Fair Representation Learning for Continuous Sensitive Attributes using Expectation of Integral Probability Metrics

TL;DR

This work addresses fair representation learning when the sensitive attribute is continuous by introducing the Expectation of IPMs (EIPM) as a principled fairness metric and developing FREM (Fair Representation using EIPM). EIPM compares the conditional and marginal distributions of the learned representation across the continuous sensitive attribute, and is estimated with a kernel-smoothed, finite-sample approach using MMD as the discriminator. The paper proves convergence and asymptotic fairness guarantees for the estimator and demonstrates, through extensive experiments on tabular and graph data, that FREM achieves superior fairness-accuracy trade-offs relative to both continuous-attribute baselines and traditional FRL methods that rely on binning. The results highlight the practical potential of EIPM-based FRL to deliver fair downstream predictions without requiring discretization of the sensitive attribute, with robust performance across kernels and data domains.

Abstract

AI fairness, also known as algorithmic fairness, aims to ensure that algorithms operate without bias or discrimination towards any individual or group. Among various AI algorithms, the Fair Representation Learning (FRL) approach has gained significant interest in recent years. However, existing FRL algorithms have a limitation: they are primarily designed for categorical sensitive attributes and thus cannot be applied to continuous sensitive attributes, such as age or income. In this paper, we propose an FRL algorithm for continuous sensitive attributes. First, we introduce a measure called the Expectation of Integral Probability Metrics (EIPM) to assess the fairness level of representation space for continuous sensitive attributes. We demonstrate that if the distribution of the representation has a low EIPM value, then any prediction head constructed on the top of the representation become fair, regardless of the selection of the prediction head. Furthermore, EIPM possesses a distinguished advantage in that it can be accurately estimated using our proposed estimator with finite samples. Based on these properties, we propose a new FRL algorithm called Fair Representation using EIPM with MMD (FREM). Experimental evidences show that FREM outperforms other baseline methods.

Figures (30)

• Figure 1: A framework diagram of FRL for binary sensitive attributes.
• Figure 2: A framework diagram of FRL for continuous sensitive attributes.
• Figure 3: Simulation results: Box plots of the differences between the true EIPM and the two estimators (the best binning estimator and the best proposed estimator). It is clear that our proposed estimator is more accurate. (Left)$w_{1} = \sqrt{0.2}, w_{2} = \sqrt{0.8}.$(Center)$w_{1} = \sqrt{0.5}, w_{2} = \sqrt{0.5}.$(Right)$w_{1} = \sqrt{0.8}, w_{2} = \sqrt{0.2}.$
• Figure 4: Demographic Parity: Pareto-front lines for fairness-prediction trade-off. (Top) Adult dataset, $\Delta \texttt{GDP}$ vs. ACC. (Bottom) Crime dataset, $\Delta \texttt{GDP}$ vs. 1 - MAE. $\bullet$: Unfair, --$\blacktriangleleft$--: Reg-GDP, --$\blacktriangleleft$--: Reg-HGR, --$\blacktriangleleft$--: ADV, --$\blacktriangleleft$--: sIPM-LFR, --$\blacktriangleleft$--: MMD, --$\blacktriangleleft$--: LAFTR, --$\mathbf{\filledstar}$--: FREM.
• Figure 5: Comparison of FRL methods in terms of fairness of learned representations: Mutual information between $\bm{Z}$ and $S$ for five FRL methods. (Left) Adult (Right) Crime. Categorized by median for LAFTR, MMD, and sIPM-LFR.

Theorems & Definitions (33)

• Theorem 1: Perfect fairness
• Theorem 2: Controlling the level of fairness by EIPM
• Example 1: Hölder smooth functions
• Example 2: Lipschitz continuous functions
• Remark 1
• Remark 2
• Proposition 3
• Theorem 4: Convergence of proposed estimator
• Remark 3
• Theorem 5: Asymptotically equivalence of the constraints