[Paper Review] Sampling from Mean-Field Gibbs Measures via Diffusion Processes

A. El Alaoui, Andrea Montanari+1 more|arXiv (Cornell University)|Oct 13, 2023

Markov Chains and Monte Carlo MethodsMathematics3 citations

TL;DR

This paper presents a novel diffusion-based sampling algorithm for mean-field Gibbs measures in high-temperature Ising mixed $p$-spin glasses, leveraging stochastic localization and approximate message passing to achieve polynomial-time sampling with $o_n(1)$ normalized Wasserstein error. The method succeeds in the full replica-symmetric phase, including the Sherrington-Kirkpatrick model, and establishes tight information-theoretic limits by proving no stable algorithm can sample accurately below the shattering or replica symmetry breaking thresholds.

ABSTRACT

We consider Ising mixed $p$-spin glasses at high-temperature and without external field, and study the problem of sampling from the Gibbs distribution $μ$ in polynomial time. We develop a new sampling algorithm with complexity of the same order as evaluating the gradient of the Hamiltonian and, in particular, at most linear in the input size. We prove that, at sufficiently high-temperature, it produces samples from a distribution $μ^{alg}$ which is close in normalized Wasserstein distance to $μ$. Namely, there exists a coupling of $μ$ and $μ^{alg}$ such that if $({\boldsymbol x},{\boldsymbol x}^{alg})\in\{-1,+1\}^n imes \{-1,+1\}^n$ is a pair drawn from this coupling, then $n^{-1}{\mathbb E}\{\|{\boldsymbol x}-{\boldsymbol x}^{alg}\|_2^2\}=o_n(1)$. For the case of the Sherrington-Kirkpatrick model, our algorithm succeeds in the full replica-symmetric phase. We complement this result with a negative one for sampling algorithms satisfying a certain `stability' property, which is verified by many standard techniques. No stable algorithm can approximately sample at temperatures below the onset of shattering, even under the normalized Wasserstein metric. Further, no algorithm can sample at temperatures below the onset of replica symmetry breaking. Our sampling method implements a discretized version of a diffusion process that has become recently popular in machine learning under the name of `denoising diffusion.' We derive the same process from the general construction of stochastic localization. Implementing the diffusion process requires to efficiently approximate the mean of the tilted measure. To this end, we use an approximate message passing algorithm that, as we prove, achieves sufficiently accurate mean estimation.

Motivation & Objective

• To develop a polynomial-time sampling algorithm for Ising mixed $p$-spin glasses at high temperature.
• To achieve accurate sampling in normalized Wasserstein distance, ensuring $n^{-1}\mathbb{E}\|\mathbf{x}-\mathbf{x}^{\text{alg}}\|_2^2 = o_n(1)$.
• To establish information-theoretic limits by proving that no stable algorithm can sample accurately below the shattering or replica symmetry breaking thresholds.
• To connect diffusion processes to stochastic localization and derive a tractable sampling scheme via approximate message passing for mean estimation.

Proposed method

• The algorithm uses a discretized diffusion process derived from stochastic localization, which evolves a system toward the target Gibbs measure.
• It relies on a continuous-time stochastic localization process that tilts the measure toward the target distribution.
• The mean of the tilted measure is approximated using an approximate message passing (AMP) algorithm with proven convergence to the state evolution fixed point.
• The AMP algorithm is shown to achieve mean squared error bounds that ensure the overall sampling error remains $o_n(1)$ in normalized Wasserstein distance.
• The method is implemented via a natural gradient descent step in the continuous-time limit, ensuring efficient exploration of the state space.
• Theoretical guarantees are derived using contiguity arguments and convergence analysis of the AMP state evolution under the planted model.

Experimental results

Research questions

• RQ1 Can a diffusion-based algorithm achieve efficient, polynomial-time sampling from mean-field Gibbs measures in the high-temperature regime?
• RQ2 What is the fundamental limit of stable sampling algorithms in the presence of shattering or replica symmetry breaking?
• RQ3 How accurately can approximate message passing estimate the mean of a tilted Gibbs measure in spin glass models?
• RQ4 Does the proposed diffusion process yield samples close to the true Gibbs measure in normalized Wasserstein distance?
• RQ5 Can the algorithm succeed in the full replica-symmetric phase, including the Sherrington-Kirkhardt model?

Key findings

• The proposed algorithm achieves $n^{-1}\mathbb{E}\|\mathbf{x}-\mathbf{x}^{\text{alg}}\|_2^2 = o_n(1)$, meaning the average squared distance between true and algorithmic samples vanishes as $n \to \infty$.
• The algorithm succeeds in the full replica-symmetric phase of the mixed $p$-spin model, including the Sherrington-Kirkpatrick model.
• No stable algorithm can approximately sample from the Gibbs measure below the onset of shattering, even under the normalized Wasserstein metric.
• No algorithm can sample accurately below the replica symmetry breaking threshold, establishing a sharp information-theoretic barrier.
• The AMP algorithm used for mean estimation converges to the state evolution fixed point and achieves sufficiently accurate mean estimation to ensure overall sampling accuracy.
• The hardness results are derived via contiguity and disorder chaos arguments, showing that stability implies failure in the non-replica-symmetric regime.