The Pokémon Theorem and other Fairness Impossibility Results
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917477656559616 |
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| author | Smola, Daniel Matsui Smola, Alex |
| author_facet | Smola, Daniel Matsui Smola, Alex |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09221 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Pokémon Theorem and other Fairness Impossibility Results Smola, Daniel Matsui Smola, Alex Machine Learning Artificial Intelligence 68T05 (Primary) 46E22, 47B32, 62H30, 41A46, 62G10 (Secondary) I.2.6; G.3; K.4.1 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. |
| title | The Pokémon Theorem and other Fairness Impossibility Results |
| topic | Machine Learning Artificial Intelligence 68T05 (Primary) 46E22, 47B32, 62H30, 41A46, 62G10 (Secondary) I.2.6; G.3; K.4.1 |
| url | https://arxiv.org/abs/2605.09221 |