用于选区重划方案快速采样的多尺度并行回火算法
Multiscale Parallel Tempering for Fast Sampling on Redistricting Plans
AI总结：
针对选区重划方案审计中现有采样算法在大规模图上混合速度慢的问题，提出多尺度并行回火算法，通过各尺度局部移动适配多种政策指标，在康涅狄格州实现了该规模下从未有过的政策分布快速采样。
摘要：
在审计选区重划方案时，一种有说服力的方法是将该方案与一组中立生成的选区重划方案集合进行对比。方案集合通过对平衡图分区上的分布进行采样的算法生成。要审计方案集合与给定方案之间的党派差异，必须确保非党派标准相匹配，这样我们才能得出党派差异源于偏见，而非例如紧凑度水平或社区保留差异等因素的结论。某些采样算法允许人们明确设定方案上基于政策的概率分布，但这些算法在大规模图（即选区重划空间）上的混合时间表现很差，仅适用于少数特殊指标。在本研究中，我们提出了一种多尺度并行回火方法，该方法在每个尺度上执行局部移动。这些局部移动使我们能够采用多种基于政策的指标。我们在康涅狄格州对方法进行了检验，并成功在一种基于政策的分布上实现了快速混合，而这种分布在此规模下从未被采样过。我们的算法有望扩展到更广泛的指标类别，这将（i）支持更具原则性、基于具体情境的比较，以及（ii）探究政策对选区重划可能产生的典型党派影响。
英文摘要：
When auditing a redistricting plan, a persuasive method is to compare the plan with an ensemble of neutrally drawn redistricting plans. Ensembles are generated via algorithms that sample distributions on balanced graph partitions. To audit the partisan difference between the ensemble and a given plan, one must ensure that the non-partisan criteria are matched so that we may conclude that partisan differences come from bias rather than, for example, levels of compactness or differences in community preservation. Certain sampling algorithms allow one to explicitly state the policy-based probability distribution on plans, however, these algorithms have shown poor mixing times for large graphs (i.e. redistricting spaces) for all but a few specialized measures. In this work, we generate a multiscale parallel tempering approach that makes local moves at each scale. The local moves allow us to adopt a wide variety of policy-based measures. We examine our method in the state of Connecticut and succeed at achieving fast mixing on a policy-based distribution that has never before been sampled at this scale. Our algorithm shows promise to expand to a significantly wider class of measures that will (i) allow for more principled and situation-based comparisons and (ii) probe for the typical partisan impact that policy can have on redistricting.
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