research-article

Simple parallel algorithms for single-site dynamics

Authors:

Yitong YinAuthors Info & Claims

STOC 2022: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing

Pages 1431 - 1444

https://doi.org/10.1145/3519935.3519999

Published: 10 June 2022 Publication History

Abstract

The single-site dynamics are a canonical class of Markov chains for sampling from high-dimensional probability distributions, e.g. the ones represented by graphical models.

We give a simple and generic parallel algorithm that can faithfully simulate single-site dynamics. When the chain asymptotically satisfies the ℓ_p-Dobrushin’s condition, specifically, when the Dobrushin’s influence matrix has constantly bounded ℓ_p-induced operator norm for an arbitrary p∈[1, ∞], the parallel simulation of N steps of single-site updates succeeds within O(N/n+logn) depth of parallel computing using Õ(m) processors, where n is the number of sites and m is the size of graphical model. Since the Dobrushin’s condition is almost always satisfied asymptotically by mixing chains, this parallel simulation algorithm essentially transforms single-site dynamics with optimal O(nlogn) mixing time to algorithms for sampling. In particular we obtain samplers, for the Ising models on general graphs in the uniqueness regime, and for satisfying solutions of CNF formulas in a local lemma regime. With non-adaptive simulated annealing, these samplers can be transformed routinely to algorithms for approximate counting.

A key step in our parallel simulation algorithm, is a so-called “universal coupling” procedure, which tries to simultaneously couple all distributions over the same sample space. We construct such a universal coupling, that for every pair of distributions the coupled probability is at least their Jaccard similarity. We also prove that this is optimal in the worst case. The universal coupling and its applications are of independent interests.

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Cited By

Anari NHuang YLiu TVuong TXu BYu KSaha BServedio R(2023)Parallel Discrete Sampling via Continuous WalksProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585207(103-116)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585207

Index Terms

Simple parallel algorithms for single-site dynamics
1. Theory of computation
  1. Design and analysis of algorithms
    1. Parallel algorithms

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cover image ACM Conferences

STOC 2022: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing

June 2022

1698 pages

ISBN:9781450392648

DOI:10.1145/3519935

General Chair:
Stefano Leonardi
Sapienza University of Rome, Italy
,
Program Chair:
Anupam Gupta
Carnegie Mellon University, USA

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Published: 10 June 2022

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STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing

June 20 - 24, 2022

Rome, Italy

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Cited By

Anari NHuang YLiu TVuong TXu BYu KSaha BServedio R(2023)Parallel Discrete Sampling via Continuous WalksProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585207(103-116)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585207

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