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Geometry of Relaxed Fair Regression: A Unified Framework for Aware and Unaware Settings

This paper presents a unified optimal transport framework for relaxed fair regression that bridges aware and unaware settings by characterizing optimal predictors under Wasserstein-2 and Total Variation penalties, revealing distinct fairness philosophies and enabling a simple, efficient algorithm that outperforms state-of-the-art baselines.

Original authors: M. Generali Lince, V. Divol, R. Flamary, S. Gaucher, P. Loiseau

Published 2026-05-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: M. Generali Lince, V. Divol, R. Flamary, S. Gaucher, P. Loiseau

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Problem: The "Fairness vs. Accuracy" Tug-of-War

Imagine you are a coach trying to predict how well a player will perform in a game. You have a lot of data about their skills. However, you also know that historically, players from one specific team (let's call them "Team A") have been treated unfairly by the system, leading to lower scores in the past, even if their actual skills are the same as players from "Team B."

You want your predictions to be accurate (predicting the true skill) but also fair (not letting the player's team history drag their score down).

The paper tackles a specific, tricky situation: What if you don't know which team the player is on when you are making the prediction?

The "Aware" setting: You know the player is from Team A or Team B. You can adjust the score directly.
The "Unaware" setting: You don't know the team. You only see their stats. This is common in real life (e.g., a loan officer doesn't know a customer's race, or a hiring algorithm doesn't know a candidate's gender).

Most existing methods either ignore the problem (unfair) or try to guess the team first and then adjust (which often fails because the guess is wrong). This paper says: "We can fix this without guessing the team, using a new mathematical map."

The Solution: A New Way to "Redistribute" Predictions

The authors propose a method that treats predictions like a moving puzzle. They use a branch of math called Optimal Transport.

The Analogy: Moving Sand
Imagine you have two piles of sand.

Pile A represents the predictions for Team A.
Pile B represents the predictions for Team B.

If the piles are in different shapes or locations, the system is unfair. To make it fair, you need to move the sand grains from one pile to the other until they look exactly the same. The "cost" of moving a grain is how far you have to drag it.

The paper introduces a unified framework that works even if you don't know which grain belongs to which pile initially. They figure out the most efficient way to move the sand grains so that the final shape is identical for everyone, while moving them as little as possible to keep the original accuracy.

Two Different Philosophies: "Smooth Compromise" vs. "Hard Cutoff"

The paper discovers that there are two distinct ways to move this sand, depending on the "penalty" you choose. This is a major finding: The math you choose changes the philosophy of fairness.

1. The "Wasserstein" Penalty (The Smooth Compromise)

How it works: Imagine you have a group of people standing at different heights. To make them equal, you gently nudge everyone a little bit toward the middle.
The Metaphor: It's like a group hug. Everyone moves slightly closer together. No one stays exactly where they were, but no one is pushed too hard.
The Result: This creates a "smooth" fairness. It accepts that everyone is slightly less accurate than before, but the error is shared equally across the whole population. It's an egalitarian approach: "We all suffer a tiny bit to be equal."

2. The "Total Variation" Penalty (The Hard Cutoff)

How it works: Imagine you have a group of people. You only move the people who are very different from each other. If two people are already close, you leave them exactly where they are.
The Metaphor: It's like a traffic light. If the gap is small, the light is green (do nothing). If the gap is too big, the light turns red, and you force a massive, sudden jump to make them match.
The Result: This creates "exact parity" for a large group of people, but leaves the outliers (the extreme cases) completely untouched. It's a sufficientarian approach: "We ensure a huge chunk of the population is perfectly equal, even if the extremes remain unfair."

Why This Matters (The "Unaware" Breakthrough)

The paper's biggest claim is that they solved the "Unaware" problem (where you don't know the team) using these smooth geometric maps.

Old Way: Try to guess the team first (e.g., "This person looks like Team A"), then apply the fix. If your guess is wrong, you make the fairness worse.
New Way: The authors created a mathematical "map" that looks at the data and the probability of being in a group, rather than a hard guess. They move the predictions based on this probability.
The Result: Their method is faster and more accurate than the current state-of-the-art methods. It doesn't need to guess the team; it just rearranges the predictions geometrically to ensure fairness automatically.

The Takeaway

This paper provides a "toolbox" for making AI predictions fair when you don't have all the sensitive information (like race or gender).

It unifies the problem: It treats "knowing the team" and "not knowing the team" as part of the same geometric puzzle.
It offers a choice: You can choose between a smooth, gentle fairness (everyone moves a little) or a strict, hard fairness (some people stay put, others jump to match).
It works better: In tests on real-world data (like predicting law school grades or crime rates), their method was more accurate and faster than existing tools, proving that you don't need to sacrifice accuracy to be fair—you just need the right geometric map.

Technical Summary: Geometry of Relaxed Fair Regression

Problem Statement

The paper addresses the challenge of enforcing Demographic Parity (DP) in regression tasks, specifically within the unaware setting where sensitive attributes ( $S$ ) are unavailable at inference time. While post-processing methods for exact fairness exist in the "aware" setting (where $S$ is observed), principled methods for achieving relaxed fairness (balancing accuracy and fairness) in the unaware setting are largely missing. Existing approaches often rely on noisy proxy classifiers or grid-based discretizations that fail to capture the continuous geometric structure of the problem, leading to suboptimal accuracy-fairness trade-offs.

Methodology

The authors propose a unified framework that formulates relaxed fair regression as an Optimal Transport (OT) problem. The core insight is to characterize the optimal prediction function via OT maps that transport the conditional distributions of the base predictor to a common target distribution, while minimizing a penalty for unfairness.

1. Unified Framework

The framework unifies both aware and unaware settings by defining the regression problem in terms of the signed measure $\chi_+ - \chi_-$ (the difference between the laws of features in the positive and negative groups).

Unaware Setting: The optimal predictor relies on the signed group probability $\Delta(x) = \frac{P(S=+|X=x)}{p_+} - \frac{P(S=-|X=x)}{p_-}$ . The problem reduces to transporting 2D measures $\mu^{\pm}_{\eta, \Delta}$ (defined on the space of base predictions $\eta(x)$ and $\Delta(x)$ ) to a 1D prediction space.
Aware Setting: This is recovered as a special case where $\Delta(x)$ is deterministic based on the observed $S$ , reducing the problem to 1D transport.

2. Relaxed Fairness via Penalties

Instead of enforcing exact DP ( $\lambda \to \infty$ ), the authors minimize a penalized risk:
$\min_f \mathbb{E}[(f(X) - \eta(X))^2] + \lambda m U_D(f)$
where $U_D(f)$ is a discrepancy measure between the pushforwards of the positive and negative variations of the signed measure. The paper analyzes two specific penalties:

Squared Wasserstein-2 ( $W_2^2$ ): Penalizes the magnitude of displacement in the label space.
Total Variation (TV): Penalizes non-overlapping probability mass, ignoring distances between distinct values.

3. Closed-Form Solutions

A key theoretical contribution is the derivation of exact, closed-form expressions for the optimal almost-fair prediction functions.

$W_2^2$ Relaxation: The solution involves a weighted interpolation between the base prediction and the target. The optimal transport plan is characterized by a specific cost function $C_{\lambda, c_2}$ , and the prediction maps $T^\pm$ are linear combinations of the inputs weighted by the local density ratios.
TV Relaxation: The solution exhibits a hard-thresholding behavior. If the cost of merging two predictions is below a threshold $\lambda$ , they are merged to a weighted barycenter; otherwise, they remain unchanged. This leads to a piecewise-defined map.

4. Algorithmic Implementation

The authors propose a practical algorithm (Algorithm 1) that:

Estimates the base regressor $\hat{\eta}$ and the posterior probabilities to compute $\hat{\Delta}$ .
Constructs empirical 2D measures from the training data.
Solves a discrete OT problem using the derived closed-form cost matrices.
Uses barycentric projection to generate pseudo-labels for the training set.
Fits a continuous regressor (e.g., Random Forest) on the pseudo-labels to enable out-of-sample inference.

Key Contributions

Unified Framework: A general penalized OT objective that unifies aware and unaware relaxed fair regression, accommodating both $W_2^2$ and TV penalties.
Closed-Form Characterizations: The first exact, continuous geometric reductions for relaxed unaware fairness. The paper provides explicit formulas for the optimal prediction maps, moving beyond the grid-based approximations of prior work (e.g., FairReg).
Philosophical Distinction: The paper reveals that the choice of penalty reflects fundamentally different fairness philosophies:
- $W_2^2$ induces a smooth, population-wide compromise (egalitarian view), shifting all individuals slightly toward a fair center.
- TV enforces exact parity for a subset of individuals (sufficientarian view), leaving extreme predictions unchanged while perfectly merging those where it is "affordable."
Efficient Algorithm: A deterministic, computationally efficient post-processing algorithm that avoids stochastic gradient descent.

Experimental Results

The framework was evaluated on synthetic data and real-world benchmarks (Law School Admissions, Communities and Crime, and Adult Census Income).

Performance: The proposed "OT Unaware" method consistently matches or outperforms the state-of-the-art baseline FairReg (Taturyan et al., 2024) across all fairness metrics ( $W_2$ , TV, and Kolmogorov-Smirnov distance) while maintaining competitive Mean Squared Error (MSE).
Comparison to Proxies: Naive approaches that estimate $S$ via a proxy classifier and apply aware OT mappings (plug-in estimators) perform significantly worse, often failing to achieve fairness due to classification errors. The proposed 2D framework successfully absorbs this uncertainty.
Efficiency: The OT Unaware method is significantly faster than FairReg (e.g., ~26x speedup on the Communities dataset) because it relies on exact discrete OT solvers rather than iterative stochastic optimization.
Robustness: The method demonstrates robustness across different base estimators (Linear, Random Forest, Gradient Boosting).

Significance and Claims

The paper claims to bridge a significant gap in the literature by providing a theoretically grounded, continuous geometric solution for relaxed fair regression in the unaware setting.

Theoretical Rigor: Unlike previous methods that rely on discretization, this work offers exact closed-form reductions, clarifying the geometric nature of fairness constraints.
Practical Utility: The derived algorithms are simple to implement and computationally efficient, making them suitable for real-world deployment where models are black-box and sensitive attributes are missing at inference.
Philosophical Insight: By contrasting $W_2$ and TV penalties, the work highlights that "relaxed fairness" is not a monolithic concept but depends on the specific trade-off philosophy (global smoothing vs. local exactness) chosen by the practitioner.

The authors note limitations, including the restriction to binary sensitive attributes and specific penalty types ( $W_2^2$ and TV), and the dependence of performance on the quality of the base estimators $\hat{\eta}$ and $\hat{\Delta}$ . However, they assert that their approach provides a superior baseline for accuracy-fairness trade-offs compared to existing state-of-the-art methods.