# Fairness with Continuous Optimal Transport

arXiv: https://arxiv.org/abs/2101.02084 · HTML: https://mlsanity.com/paper/2101.02084/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 43 references that are in the map.
- First posted on arXiv 6 Jan 2021.

## Builds on (most cited first, 23 of 23)

- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Mitigating Unwanted Biases with Adversarial Learning](https://mlsanity.com/paper/1801.07593/) (2018)
- [Fairness in Machine Learning](https://mlsanity.com/paper/2012.15816/) (2020)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [Delayed Impact of Fair Machine Learning](https://mlsanity.com/paper/1803.04383/) (2018)
- [Fairness in Learning: Classic and Contextual Bandits](https://mlsanity.com/paper/1605.07139/) (2016)
- [Wasserstein Fair Classification](https://mlsanity.com/paper/1907.12059/) (2019)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [Fair Regression with Wasserstein Barycenters](https://mlsanity.com/paper/2006.07286/) (2020)
- [Fairness in Reinforcement Learning](https://mlsanity.com/paper/1611.03071/) (2016)
- [Training individually fair ML models with Sensitive Subspace Robustness](https://mlsanity.com/paper/1907.00020/) (2019)
- [On Formalizing Fairness in Prediction with Machine Learning](https://mlsanity.com/paper/1710.03184/) (2017)
- [Online Learning with an Unknown Fairness Metric](https://mlsanity.com/paper/1802.06936/) (2018)
- [Repairing without Retraining: Avoiding Disparate Impact with Counterfactual Distributions](https://mlsanity.com/paper/1901.10501/) (2019)
- [Causal Modeling for Fairness in Dynamical Systems](https://mlsanity.com/paper/1909.09141/) (2019)
- [Fairness in Learning-Based Sequential Decision Algorithms: A Survey](https://mlsanity.com/paper/2001.04861/) (2020)
- [Ensuring Fairness Beyond the Training Data](https://mlsanity.com/paper/2007.06029/) (2020)
- [Improved Adversarial Learning for Fair Classification](https://mlsanity.com/paper/1901.10443/) (2019)
- [Review of Mathematical frameworks for Fairness in Machine Learning](https://mlsanity.com/paper/2005.13755/) (2020)
- [On preserving non-discrimination when combining expert advice](https://mlsanity.com/paper/1810.11829/) (2018)
- [Projection to Fairness in Statistical Learning](https://mlsanity.com/paper/2005.11720/) (2020)
- [Consequential Ranking Algorithms and Long-term Welfare](https://mlsanity.com/paper/1905.05305/) (2019)
- [Convergence Rates of Smooth Message Passing with Rounding in Entropy-Regularized MAP Inference](https://mlsanity.com/paper/1907.01127/) (2019)

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

- [Fair learning with Wasserstein barycenters for non-decomposable performance measures](https://mlsanity.com/paper/2209.00427/) (2022)
- [An improved central limit theorem and fast convergence rates for entropic transportation costs](https://mlsanity.com/paper/2204.09105/) (2022)
- [Learning Counterfactually Invariant Predictors](https://mlsanity.com/paper/2207.09768/) (2022)
- [Obtaining Dyadic Fairness by Optimal Transport](https://mlsanity.com/paper/2202.04520/) (2022)
- [Everything is Relative: Understanding Fairness with Optimal Transport](https://mlsanity.com/paper/2102.10349/) (2021)
- [fAux: Testing Individual Fairness via Gradient Alignment](https://mlsanity.com/paper/2210.06288/) (2022)
- [Adversarial Reweighting Guided by Wasserstein Distance for Bias Mitigation](https://mlsanity.com/paper/2311.12684/) (2023)
- [Auditing and Enforcing Conditional Fairness via Optimal Transport](https://mlsanity.com/paper/2410.14029/) (2024)
- [Evaluating Fairness and Mitigating Bias in Machine Learning: A Novel Technique using Tensor Data and Bayesian Regression](https://mlsanity.com/paper/2506.11627/) (2025)
- [Counterfactually Fair Regression via Optimal Transport](https://mlsanity.com/paper/2605.28251/) (2026)

## Abstract (arXiv)

Whilst optimal transport (OT) is increasingly being recognized as a powerful and flexible approach for dealing with fairness issues, current OT fairness methods are confined to the use of discrete OT. In this paper, we leverage recent advances from the OT literature to introduce a stochastic-gradient fairness method based on a dual formulation of continuous OT. We show that this method gives superior performance to discrete OT methods when little data is available to solve the OT problem, and similar performance otherwise. We also show that both continuous and discrete OT methods are able to continually adjust the model parameters to adapt to different levels of unfairness that might occur in real-world applications of ML systems.