什么样的产品可以在当地进行采样？
What can be sampled locally?
摘要：
Linial [FOCS'87]和Naor和Stockmeyer [STOC'93]的局部计算关注的问题是，局部可定义的分布式计算问题是否可以局部求解：对于给定的局部CSP，CSP解是否可以通过使用局部信息的分布式算法构造。本文考虑了用分布式算法对一致CSP解进行抽样的问题，并提出了一个局部可定义的联合分布是否可以从局部抽样的问题。更广泛地说，我们考虑从吉布斯分布中抽样加权本地CSP在ESTA模型。我们给出了两个马尔可夫链为基础的分布式算法，我们相信代表两个基本的方法，通过分布式算法从吉布斯分布采样。第一种算法通过并行迭代更新一组随机独立的变量，将单站点序列马尔可夫链一般并行化，并在满足吉布斯分布的Dobrushin条件时，在Markov模型中实现了$O(Δ\log n)$时间上界，其中$Δ$是最大度.第二种算法是一种新的并行马尔可夫链，它建议同时更新所有变量，但仍然保证正确收敛，没有偏见。它令人惊讶地并行化了一个内在的顺序过程：稳定到一个具有大量局部依赖性的联合分布，并且可以在更强的混合条件下实现与最大程度$Δ$无关的最佳$O(\log n)$时间上限。我们还证明了一个强$Ω(diam)$下界的最大程度的图$Δ\ge 6$的采样独立集。即使每个节点都知道图，这个下界也成立。这给出了采样和构造局部可检查标签之间的强分离。
英文摘要：
The local computation of Linial [FOCS'87] and Naor and Stockmeyer [STOC'93] concerns with the question of whether a locally definable distributed computing problem can be solved locally: for a given local CSP whether a CSP solution can be constructed by a distributed algorithm using local information. In this paper, we consider the problem of sampling a uniform CSP solution by distributed algorithms, and ask whether a locally definable joint distribution can be sampled from locally. More broadly, we consider sampling from Gibbs distributions induced by weighted local CSPs in the LOCAL model. We give two Markov chain based distributed algorithms which we believe to represent two fundamental approaches for sampling from Gibbs distributions via distributed algorithms. The first algorithm generically parallelizes the single-site sequential Markov chain by iteratively updating a random independent set of variables in parallel, and achieves an $O(Δ\log n)$ time upper bound in the LOCAL model, where $Δ$ is the maximum degree, when the Dobrushin's condition for the Gibbs distribution is satisfied. The second algorithm is a novel parallel Markov chain which proposes to update all variables simultaneously yet still guarantees to converge correctly with no bias. It surprisingly parallelizes an intrinsically sequential process: stabilizing to a joint distribution with massive local dependencies, and may achieve an optimal $O(\log n)$ time upper bound independent of the maximum degree $Δ$ under a stronger mixing condition. We also show a strong $Ω(diam)$ lower bound for sampling independent set in graphs with maximum degree $Δ\ge 6$. This lower bound holds even when every node is aware of the graph. This gives a strong separation between sampling and constructing locally checkable labelings.
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