发表机构
Massachusetts Institute of Technology（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对PJR+和EJR+两种比例代表概念，分析了全见证与仅候选者两种违规反馈下的提案拒绝复杂度，给出了确定性与随机化场景下的拒绝界及对应算法。
AI 中文摘要
我们研究基于违规反馈的比例代表批准委员会学习问题。每一轮中，学习器会提出一个规模为$k$的委员会，预言机要么接受该提案，要么针对单个固定的隐藏批准概要以对抗方式选择一个代表违规情况。我们对比两种反馈：全见证反馈，会揭示违规程度、一个被遗漏的候选者以及受影响的选民群体；仅候选者反馈，仅揭示该候选者。目标概念是比例正当代表加（PJR+）和扩展正当代表加（EJR+）。在我们研究的所有场景中，被拒绝提案的数量仅可由$k$界定，与选民和候选者的数量无关。对于PJR+，两种反馈模型下最优的确定性和随机拒绝复杂度均等于$k$。对于EJR+，情况更微妙：在全见证反馈下，我们证明确定性下界为$\boldsymbol{\u03a9}(k^{3/2})$，并给出一个确定性多项式时间算法，使用$O(k^2\boldsymbol{\u006cog}k)$次拒绝；在仅候选者反馈下，随机化通过均匀随机删除实现$O(k^2\boldsymbol{\u006cog}k)$次预期拒绝，而确定性穷举分支给出$2^{O(k^2(\boldsymbol{\u006cog}k)^2)}$的拒绝界。即使有全见证反馈，随机学习器也可能需要$k$次拒绝。
英文摘要
We study violation-feedback learning of proportionally representative approval-based committees. In each round, a learner proposes a committee of size $k$. An oracle either accepts the proposal or adversarially selects a representation violation with respect to a single fixed hidden approval profile. We compare \emph{full-witness feedback}, which reveals the violation level, an omitted candidate, and the affected voter group, with \emph{candidate-only feedback}, which reveals only that candidate. The target notions are proportional justified representation plus (PJR+) and extended justified representation plus (EJR+). In every setting we study, the number of rejected proposals can be bounded solely in terms of $k$, with no dependence on the numbers of voters and candidates. For PJR+, the optimal deterministic and randomized rejection complexities equal $k$ under both feedback models. For EJR+, the picture is more nuanced. Under full-witness feedback, we prove an $Ω(k^{3/2})$ deterministic lower bound and give a deterministic polynomial-time algorithm using $O(k^2\log k)$ rejections. Under candidate-only feedback, randomization achieves $O(k^2\log k)$ expected rejections via uniform random deletion, while deterministic exhaustive branching gives a $2^{O(k^2(\log k)^2)}$ rejection bound. Even with full-witness feedback, randomized learners may require $k$ rejections.
