# Costs and Benefits of Fair Regression

arXiv: https://arxiv.org/abs/2106.08812 · HTML: https://mlsanity.com/paper/2106.08812/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 58 references that are in the map.
- First posted on arXiv 16 Jun 2021.

## Builds on (most cited first, 27 of 27)

- [Domain-Adversarial Training of Neural Networks](https://mlsanity.com/paper/1505.07818/) (2015)
- [Equality of Opportunity in Supervised Learning](https://mlsanity.com/paper/1610.02413/) (2016)
- [Certifying and removing disparate impact](https://mlsanity.com/paper/1412.3756/) (2014)
- [Mitigating Unwanted Biases with Adversarial Learning](https://mlsanity.com/paper/1801.07593/) (2018)
- [Wasserstein GAN](https://mlsanity.com/paper/1701.07875/) (2017)
- [Algorithmic decision making and the cost of fairness](https://mlsanity.com/paper/1701.08230/) (2017)
- [Learning Adversarially Fair and Transferable Representations](https://mlsanity.com/paper/1802.06309/) (2018)
- [Learning Representations for Counterfactual Inference](https://mlsanity.com/paper/1605.03661/) (2016)
- [Differential Privacy Has Disparate Impact on Model Accuracy](https://mlsanity.com/paper/1905.12101/) (2019)
- [Joint Distribution Optimal Transportation for Domain Adaptation](https://mlsanity.com/paper/1705.08848/) (2017)
- [The Variational Fair Autoencoder](https://mlsanity.com/paper/1511.00830/) (2015)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Fairness in Machine Learning](https://mlsanity.com/paper/2012.15816/) (2020)
- [Censoring Representations with an Adversary](https://mlsanity.com/paper/1511.05897/) (2015)
- [Fair Regression: Quantitative Definitions and Reduction-based Algorithms](https://mlsanity.com/paper/1905.12843/) (2019)
- [Data Decisions and Theoretical Implications when Adversarially Learning Fair Representations](https://mlsanity.com/paper/1707.00075/) (2017)
- [A Convex Framework for Fair Regression](https://mlsanity.com/paper/1706.02409/) (2017)
- [Inherent Tradeoffs in Learning Fair Representations](https://mlsanity.com/paper/1906.08386/) (2019)
- [Fair Regression with Wasserstein Barycenters](https://mlsanity.com/paper/2006.07286/) (2020)
- [Conditional Learning of Fair Representations](https://mlsanity.com/paper/1910.07162/) (2019)
- [Generalization Bounds and Representation Learning for Estimation of Potential Outcomes and Causal Effects](https://mlsanity.com/paper/2001.07426/) (2020)
- [Pairwise Fairness for Ranking and Regression](https://mlsanity.com/paper/1906.05330/) (2019)
- [Recovering from Biased Data: Can Fairness Constraints Improve Accuracy?](https://mlsanity.com/paper/1912.01094/) (2019)
- [On the relation between accuracy and fairness in binary classification](https://mlsanity.com/paper/1505.05723/) (2015)
- [Convergence and Concentration of Empirical Measures under Wasserstein Distance in Unbounded Functional Spaces](https://mlsanity.com/paper/1804.10556/) (2018)
- [A statistical framework for fair predictive algorithms](https://mlsanity.com/paper/1610.08077/) (2016)
- [Understanding and Mitigating Accuracy Disparity in Regression](https://mlsanity.com/paper/2102.12013/) (2021)

## Built on by (most cited first, 8 of 8)

- [Regression under demographic parity constraints via unlabeled post-processing](https://mlsanity.com/paper/2407.15453/) (2024)
- [A Theoretical Approach to Characterize the Accuracy-Fairness Trade-off Pareto Frontier](https://mlsanity.com/paper/2310.12785/) (2023)
- [Standardized Interpretable Fairness Measures for Continuous Risk Scores](https://mlsanity.com/paper/2308.11375/) (2023)
- [The Statistical Fairness-Accuracy Frontier](https://mlsanity.com/paper/2508.17622/) (2025)
- [Fairness under Covariate Shift: Improving Fairness-Accuracy tradeoff with few Unlabeled Test Samples](https://mlsanity.com/paper/2310.07535/) (2023)
- [Towards Harmless Rawlsian Fairness Regardless of Demographic Prior](https://mlsanity.com/paper/2411.02467/) (2024)
- [An Empirical Study of Self-supervised Learning with Wasserstein Distance](https://mlsanity.com/paper/2310.10143/) (2023)
- [Group Fairness Meets the Black Box: Enabling Fair Algorithms on Closed LLMs via Post-Processing](https://mlsanity.com/paper/2508.11258/) (2025)

## Abstract (arXiv)

Real-world applications of machine learning tools in high-stakes domains are often regulated to be fair, in the sense that the predicted target should satisfy some quantitative notion of parity with respect to a protected attribute. However, the exact tradeoff between fairness and accuracy with a real-valued target is not entirely clear. In this paper, we characterize the inherent tradeoff between statistical parity and accuracy in the regression setting by providing a lower bound on the error of any fair regressor. Our lower bound is sharp, algorithm-independent, and admits a simple interpretation: when the moments of the target differ between groups, any fair algorithm has to make an error on at least one of the groups. We further extend this result to give a lower bound on the joint error of any (approximately) fair algorithm, using the Wasserstein distance to measure the quality of the approximation. With our novel lower bound, we also show that the price paid by a fair regressor that does not take the protected attribute as input is less than that of a fair regressor with explicit access to the protected attribute. On the upside, we establish the first connection between individual fairness, accuracy parity, and the Wasserstein distance by showing that if a regressor is individually fair, it also approximately verifies the accuracy parity, where the gap is given by the Wasserstein distance between the two groups. Inspired by our theoretical results, we develop a practical algorithm for fair regression through the lens of representation learning, and conduct experiments on a real-world dataset to corroborate our findings.