in total variation distance.

Moreover, the paper provides a critical analysis contrasting the conditions for efficient algorithmic sampling with theoretical limits dictated by phase transitions such as shattering and replica symmetry breaking. This comparison elucidates the efficacy boundaries of stochastic localization vis-à-vis conjectured computational hardness in disordered systems.

Empirical and Practical Implications

Empirically, the implementation anticipates improvements in sampling complexity over prior models, aided by a significant reduction in the computational requirement for simulating non-trivial spin glass instances. Practically, the results suggest broader applicability to other models beyond spin glasses, potentially influencing sampling strategies in machine learning and Bayesian inference.

Prospects for Future Research

This work opens several avenues for future research, particularly in refining the algorithm's efficiency in the sampling of strongly correlated systems and extending the stochastic localization technique to a wider class of models. Additionally, deeper investigations into the performance guarantees across different models, particularly those exhibiting intricate energy landscapes, could further solidify the algorithm's utility in computational physics and statistical modeling.

In conclusion, the paper makes a consequential step forward in the efficient sampling from complex probabilistic models via stochastic localization, underpinning its potential both as a theoretical framework and a practical tool in scientific computing.