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Enforcing Fairness Where It Matters: An Approach Based on Difference-of-Convex Constraints

Yutian He, Yankun Huang, Yao Yao, Qihang Lin

TL;DR

This work tackles fairness by restricting constraints to a contested score interval $\mathcal{I}$ rather than the full score distribution. It formulates partial fairness as a DC-constrained in-processing optimization and solves it with an inexact difference-of-convex algorithm (IDCA), providing the first complexity analysis for IDCA in DC-constrained problems. The approach yields high predictive performance while enforcing partial fairness via $\text{DP}^{\mathcal{I}}$ and $\text{WDP}^{\mathcal{I}}$ with tolerance $\kappa$, demonstrated on three real datasets and multiple fairness levels. Overall, the framework offers a practical, scalable means to achieve fairness where it matters most in high-stakes decisions, without sacrificing global accuracy.

Abstract

Fairness in machine learning has become a critical concern, particularly in high-stakes applications. Existing approaches often focus on achieving full fairness across all score ranges generated by predictive models, ensuring fairness in both high and low-scoring populations. However, this stringent requirement can compromise predictive performance and may not align with the practical fairness concerns of stakeholders. In this work, we propose a novel framework for building partially fair machine learning models, which enforce fairness within a specific score range of interest, such as the middle range where decisions are most contested, while maintaining flexibility in other regions. We introduce two statistical metrics to rigorously evaluate partial fairness within a given score range, such as the top 20%-40% of scores. To achieve partial fairness, we propose an in-processing method by formulating the model training problem as constrained optimization with difference-of-convex constraints, which can be solved by an inexact difference-of-convex algorithm (IDCA). We provide the complexity analysis of IDCA for finding a nearly KKT point. Through numerical experiments on real-world datasets, we demonstrate that our framework achieves high predictive performance while enforcing partial fairness where it matters most.

Enforcing Fairness Where It Matters: An Approach Based on Difference-of-Convex Constraints

TL;DR

This work tackles fairness by restricting constraints to a contested score interval rather than the full score distribution. It formulates partial fairness as a DC-constrained in-processing optimization and solves it with an inexact difference-of-convex algorithm (IDCA), providing the first complexity analysis for IDCA in DC-constrained problems. The approach yields high predictive performance while enforcing partial fairness via and with tolerance , demonstrated on three real datasets and multiple fairness levels. Overall, the framework offers a practical, scalable means to achieve fairness where it matters most in high-stakes decisions, without sacrificing global accuracy.

Abstract

Fairness in machine learning has become a critical concern, particularly in high-stakes applications. Existing approaches often focus on achieving full fairness across all score ranges generated by predictive models, ensuring fairness in both high and low-scoring populations. However, this stringent requirement can compromise predictive performance and may not align with the practical fairness concerns of stakeholders. In this work, we propose a novel framework for building partially fair machine learning models, which enforce fairness within a specific score range of interest, such as the middle range where decisions are most contested, while maintaining flexibility in other regions. We introduce two statistical metrics to rigorously evaluate partial fairness within a given score range, such as the top 20%-40% of scores. To achieve partial fairness, we propose an in-processing method by formulating the model training problem as constrained optimization with difference-of-convex constraints, which can be solved by an inexact difference-of-convex algorithm (IDCA). We provide the complexity analysis of IDCA for finding a nearly KKT point. Through numerical experiments on real-world datasets, we demonstrate that our framework achieves high predictive performance while enforcing partial fairness where it matters most.
Paper Structure (36 sections, 7 theorems, 103 equations, 10 figures, 1 table, 3 algorithms)

This paper contains 36 sections, 7 theorems, 103 equations, 10 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Preliminary and Notations
  4. Partial Fairness
  5. Optimization Models
  6. Inexact Difference-of-Convex Algorithm
  7. Numerical Experiments
  8. Partial demographic parity
  9. Weak partial demographic parity
  10. Changes of score densities with $\kappa$ and optimization performances
  11. Conclusion
  12. Equivalent formulations of strong and weak partial fairness constraints
  13. Reformulate Training Problems with Partial Fairness Constraints as DC Optimization
  14. DC objective function in \ref{['eq:inprocess_pdp_approx']} and \ref{['eq:inprocess_wpdp_approx']}
  15. DC constraints in \ref{['eq:inprocess_pdp_approx']}
  16. ...and 21 more sections

Key Result

Lemma 5.1

A solution ${\mathbf{w}}\in{\mathcal{W}}$ is feasible to eq:inprocess_pdp if and only if, for any $p\in[\alpha,\beta-\kappa(\beta-\alpha))$, there exists $\theta_p\in\mathbb{R}$ such that

Figures (10)

  • Figure 1: Pareto frontiers showing the trade-off between classification accuracy and pDP fairness (the first row) and the trade-off between classification accuracy and weak pDP fairness (the second row) by each method. $\mathcal{I}$ is $[5\%, 30\%]$ for a9a, $[0\%, 25\%]$ for bank, and $[70\%, 100\%]$ for law school.
  • Figure 2: How the predicted positive rates in both groups ($\bar{F}_{{\mathbf{w}},1}(\theta)$ and $\bar{F}_{{\mathbf{w}},2}(\theta)$) change with threshold $\theta$ when using different $\kappa$ in \ref{['eq:inprocess_pdp_approx']} by the proposed method. $\mathcal{I}$ is $[5\%, 30\%]$ for a9a, $[0\%, 25\%]$ for bank, and $[70\%, 100\%]$ for law school (between the two red numbers marked on both axises).
  • Figure 3: Distributions of predicted scores of different sensitive groups on a9a dataset by the unconstrained model and the models solved from \ref{['eq:inprocess_pdp_approx']} (pDP constraints) with different $\kappa$'s. The interval $\mathcal{I}$ is $[5\%, 30\%]$ and is highlighted in red.
  • Figure 4: Distributions of predicted scores of different sensitive groups on bank dataset by the unconstrained model and the models solved from \ref{['eq:inprocess_pdp_approx']} (pDP constraints) with different $\kappa$'s. The interval $\mathcal{I}$ is $[0\%, 25\%]$ and is highlighted in red.
  • Figure 5: Distributions of predicted scores of different sensitive groups on law school dataset by the unconstrained model and the models solved from \ref{['eq:inprocess_pdp_approx']} (pDP constraints) with different $\kappa$'s. The interval $\mathcal{I}$ is $[70\%, 100\%]$ and is highlighted in red.
  • ...and 5 more figures

Theorems & Definitions (25)

  • Definition 4.1
  • Definition 4.2
  • Definition 4.3
  • Lemma 5.1
  • Lemma 5.2
  • Definition 6.1
  • Definition 6.2
  • Proposition 6.4
  • Theorem 6.5
  • proof : Proof of Lemma \ref{['eq:pdp_equiv']}
  • ...and 15 more