# Metrizing Fairness

arXiv: https://arxiv.org/abs/2205.15049 · HTML: https://mlsanity.com/paper/2205.15049/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 44 of its 103 references that are in the map.
- First posted on arXiv 30 May 2022.

## Builds on (most cited first, 30 of 44)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Equality of Opportunity in Supervised Learning](https://mlsanity.com/paper/1610.02413/) (2016)
- [A Survey on Bias and Fairness in Machine Learning](https://mlsanity.com/paper/1908.09635/) (2019)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Counterfactual Fairness](https://mlsanity.com/paper/1703.06856/) (2017)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [Certifying and removing disparate impact](https://mlsanity.com/paper/1412.3756/) (2014)
- [Fair prediction with disparate impact: A study of bias in recidivism prediction instruments](https://mlsanity.com/paper/1610.07524/) (2016)
- [On Fairness and Calibration](https://mlsanity.com/paper/1709.02012/) (2017)
- [Algorithmic decision making and the cost of fairness](https://mlsanity.com/paper/1701.08230/) (2017)
- [Fairness Beyond Disparate Treatment & Disparate Impact: Learning Classification without Disparate Mistreatment](https://mlsanity.com/paper/1610.08452/) (2016)
- [Fairness in Criminal Justice Risk Assessments: The State of the Art](https://mlsanity.com/paper/1703.09207/) (2017)
- [Empirical Risk Minimization under Fairness Constraints](https://mlsanity.com/paper/1802.08626/) (2018)
- [A Rewriting System for Convex Optimization Problems](https://mlsanity.com/paper/1709.04494/) (2017)
- [Fair Regression: Quantitative Definitions and Reduction-based Algorithms](https://mlsanity.com/paper/1905.12843/) (2019)
- [FairGAN: Fairness-aware Generative Adversarial Networks](https://mlsanity.com/paper/1805.11202/) (2018)
- [Counterfactual Fairness in Text Classification through Robustness](https://mlsanity.com/paper/1809.10610/) (2018)
- [A Convex Framework for Fair Regression](https://mlsanity.com/paper/1706.02409/) (2017)
- [Identifying and Correcting Label Bias in Machine Learning](https://mlsanity.com/paper/1901.04966/) (2019)
- [Fair Inference On Outcomes](https://mlsanity.com/paper/1705.10378/) (2017)
- [Wasserstein Fair Classification](https://mlsanity.com/paper/1907.12059/) (2019)
- [Distributionally Robust Logistic Regression](https://mlsanity.com/paper/1509.09259/) (2015)
- [The Cramer Distance as a Solution to Biased Wasserstein Gradients](https://mlsanity.com/paper/1705.10743/) (2017)
- [Revisiting Small Batch Training for Deep Neural Networks](https://mlsanity.com/paper/1804.07612/) (2018)
- [Fair Regression with Wasserstein Barycenters](https://mlsanity.com/paper/2006.07286/) (2020)
- [FairBatch: Batch Selection for Model Fairness](https://mlsanity.com/paper/2012.01696/) (2020)
- [Fairness risk measures](https://mlsanity.com/paper/1901.08665/) (2019)
- [On the Almost Sure Convergence of Stochastic Gradient Descent in Non-Convex Problems](https://mlsanity.com/paper/2006.11144/) (2020)

## Built on by (most cited first, 6 of 6)

- [Fair and Accurate Regression: Strong Formulations and Algorithms](https://mlsanity.com/paper/2412.17116/) (2024)
- [Global Group Fairness in Federated Learning via Function Tracking](https://mlsanity.com/paper/2503.15163/) (2025)
- [Risk-averse Fair Multi-class Classification](https://mlsanity.com/paper/2509.05771/) (2025)
- [Kernel-based Equalized Odds: A Quantification of Accuracy-Fairness Trade-off in Fair Representation Learning](https://mlsanity.com/paper/2508.15084/) (2025)
- [Convergence of empirical subgradients for optimal transport-based objectives](https://mlsanity.com/paper/2605.28134/) (2026)
- [Fairness in multi-class multi-group classification problems via contextial coherent risk measures](https://mlsanity.com/paper/2608.30223/) (2026)

## Abstract (arXiv)

We study supervised learning problems that have significant effects on individuals from two demographic groups, and we seek predictors that are fair with respect to a group fairness criterion such as statistical parity (SP). A predictor is SP-fair if the distributions of predictions within the two groups are close in Kolmogorov distance, and fairness is achieved by penalizing the dissimilarity of these two distributions in the objective function of the learning problem. In this paper, we identify conditions under which hard SP constraints are guaranteed to improve predictive accuracy. We also showcase conceptual and computational benefits of measuring unfairness with integral probability metrics (IPMs) other than the Kolmogorov distance. Conceptually, we show that the generator of any IPM can be interpreted as a family of utility functions and that unfairness with respect to this IPM arises if individuals in the two demographic groups have diverging expected utilities. We also prove that the unfairness-regularized prediction loss admits unbiased gradient estimators, which are constructed from random mini-batches of training samples, if unfairness is measured by the squared mathcal L2-distance or by a squared maximum mean discrepancy. In this case, the fair learning problem is susceptible to efficient stochastic gradient descent (SGD) algorithms. Numerical experiments on synthetic and real data show that these SGD algorithms outperform state-of-the-art methods for fair learning in that they achieve superior accuracy-unfairness trade-offs -- sometimes orders of magnitude faster.