SLIDE: a surrogate fairness constraint to ensure fairness consistency

TL;DR

This work tackles fair classification by replacing the intractable 0-1 fairness indicator with a surrogate while preserving asymptotic fairness guarantees. It introduces SLIDE, a non-convex surrogate $ u_ au$ for the indicator, and proves that estimators trained under the SLIDE constraint converge to the truly fair predictor and achieve competitive risk performance. Theoretical results quantify fairness and risk convergence rates in terms of the surrogate parameter $ au$ and model complexity, and experiments on the Adult, Bank, and Law datasets show SLIDE consistently outperforms the common hinge surrogate in DI and UIF settings. The approach offers a scalable, theoretically sound path for in-processing fairness with practical impact for real-world decision systems, and it highlights the idea of partial fairness regions such as the DI-boundary for future work.

Abstract

As they have a vital effect on social decision makings, AI algorithms should be not only accurate and but also fair. Among various algorithms for fairness AI, learning a prediction model by minimizing the empirical risk (e.g., cross-entropy) subject to a given fairness constraint has received much attention. To avoid computational difficulty, however, a given fairness constraint is replaced by a surrogate fairness constraint as the 0-1 loss is replaced by a convex surrogate loss for classification problems. In this paper, we investigate the validity of existing surrogate fairness constraints and propose a new surrogate fairness constraint called SLIDE, which is computationally feasible and asymptotically valid in the sense that the learned model satisfies the fairness constraint asymptotically and achieves a fast convergence rate. Numerical experiments confirm that the SLIDE works well for various benchmark datasets.

Figures (8)

• Figure 1: Comparison of the 0-1, hinge and SLIDE ($\tau = 0.25$) functions (black long dotted line: the $0$-$1$, blue short dotted line: the hinge, and green solid line: the SLIDE).
• Figure 2: (Left) The scatter plot of the 2-D toy dataset (2-component Gaussian Mixture). (Right) Plot of $\alpha$ vs. $d^{\textup{hinge}}_\alpha$ and $d^{\textup{slide}}_{\alpha,\tau}$ for $\tau\in \{0.01, 0.1\}$.
• Figure 3: Box plots of levels of fairness (upper - Con., lower - DI) of fair models learned by the SLIDE-surrogate constraint with $\tau \in \{ 0.01, 0.05, 0.1, 0.2 \}$ and the hinge-surrogate constraint on Adult, Bank, and Law test datasets.
• Figure 4: (Left) Group fairness (DI) Pareto-front lines between DI and Acc on (upper): Adult, (center): Bank, and (lower): Law test datasets. (Right) Individual fairness (UIF) Pareto-front lines between Con. and Acc on (upper): Adult, (center): Bank, and (lower): Law test datasets. These results are averaged on each hyperparameter $\lambda.$ The blue lines are the results of learned models by the SLIDE-surrogate constraint, and the orange lines are those by the hinge-surrogate constraint.
• Figure 5: Individual fairness (UIF) Pareto-front lines between (left) Con. and Acc, and (right) Con. and BA on Adult test dataset. Blue and skyblue lines are those for the SLIDE-surrogate with different $\tau = 0.05, 0.20,$ and the orange lines are those for the hinge-surrogate.

Theorems & Definitions (20)

• Theorem 1: Fairness convergence rate for DI
• Theorem 2: Fairness convergence rate for UIF
• Theorem 3: Risk convergence rates for DI and UIF
• Definition 1: Smooth functions
• Definition 2: Hölder space
• Definition 3: The $\epsilon$-covering number
• Definition 4: Metric entropy
• Definition 5: Rademacher complexity
• Lemma 1
• Lemma 2: Theorem 26.5 of shalev2014understanding