AI 中文总结
研究从高温伊辛混合p-自旋吉布斯测度采样的并行复杂度，提出两种算法。一种结合块动力学与近似拒绝采样，另一种用皮卡迭代并行化ASL过程，相比朴素ASL在运行时和工作量的\(\varepsilon\)依赖性上有显著改进。
AI 中文摘要
我们研究了从高温伊辛混合p-自旋吉布斯测度采样的并行复杂度，这是超立方体{\pm 1}^n上平均场自旋玻璃的典型实例。我们提出了两种不同算法，对应两种不同精度 regime。第一种算法是马尔可夫链块动力学的并行实现，结合近似拒绝采样步骤，用伊辛模型以新方式作为提议分布近似p-自旋哈密顿量的二次相互作用项。对于任意\(\varepsilon >\) 0，该算法在\(n^{\tfrac{1}{3}}\operatorname{polylog}(\tfrac{n}{\varepsilon})\)并行时间内运行，工作量为\(\operatorname{poly}(n,\log(\tfrac{1}{\varepsilon}))\)，输出的样本在总变差距离上与p-自旋测度\(\varepsilon -\)接近。第二种算法使用皮卡迭代并行化El Alaoui、Montanari和Sellke（2025）的算法随机局部化（ASL）过程，对于任意\(\varepsilon >\varepsilon_n\)，在\(\operatorname{polylog}(\tfrac{n}{\varepsilon})\)并行时间和\(\operatorname{poly}(\tfrac{n}{\varepsilon})\)工作量下产生一个在归一化2-瓦瑟斯坦度量下与p-自旋测度\(\varepsilon -\)接近的样本。这里，\(\varepsilon_n > 0\)是一个当\(n\to\infty\)时趋于0的阈值。与朴素ASL相比，我们在运行时的\(\varepsilon\)依赖性上有双指数改进，在总工作量的\(\varepsilon\)依赖性上有指数改进，朴素ASL的运行时规模为\(\exp(\operatorname{poly}(\tfrac{1}{\varepsilon}))\)。
英文摘要
We study the parallel complexity of sampling from the high-temperature Ising mixed $p$-spin Gibbs measure, a canonical instance of a mean-field spin glass on the hypercube $\{\pm 1\}^n$. We propose two different algorithms for this problem, corresponding to two different regimes of accuracy. Our first algorithm is a parallel implementation of a Markov chain known as block dynamics, combined with an approximate rejection sampling step that uses an Ising model in a novel way as a proposal distribution to approximate the quadratic interaction terms of the $p$-spin Hamiltonian. For any $\varepsilon > 0$, this algorithm runs in $n^{\tfrac{1}{3}}\operatorname{polylog}(\tfrac{n}{\varepsilon})$ parallel time with $\operatorname{poly}(n, \log(\tfrac{1}{\varepsilon}))$ work, and outputs a sample whose law is $\varepsilon$-close to the $p$-spin measure in total variation distance. Our second algorithm uses Picard iterations to parallelize the Algorithmic Stochastic Localization (ASL) process of El Alaoui, Montanari, and Sellke (2025), and for any $\varepsilon > \varepsilon_n$, takes $\operatorname{polylog}(\tfrac{n}{\varepsilon})$ parallel time and $\operatorname{poly}(\tfrac{n}{\varepsilon})$ work to produce a sample that is $\varepsilon$-close to the $p$-spin measure in the normalized 2-Wasserstein metric. Here, $\varepsilon_n > 0$ is a threshold that goes to $0$ as $n \to \infty$. Our result constitutes a doubly exponential improvement in the $\varepsilon$ dependence of the runtime and an exponential improvement in the $\varepsilon$ dependence of the total work when compared to naïve ASL, whose runtime scales as $\exp(\operatorname{poly}(\tfrac{1}{\varepsilon}))$.
CommentsRANDOM 2026, to appear
