Fair regression under localized demographic parity constraints
Authors: Arthur Charpentier, Christophe Denis, Romuald Elie, Mohamed Hebiri, François HU
Organizations: UQAM · Université du Québec à Montréal, UQAM, Canada · Kyoto University, Japan · SAMM · Université Paris 1 Panthéon-Sorbonne, Paris, France · LAMA · Université Gustave Eiffel, Paris, France · UdeM · Milliman Paris, France · Université Claude Bernard, Lyon, France
Demographic parity (DP) is a widely used group fairness criterion requiring predictive distributions to be invariant across sensitive groups. While natural in classification, full distributional DP is often overly restrictive in regression and can lead to substantial accuracy loss. We propose a relaxation of DP tailored to regression, enforcing parity only at a finite set of quantile levels and/or score thresholds. Concretely, we introduce a novel (ℓ, Z)-fair predictor, which imposes groupwise CDF constraints of the form F f |S=s (z m ) = ℓ m for prescribed pairs (ℓ m , z m ). For this setting, we derive closed-form characterizations of the optimal fair discretized predictor via a Lagrangian dual formulation and quantify the discretization cost, showing that the risk gap to the continuous optimum vanishes as the grid is refined. We further develop a model-agnostic post-processing algorithm based on two samples (labeled for learning a base regressor and unlabeled for calibration), and establish finite-sample guarantees on constraint violation and excess penalized risk. In addition, we introduce two alternative frameworks where we match group and marginal CDF values at selected score thresholds. In both settings, we provide closed-form solutions for the optimal fair discretized predictor. Experiments on synthetic and real datasets illustrate an interpretable fairness-accuracy trade-off, enabling targeted corrections at decision-relevant quantiles or thresholds while preserving predictive performance.
Figures & tables
Figure 1: Comparison of localized constraints and full distribution matching on CRIME data using a LightGBM model with default scikit-learn parameters.
Figure 2: Analysis of (ℓ,Z) -fair methods on synthetic data. We compare three prescriptions for Z (Global, Target-A, Target-B) with M=3 and ℓ=(0.25,0.50,0.75) .
Figure 3: Analysis of Z -fair methods on synthetic data. We compare enforcing parity at M=3 thresholds, enforcing parity only over a selected range, and full distribution matching (strong DP, full grid).
Appendix figures & tables1 asset
Supplementary material from the paper’s appendix.
Appendix
Figure 4: Fairness–accuracy trade-off on synthetic data for increasingly global constraints.
Algorithmic fairness studies how automated decisions distribute errors, opportunities, and harms across groups, and how to measure and reduce disparities.
Demographic parity (DP) is a widely studied fairness criterion in regression, enforcing independence between the predictions and sensitive attributes. However, constraining the entire distribution can degrade predictive accuracy and may be unnecessary for many applications, where fairness concerns are localized to specific regions of the distribution. To overcome this issue, we propose a new framework for regression under DP that focuses on the tails of target distribution across sensitive groups. Our methodology builds on optimal transport theory. By enforcing fairness constraints only over targeted regions of the distribution, our approach enables more nuanced and context-sensitive interventions. Leveraging recent advances, we develop an interpretable and flexible algorithm that leverages the geometric structure of optimal transport. We provide theoretical guarantees, including risk bounds and fairness properties, and validate the method through experiments in regression settings.
Naht Sinh Le, Christophe Denis, Mohamed Hebiri
LAMA · Université Gustave Eiffel, Paris, France · SAMM +1
Exact demographic parity requires the same prediction distribution across sensitive groups. In regression, this can be too restrictive and reduce predictive accuracy. We instead bound every pairwise Wasserstein distance between group-specific prediction distributions by a chosen tolerance, allowing the groups to differ while keeping fairness tunable. On the real line, the prediction and constrained transport problems have the same infimum. If the transport problem attains this value and suitable Monge maps exist, a deterministic transport-form predictor realizes it. Under a standard one-dimensional condition, the problem becomes a bounded quantile optimization that is convex when the conditional expected loss is convex in prediction. Under squared loss with two positively weighted groups, the W1, W2, and W∞ constraints yield soft-thresholding, shrinkage, and clipping. Under the stated conditions, we establish rates for the known-cost plug-in estimator. When the cost is unavailable, we describe an observed-loss computational proxy. Synthetic and California 2018 ACS studies illustrate the tunable fairness-accuracy trade-off and distinguish calibration feasibility from held-out fairness and accuracy.
We consider the problem of learning a counterfactually fair regressor. We adopt a causal uncertainty view in which counterfactual fairness is defined with resampled noise. We focus on obtaining theoretical fairness guarantees for a new post-processing estimator. We begin by showing that counterfactual fairness is equivalent to satisfying demographic parity conditional on the latent variable. This allows us to provide a closed-form expression of the optimal fair regressor via a barycentric quantile map. In order to handle continuous latent variables, we propose a discretized post-processing method. Then, under mild regularity assumptions, we prove high-probability finite-sample fairness guarantees for our estimator, providing an unfairness decay at rate O~(n−1/3), and establishing a matching risk bound of order O~(n−1/3). We provide a matching lower bound on the excess risk of almost fair predictions. Finally, we extend our results to the setting of relaxed counterfactual fairness. We validate our approach on real-world and synthetic data.
M. Generali Lince, S. Gaucher, J-J. Vie +1
Inria, Soda team · CREST, ENSAE, IP Paris · CMAP, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris +1