[Paper Review] Certifying Some Distributional Fairness with Subpopulation Decomposition

Mintong Kang, Linyi Li+4 more|arXiv (Cornell University)|May 31, 2022

Health Systems, Economic Evaluations, Quality of Life4 citations

TL;DR

This paper proposes a novel framework for certifying end-to-end fairness of machine learning models by bounding prediction loss on a fairness-constrained distribution within Hellinger distance of the training distribution. It uses subpopulation decomposition and convex optimization to compute tight, certified upper bounds on model loss under base rate constraints, demonstrating tighter bounds than existing distributional robustness methods and validating its effectiveness across six real-world datasets with both sensitive and general distribution shifts.

ABSTRACT

Extensive efforts have been made to understand and improve the fairness of machine learning models based on observational metrics, especially in high-stakes domains such as medical insurance, education, and hiring decisions. However, there is a lack of certified fairness considering the end-to-end performance of an ML model. In this paper, we first formulate the certified fairness of an ML model trained on a given data distribution as an optimization problem based on the model performance loss bound on a fairness constrained distribution, which is within bounded distributional distance with the training distribution. We then propose a general fairness certification framework and instantiate it for both sensitive shifting and general shifting scenarios. In particular, we propose to solve the optimization problem by decomposing the original data distribution into analytical subpopulations and proving the convexity of the subproblems to solve them. We evaluate our certified fairness on six real-world datasets and show that our certification is tight in the sensitive shifting scenario and provides non-trivial certification under general shifting. Our framework is flexible to integrate additional non-skewness constraints and we show that it provides even tighter certification under different real-world scenarios. We also compare our certified fairness bound with adapted existing distributional robustness bounds on Gaussian data and demonstrate that our method is significantly tighter.

Motivation & Objective

• To define and compute certified fairness for end-to-end ML models, ensuring worst-case performance bounds on fairness-constrained test distributions.
• To address the lack of certified fairness in high-stakes ML applications where training data may contain biases or distributional shifts.
• To provide a flexible framework that integrates fairness constraints, such as base rate conditions, and supports additional non-skewness constraints.
• To improve upon existing distributional robustness bounds by incorporating fairness constraints and achieving tighter certification.
• To validate the framework’s tightness and practicality across diverse real-world datasets under both sensitive and general distribution shifts.

Proposed method

• Formulates certified fairness as a constrained optimization problem: minimizing the worst-case prediction loss on a fairness-constrained distribution within bounded Hellinger distance from the training distribution.
• Decomposes the training and test distributions into subpopulations based on sensitive attributes and labels to encode base rate fairness constraints.
• Proves convexity of the resulting low-dimensional optimization problems in the sensitive shifting scenario, enabling efficient solution via convex optimization.
• Applies a divide-and-conquer strategy in the general shifting scenario, partitioning the feasible space and solving relaxed sub-problems with proven convexity.
• Uses Hellinger distance as the divergence measure to bound distributional shifts, enabling generalization to any black-box model.
• Integrates additional non-skewness constraints (on labels or sensitive attributes) to further tighten the fairness certificates.

Experimental results

Research questions

• RQ1 Can we define and compute a certified upper bound on model prediction loss under fairness constraints within a bounded distributional shift?
• RQ2 How does subpopulation decomposition enable convex optimization and tight fairness certification under sensitive and general distribution shifts?
• RQ3 Does integrating non-skewness constraints on labels or sensitive attributes improve the tightness of the fairness certificate?
• RQ4 How does the proposed framework compare in tightness to existing distributional robustness bounds under the same conditions?
• RQ5 Can the fairness certificate serve as a reliable indicator of actual model fairness across real-world datasets?

Key findings

• The proposed framework achieves significantly tighter fairness certificates than existing distributional robustness bounds, particularly when fairness constraints are integrated.
• In the sensitive shifting scenario, the certification is tight and consistently bounds the worst-case loss across all Hellinger distance levels tested.
• For the general shifting scenario, the framework provides non-trivial, progressively increasing upper bounds on loss as the distributional shift increases.
• Adding non-skewness constraints on labels or sensitive attributes results in tighter fairness certificates across all evaluated datasets.
• The fairness certificate correlates strongly with actual model fairness: the perfectly fair model consistently achieves lower certified loss than the unfair (vanilla) model on Adult, Compas, Health, and Law School datasets.
• On the Adult dataset, the vanilla model’s certified loss under general shifting increases from 0.274 at ρ=0.1 to 0.828 at ρ=0.5, while the fair model’s loss increases from 0.266 to 0.824, showing the certificate reflects relative fairness levels.