Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fair regression under localized demographic parity constraints

Arthur Charpentier, Christophe Denis, Romuald Elie, Mohamed Hebiri, François HU

Abstract

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 (${\ell}$, Z)-fair predictor, which imposes groupwise CDF constraints of the form F f |S=s (z m ) = ${\ell}$ m for prescribed pairs (${\ell}$ 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.

Fair regression under localized demographic parity constraints

Abstract

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 (, Z)-fair predictor, which imposes groupwise CDF constraints of the form F f |S=s (z m ) = m for prescribed pairs ( 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.

Paper Structure

This paper contains 54 sections, 10 theorems, 99 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related work.
  3. Our framework.
  4. Why quantiles rather than optimal transport?
  5. Contributions.
  6. Statistical setting
  7. Data and risk.
  8. Non-atomicity.
  9. Predictor classes and discretization.
  10. $(\boldsymbol{\ell},\mathcal{Z})$-fair predictor
  11. Quantile/threshold constraints.
  12. Lagrangian form.
  13. Penalized-risk interpretation.
  14. Discretization cost.
  15. Data-driven algorithm
  16. ...and 39 more sections

Key Result

Theorem 2.2

Under Assumption ass:cont, there exists $\lambda^\star$ such that the predictor $f^\star_{(\boldsymbol{\ell},\mathcal{Z})\text{-fair}}$ admits the pointwise form Moreover, $\lambda^\star$ can be chosen as a minimizer of the dual objective where $V_s(\lambda;x):=\max_{y\in\mathcal{Y}_K}\lbrace\Phi_{s,\lambda}(x,y)\rbrace$ and

Figures (4)

  • 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 $(\ell,\mathcal{Z})$-fair methods on synthetic data. We compare three prescriptions for $\mathcal{Z}$ (Global, Target-A, Target-B) with $M=3$ and $\boldsymbol{\ell}=(0.25,0.50,0.75)$.
  • Figure 3: Analysis of $\mathcal{Z}$-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).
  • Figure 4: Fairness--accuracy trade-off on synthetic data for increasingly global constraints.

Theorems & Definitions (13)

  • Theorem 2.2: Optimal $({\boldsymbol{\ell}},\mathcal{Z})$-fair discretized predictor
  • Corollary 2.3
  • Proposition 2.4: Cost of discretization
  • Theorem 3.1: Rate for fairness violation
  • Theorem 3.2: High-probability fairness violation
  • Theorem 3.3: Excess penalized risk
  • Theorem 3.4: High-probability excess penalized risk
  • Remark 3.5: High-probability variants
  • Corollary 4.1: Recovery of discretized strong DP
  • Theorem 4.2: Optimal $\mathcal{Z}$-DP-fair discretized predictor
  • ...and 3 more