# Impossibility results for fair representations

arXiv: https://arxiv.org/abs/2107.03483 · HTML: https://mlsanity.com/paper/2107.03483/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 16 references that are in the map.
- First posted on arXiv 7 Jul 2021.

## Builds on (most cited first, 9 of 9)

- [Equality of Opportunity in Supervised Learning](https://mlsanity.com/paper/1610.02413/) (2016)
- [Mitigating Unwanted Biases with Adversarial Learning](https://mlsanity.com/paper/1801.07593/) (2018)
- [Inherent Trade-Offs in the Fair Determination of Risk Scores](https://mlsanity.com/paper/1609.05807/) (2016)
- [Learning Adversarially Fair and Transferable Representations](https://mlsanity.com/paper/1802.06309/) (2018)
- [Flexibly Fair Representation Learning by Disentanglement](https://mlsanity.com/paper/1906.02589/) (2019)
- [Censoring Representations with an Adversary](https://mlsanity.com/paper/1511.05897/) (2015)
- [Data Decisions and Theoretical Implications when Adversarially Learning Fair Representations](https://mlsanity.com/paper/1707.00075/) (2017)
- [Learning Controllable Fair Representations](https://mlsanity.com/paper/1812.04218/) (2018)
- [Learning Fair and Transferable Representations](https://mlsanity.com/paper/1906.10673/) (2019)

## Built on by (most cited first, 10 of 10)

- [The Impossibility of Fair LLMs](https://mlsanity.com/paper/2406.03198/) (2024)
- [FARE: Provably Fair Representation Learning with Practical Certificates](https://mlsanity.com/paper/2210.07213/) (2022)
- [Fair Representation: Guaranteeing Approximate Multiple Group Fairness for Unknown Tasks](https://mlsanity.com/paper/2109.00545/) (2021)
- [f-FERM: A Scalable Framework for Robust Fair Empirical Risk Minimization](https://mlsanity.com/paper/2312.03259/) (2023)
- [Information-Theoretic Bounds on The Removal of Attribute-Specific Bias From Neural Networks](https://mlsanity.com/paper/2310.04955/) (2023)
- [On Optimal Steering to Achieve Exact Fairness](https://mlsanity.com/paper/2509.15759/) (2025)
- [Dr. FERMI: A Stochastic Distributionally Robust Fair Empirical Risk Minimization Framework](https://mlsanity.com/paper/2309.11682/) (2023)
- [SABAF: Removing Strong Attribute Bias from Neural Networks with Adversarial Filtering](https://mlsanity.com/paper/2311.07141/) (2023)
- [A Critical Review of Predominant Bias in Neural Networks](https://mlsanity.com/paper/2502.11031/) (2025)
- [The Pok'emon Theorem and other Fairness Impossibility Results](https://mlsanity.com/paper/2605.09221/) (2026)

## Abstract (arXiv)

With the growing awareness to fairness in machine learning and the realization of the central role that data representation has in data processing tasks, there is an obvious interest in notions of fair data representations. The goal of such representations is that a model trained on data under the representation (e.g., a classifier) will be guaranteed to respect some fairness constraints. Such representations are useful when they can be fixed for training models on various different tasks and also when they serve as data filtering between the raw data (known to the representation designer) and potentially malicious agents that use the data under the representation to learn predictive models and make decisions. A long list of recent research papers strive to provide tools for achieving these goals. However, we prove that this is basically a futile effort. Roughly stated, we prove that no representation can guarantee the fairness of classifiers for different tasks trained using it; even the basic goal of achieving label-independent Demographic Parity fairness fails once the marginal data distribution shifts. More refined notions of fairness, like Odds Equality, cannot be guaranteed by a representation that does not take into account the task specific labeling rule with respect to which such fairness will be evaluated (even if the marginal data distribution is known a priory). Furthermore, except for trivial cases, no representation can guarantee Odds Equality fairness for any two different tasks, while allowing accurate label predictions for both. While some of our conclusions are intuitive, we formulate (and prove) crisp statements of such impossibilities, often contrasting impressions conveyed by many recent works on fair representations.