[Paper Review] A Convex Framework for Fair Regression

Richard A. Berk, Hoda Heidari+6 more|arXiv (Cornell University)|Jun 7, 2017

Ethics and Social Impacts of AISocial Sciences19 references192 citations

TL;DR

The paper introduces a convex family of fairness regularizers for linear and logistic regression, enabling efficient optimization and the exploration of accuracy-fairness trade-offs via a Pareto frontier and Price of Fairness across six datasets.

ABSTRACT

We introduce a flexible family of fairness regularizers for (linear and logistic) regression problems. These regularizers all enjoy convexity, permitting fast optimization, and they span the rang from notions of group fairness to strong individual fairness. By varying the weight on the fairness regularizer, we can compute the efficient frontier of the accuracy-fairness trade-off on any given dataset, and we measure the severity of this trade-off via a numerical quantity we call the Price of Fairness (PoF). The centerpiece of our results is an extensive comparative study of the PoF across six different datasets in which fairness is a primary consideration.

Motivation & Objective

• Motivate and formalize fairness in regression settings beyond classification.
• Introduce a flexible convex regularizer family that enforces fairness between groups or individuals.
• Enable computation of the accuracy-fairness Pareto frontier by varying a fairness weight.
• Quantify the trade-off severity with a data-dependent Price of Fairness (PoF).
• Provide an extensive empirical study across multiple datasets to compare fairness notions and treatments of protected variables.

Proposed method

• Define a weighted objective combining regression loss, L2 regularization, and a convex fairness regularizer with weight lambda.
• Propose two core fairness penalties: f1 for individual fairness and f2 for group fairness, each operating on cross-group instance pairs and weighted by d(y_i,y_j).
• Extend to a single-model setting and a separate-model setting for the protected groups.
• Allow hybrid fairness by grouping cross-pairs by label in the binary case and applying a squared average penalty.
• Normalize datasets and select d(y_i,y_j) appropriately for real-valued and binary targets (e^(-(y_i-y_j)^2) and 1[y_i=y_j], respectively).
• Compute the Pareto frontier by solving min_w l(w,S) + lambda f(w,S) + gamma||w||^2 for varying lambda.

Experimental results

Research questions

• RQ1 How can we regularize regression to enforce fairness across protected groups or individuals in a convex optimization framework?
• RQ2 What is the impact of different fairness notions (individual, group, hybrid) on regression accuracy across datasets?
• RQ3 Does allowing separate models per group materially improve the fairness-accuracy trade-off compared to a single shared model?
• RQ4 How severe is the accuracy cost as fairness requirements tighten, and how can we quantify it across datasets?
• RQ5 How does the choice of d(y_i,y_j) influence the fairness-accuracy frontier and PoF?

Key findings

• A flexible, convex family of fairness regularizers spans from individual to group fairness in regression tasks.
• Varying the fairness weight yields a Pareto frontier showing accuracy-fairness trade-offs across datasets.
• The Price of Fairness (PoF) quantifies the relative accuracy loss under fairness constraints and varies by dataset and fairness notion.
• In the empirical study across six datasets, PoF is generally higher for individual fairness than group fairness, with notable domain-dependent variation.
• Separate models for groups often provide little to no improvement and can even worsen out-of-sample performance due to overfitting.
• Hybrid fairness can yield different trade-offs depending on the dataset and labeling regime, underscoring that there are no universal fairness prescriptions.