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Fair Regression under Demographic Parity: A Unified Framework

Yongzhen Feng, Weiwei Wang, Raymond K. W. Wong, Xianyang Zhang

TL;DR

The paper addresses fair regression under demographic parity by linking risk minimization to an optimal transport problem, specifically the Kantorovich barycenter. It derives a two-step estimation procedure that first learns group-specific latent predictors and then crafts a common, monotone quantile function to enforce fairness, transforming the constrained problem into an unconstrained risk minimization over quantiles. The authors establish consistency and convergence rates for the proposed estimator and demonstrate effectiveness across quantile, Poisson, and robust regression tasks, with empirical fairness measured by KS distance. This transport-based, loss-agnostic framework enables fair predictions across a broad range of regression problems and offers practical, scalable estimation methods with theoretical guarantees.

Abstract

We propose a unified framework for fair regression tasks formulated as risk minimization problems subject to a demographic parity constraint. Unlike many existing approaches that are limited to specific loss functions or rely on challenging non-convex optimization, our framework is applicable to a broad spectrum of regression tasks. Examples include linear regression with squared loss, binary classification with cross-entropy loss, quantile regression with pinball loss, and robust regression with Huber loss. We derive a novel characterization of the fair risk minimizer, which yields a computationally efficient estimation procedure for general loss functions. Theoretically, we establish the asymptotic consistency of the proposed estimator and derive its convergence rates under mild assumptions. We illustrate the method's versatility through detailed discussions of several common loss functions. Numerical results demonstrate that our approach effectively minimizes risk while satisfying fairness constraints across various regression settings.

Fair Regression under Demographic Parity: A Unified Framework

TL;DR

The paper addresses fair regression under demographic parity by linking risk minimization to an optimal transport problem, specifically the Kantorovich barycenter. It derives a two-step estimation procedure that first learns group-specific latent predictors and then crafts a common, monotone quantile function to enforce fairness, transforming the constrained problem into an unconstrained risk minimization over quantiles. The authors establish consistency and convergence rates for the proposed estimator and demonstrate effectiveness across quantile, Poisson, and robust regression tasks, with empirical fairness measured by KS distance. This transport-based, loss-agnostic framework enables fair predictions across a broad range of regression problems and offers practical, scalable estimation methods with theoretical guarantees.

Abstract

We propose a unified framework for fair regression tasks formulated as risk minimization problems subject to a demographic parity constraint. Unlike many existing approaches that are limited to specific loss functions or rely on challenging non-convex optimization, our framework is applicable to a broad spectrum of regression tasks. Examples include linear regression with squared loss, binary classification with cross-entropy loss, quantile regression with pinball loss, and robust regression with Huber loss. We derive a novel characterization of the fair risk minimizer, which yields a computationally efficient estimation procedure for general loss functions. Theoretically, we establish the asymptotic consistency of the proposed estimator and derive its convergence rates under mild assumptions. We illustrate the method's versatility through detailed discussions of several common loss functions. Numerical results demonstrate that our approach effectively minimizes risk while satisfying fairness constraints across various regression settings.
Paper Structure (17 sections, 9 theorems, 137 equations, 4 figures)

This paper contains 17 sections, 9 theorems, 137 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related works
  3. Problem setup
  4. Characterization of optimal fair prediction functions
  5. Empirical risk minimization with fairness constraints
  6. Group-wise estimators
  7. Numerical implementation of Step 2
  8. Statistical theory
  9. Numerical experiments
  10. Benchmark data applications
  11. Quantile regression on CRIME dataset
  12. Poisson Regression on HRS Dataset
  13. Simulation Study: Robust Regression
  14. Discussion
  15. Appendix
  16. ...and 2 more sections

Key Result

Proposition 1

Under Assumptions assum:f_star and assum:cost_function, the solution to the FRP belongs to the class:

Figures (4)

  • Figure 1: Pinball loss and KS distance comparison for $\tau=0.25,0.5,0.75$ on the CRIME dataset. $95\%$ confidence intervals for the mean are plotted. QR represents standard quantile regression.
  • Figure 2: Comparison of methods on the Health & Retirement Survey Dataset using Poisson regression. $95\%$ confidence intervals for the mean are plotted. PR denotes standard Poisson regression.
  • Figure 3: Huber loss and KS distance for robust regression simulation with moderate bias ($a = 1$). Approximated $95\%$ CIs are plotted. RR denotes standard robust regression.
  • Figure 4: Huber loss and KS distance for robust regression simulation with high bias ($a = 5$). Approximated $95\%$ CIs are plotted. RR denotes standard robust regression.

Theorems & Definitions (30)

  • Definition 1: Demographic parity
  • Remark 1: Extension to Multiple Groups
  • Proposition 1
  • Proposition 2
  • Example 1: Regression with convex loss
  • Example 2: Binary classification with cross-entropy loss
  • Example 3: Generalized regression
  • Proposition 3: Risk decomposition and pointwise characterization of $\widetilde{Q}$
  • Remark 2: Mixture view
  • Example 4: Regression with $l_2$ loss
  • ...and 20 more