Multiscale Parallel Tempering for Fast Sampling on Redistricting Plans
CoRR(2024)
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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