Single-site dynamics are canonical Markov chain based algorithms for sampling from high-dimensional distributions, such as the Gibbs distributions of graphical models. We introduce a simple and generic parallel algorithm that faithfully simulates single-site dynamics. Under a much relaxed, asymptotic variant of the $\ell_p$-Dobrushin's condition -- where the Dobrushin's influence matrix has a bounded $\ell_p$-induced operator norm for an arbitrary $p\in[1, \infty]$ -- our algorithm simulates $N$ steps of single-site updates within a parallel depth of $O\left({N}/{n}+\log n\right)$ on $\tilde{O}(m)$ processors, where $n$ is the number of sites and $m$ is the size of the graphical model. For Boolean-valued random variables, if the $\ell_p$-Dobrushin's condition holds -- specifically, if the $\ell_p$-induced operator norm of the Dobrushin's influence matrix is less than~$1$ -- the parallel depth can be further reduced to $O(\log N+\log n)$, achieving an exponential speedup. These results suggest that single-site dynamics with near-linear mixing times can be parallelized into $\mathsf{RNC}$ sampling algorithms, independent of the maximum degree of the underlying graphical model, as long as the Dobrushin influence matrix maintains a bounded operator norm. We show the effectiveness of this approach with $\mathsf{RNC}$ samplers for the hardcore and Ising models within their uniqueness regimes, as well as an $\mathsf{RNC}$ SAT sampler for satisfying solutions of CNF formulas in a local lemma regime. Furthermore, by employing non-adaptive simulated annealing, these $\mathsf{RNC}$ samplers can be transformed into $\mathsf{RNC}$ algorithms for approximate counting.

翻译：单点动力学是基于马尔可夫链的经典算法，用于从高维分布（如图模型的吉布斯分布）中采样。我们提出了一种简单且通用的并行算法，能够忠实模拟单点动力学。在一个显著放宽的、渐近的 $\ell_p$-Dobrushin 条件变体下——即对于任意 $p\in[1, \infty]$，Dobrushin 影响矩阵具有有界的 $\ell_p$ 诱导算子范数——我们的算法在 $\tilde{O}(m)$ 个处理器上，以 $O\left({N}/{n}+\log n\right)$ 的并行深度模拟了 $N$ 步单点更新，其中 $n$ 是位点数，$m$ 是图模型的规模。对于布尔值随机变量，如果 $\ell_p$-Dobrushin 条件成立——具体而言，如果 Dobrushin 影响矩阵的 $\ell_p$ 诱导算子范数小于~$1$——并行深度可以进一步降至 $O(\log N+\log n)$，实现指数级加速。这些结果表明，只要 Dobrushin 影响矩阵保持有界算子范数，具有近线性混合时间的单点动力学可以被并行化为 $\mathsf{RNC}$ 采样算法，且独立于底层图模型的最大度数。我们通过 $\mathsf{RNC}$ 采样器在硬核模型和伊辛模型的唯一性区域内，以及在局部引理区域内用于 CNF 公式可满足解的 $\mathsf{RNC}$ SAT 采样器，展示了该方法的有效性。此外，通过采用非自适应模拟退火，这些 $\mathsf{RNC}$ 采样器可以转化为用于近似计数的 $\mathsf{RNC}$ 算法。