Efficient Parallel Ising Samplers via Localization Schemes

TL;DR

The paper develops parallel samplers for Ising-type distributions by leveraging localization schemes that yield global Markov chains with polylogarithmic depth and polynomial total work. It provides two main algorithms: a field-dynamics–based sampler for ferromagnetic Ising with external fields and a restricted-Gaussian-dynamics–based sampler for Ising with contracting $J$, both equipped with a coupling-with-stationary analysis to enable efficient parallelization beyond standard Dobrushin conditions. The work uses the Edwards–Sokal coupling to connect random-cluster sampling with Ising sampling, and demonstrates explicit complexity bounds: an $\mathsf{RNC}$ sampler for ferromagnetic Ising under nonzero fields with depth $(\epsilon^{-1/\log n}\log n)^{O_\delta(1)}$ and $O_\delta(m^2\log(n/\epsilon))$ processors, and an $\mathsf{RNC}$ sampler for Ising with $\|J\|_2<1$ achieving $O_\eta(\log^4(n/\epsilon))$ depth on $\widetilde{O}_\eta(n^3/\epsilon^2)$ processors. These results advance parallel sampling and partition-function estimation for Ising-like models, enabling scalable simulations in regimes previously considered challenging for parallel methods.

Abstract

We introduce efficient parallel algorithms for sampling from the Gibbs distribution and estimating the partition function of Ising models. These algorithms achieve parallel efficiency, with polylogarithmic depth and polynomial total work, and are applicable to Ising models in the following regimes: (1) Ferromagnetic Ising models with external fields; (2) Ising models with interaction matrix $J$ of operator norm $\|J\|_2<1$. Our parallel Gibbs sampling approaches are based on localization schemes, which have proven highly effective in establishing rapid mixing of Gibbs sampling. In this work, we employ two such localization schemes to obtain efficient parallel Ising samplers: the \emph{field dynamics} induced by \emph{negative-field localization}, and \emph{restricted Gaussian dynamics} induced by \emph{stochastic localization}. This shows that localization schemes are powerful tools, not only for achieving rapid mixing but also for the efficient parallelization of Gibbs sampling.

Theorems & Definitions (30)

• Theorem 1.1: ferromagnetic Ising sampler
• Theorem 1.2: random cluster sampler
• Theorem 1.3: Ising sampler
• Remark 1.4
• Theorem 3.1
• Lemma 3.2: feng2022sampling
• Lemma 3.3
• proof
• proof : Proof of \ref{['theorem:main-ising']}
• Theorem 4.1