We study the {PAC} learnability of multiwinner voting, focusing on the class of approval-based committee scoring (ABCS) rules. These are voting rules applied on profiles with approval ballots, where each voter approves some of the candidates. According to ABCS rules, each committee of $k$ candidates collects from each voter a score, which depends on the size of the voter's ballot and on the size of its intersection with the committee. Then, committees of maximum score are the winning ones. Our goal is to learn a target rule (i.e., to learn the corresponding scoring function) using information about the winning committees of a small number of sampled profiles. Despite the existence of exponentially many outcomes compared to single-winner elections, we show that the sample complexity is still low: a polynomial number of samples carries enough information for learning the target rule with high confidence and accuracy. Unfortunately, even simple tasks that need to be solved for learning from these samples are intractable. We prove that deciding whether there exists some ABCS rule that makes a given committee winning in a given profile is a computationally hard problem. Our results extend to the class of sequential Thiele rules, which have received attention recently due to their simplicity.

翻译：我们研究多优胜者投票的PAC可学习性，重点关注基于批准的委员会评分（ABCS）规则。该类投票规则适用于采用批准票的投票档案，其中每位选民批准部分候选人。根据ABCS规则，每个包含k名候选人的委员会从每位选民处获得一个分数，该分数取决于选民选票的大小及其与委员会交集的大小。随后，得分最高的委员会成为获胜委员会。我们的目标是通过利用少量抽样档案中的获胜委员会信息来学习目标规则（即学习相应的评分函数）。尽管与单优胜者选举相比，结果数量呈指数级增长，但我们发现样本复杂度仍然较低：多项式数量的样本足以高置信度、高精度地学习目标规则。不幸的是，即使是从这些样本中学习所需解决的简单任务也难以处理。我们证明，判定是否存在某种ABCS规则使给定委员会在给定档案中获胜是一个计算难题。我们的结果扩展至近期因其简洁性而备受关注的顺序蒂勒规则类。