Intersectional Fairness via Mixed-Integer Optimization

TL;DR

This work tackles the challenge of intersectional fairness in high-stakes AI by introducing a unified training and auditing framework based on Mixed-Integer Optimization. It establishes the theoretical equivalence between two intersectional-bias measures, MSD and SPSF, for identifying the most unfair subgroup, and demonstrates how MIO can reliably detect and constrain bias while producing interpretable models. The method supports conjunction-based (interpretable) subgroups and employs lazy constraints to manage the exponential number of fairness constraints, enabling scalable training with provable fairness guarantees. Experiments on US Census data show that the conjunction-based approach identifies the most unfair subgroups more effectively than linear-subgroup methods and that the training framework yields accurate, interpretable classifiers that bound intersectional bias below practical thresholds. Overall, the paper provides a principled, scalable path to certifiably fair and transparent AI suitable for regulated domains and beyond.

Abstract

The deployment of Artificial Intelligence in high-risk domains, such as finance and healthcare, necessitates models that are both fair and transparent. While regulatory frameworks, including the EU's AI Act, mandate bias mitigation, they are deliberately vague about the definition of bias. In line with existing research, we argue that true fairness requires addressing bias at the intersections of protected groups. We propose a unified framework that leverages Mixed-Integer Optimization (MIO) to train intersectionally fair and intrinsically interpretable classifiers. We prove the equivalence of two measures of intersectional fairness (MSD and SPSF) in detecting the most unfair subgroup and empirically demonstrate that our MIO-based algorithm improves performance in finding bias. We train high-performing, interpretable classifiers that bound intersectional bias below an acceptable threshold, offering a robust solution for regulated industries and beyond.

Figures (10)

• Figure 1: An illustrative example of when marginal bias is not enough. Statistical parity is achieved, suggesting marginal fairness, while blue people aged 0-18 are disproportionately more rejected. The graphic is a modification of Figure 1 in nemecek_bias_2025.
• Figure 2: Mean fairness violation (and standard deviations). In detection, finding the subgroup with more unfairness is better. The bars with slanted lines represent the optimal solutions found over conjunction subgroups ($\mathcal{S}_A$), which are a subset of linear subgroups ($\mathcal{S}_L$), explaining the difference. The blue bar with slanted lines corresponds to MSD, and the orange bar corresponds to GerryFair; the rest is our work.
• Figure 3: Comparison of trained classifiers. We compare our MIO-based approach to a classifier without fairness constraints and GerryFair. The dashed line represents the fairness threshold $\gamma$ used during training. Blue boxes indicate using SD as a proxy for FPSF to identify the violating subgroup; in training, all use the FPSF measure. We evaluate FPSF violation using MIO on conjunction subgroups ($\mathcal{S}_A$).
• Figure 4: Change in fairness, accuracy, and number of cuts, after introducing a sparsity term into the objective. A positive value means an increase in value after introducing the sparsity term.
• Figure 5: Results of bias detection with a time limit of 5 minutes.

Theorems & Definitions (7)

• Definition 1: Statistical Parity Subgroup Fairness
• Definition 2: False Positive Subgroup Fairness
• Definition 3: Maximum Subgroup Discrepancy
• Definition 4: Subgroup Discrepancy
• Theorem 1: SPSF and SD relation
• proof
• Corollary 1.1: Equivalence of SPSF and MSD