揭示公平性悖论的几何本质，证明多准则公平难以同时满足。
The Pokémon Theorem and other Fairness Impossibility Results
- 用核方法统一建模公平性约束，揭示其几何共性。
- 提出'宝可梦定理'：满足部分公平条件仍留残余不公，误差按科莫戈罗夫宽度衰减。
- 适用于研究公平机器学习的学者，尤其关注公平性权衡与不可行性分析者。
公平性不可能性结果常表现为独立的标量不相容性陈述。我们发现多个结果共享一种再生核希尔伯特空间（RKHS）几何结构：公平性标准是条件均值嵌入上的线性约束，而不同群体基数差异使全期望定律过度确定这些约束。该视角导出四个结论：Kleinberg-Mullainathan-Raghavan二分法仅需组条件无偏，无需完整校准；提出的‘宝可梦定理’表明，任意满足有限个线性均值公平准则的群体对，必然存在由最大均值差异（MMD）见证的残余违反，且在谱正则条件下以科莫戈罗夫m-宽度速率衰减；相同工具证明公平特征学习的不可能性：表示空间中平权与类条件分离强制导致类别坍塌，当基数不等时尤为显著；近似松弛形式给出信号与误差边界，实现现实估计器与公平目标间的权衡。在标准公平性基准上的实验结果与理论界限一致。
原文摘要 · Abstract (English)
Fairness impossibility results often look like distinct scalar incompatibility statements. We show that several share one RKHS geometry: fairness criteria are linear constraints on conditional mean embeddings, and unequal base rates make the law of total expectation overdetermine those constraints. This view yields four results. The Kleinberg--Mullainathan--Raghavan dichotomy needs only group-conditional unbiasedness, not full calibration. The \emph{Pokémon theorem} shows that a distinct group pair satisfying any finite collection of linear mean-fairness criteria leaves a residual violation witnessed by the MMD, decaying at the Kolmogorov $m$-width rate under spectral regularity. The same tools prove an impossibility for fair feature learning: parity and class-conditional separation in representation space force class collapse under unequal base rates. The approximate relaxations yield signal and error frontiers, allowing a trade-off between real-world estimators and fairness goals. Experiments on standard fairness benchmarks are consistent with our bounds.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。