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Does Machine Bring in Extra Bias in Learning? Approximating Fairness in Models Promptly

Yijun Bian, Yujie Luo

TL;DR

This work defines a manifold-based fairness framework that unifies data-related and algorithmic discrimination through set-distance measures. The core idea is to compute a harmonic fairness measure $\mathbf{df}(f)=\frac{\mathbf{D}_f(S_0,S_1)}{\mathbf{D}(S_0,S_1)}-1$ using distances between groups, and to make this computation practical with ApproxDist, a projection-based algorithm that reduces complexity to $\mathcal{O}(n\log n)$. The authors provide theoretical guarantees for ApproxDist and validate them empirically on multiple datasets, showing that HFM can capture discriminative risk and offers competitive or complementary insights beyond traditional group fairness metrics. The approach enables scalable, dual-perspective fairness analysis suitable for high-stakes domains and real-world deployment, with findings supported by extensive experiments and parameter analyses.

Abstract

Providing various machine learning (ML) applications in the real world, concerns about discrimination hidden in ML models are growing, particularly in high-stakes domains. Existing techniques for assessing the discrimination level of ML models include commonly used group and individual fairness measures. However, these two types of fairness measures are usually hard to be compatible with each other, and even two different group fairness measures might be incompatible as well. To address this issue, we investigate to evaluate the discrimination level of classifiers from a manifold perspective and propose a "harmonic fairness measure via manifolds (HFM)" based on distances between sets. Yet the direct calculation of distances might be too expensive to afford, reducing its practical applicability. Therefore, we devise an approximation algorithm named "Approximation of distance between sets (ApproxDist)" to facilitate accurate estimation of distances, and we further demonstrate its algorithmic effectiveness under certain reasonable assumptions. Empirical results indicate that the proposed fairness measure HFM is valid and that the proposed ApproxDist is effective and efficient.

Does Machine Bring in Extra Bias in Learning? Approximating Fairness in Models Promptly

TL;DR

This work defines a manifold-based fairness framework that unifies data-related and algorithmic discrimination through set-distance measures. The core idea is to compute a harmonic fairness measure using distances between groups, and to make this computation practical with ApproxDist, a projection-based algorithm that reduces complexity to . The authors provide theoretical guarantees for ApproxDist and validate them empirically on multiple datasets, showing that HFM can capture discriminative risk and offers competitive or complementary insights beyond traditional group fairness metrics. The approach enables scalable, dual-perspective fairness analysis suitable for high-stakes domains and real-world deployment, with findings supported by extensive experiments and parameter analyses.

Abstract

Providing various machine learning (ML) applications in the real world, concerns about discrimination hidden in ML models are growing, particularly in high-stakes domains. Existing techniques for assessing the discrimination level of ML models include commonly used group and individual fairness measures. However, these two types of fairness measures are usually hard to be compatible with each other, and even two different group fairness measures might be incompatible as well. To address this issue, we investigate to evaluate the discrimination level of classifiers from a manifold perspective and propose a "harmonic fairness measure via manifolds (HFM)" based on distances between sets. Yet the direct calculation of distances might be too expensive to afford, reducing its practical applicability. Therefore, we devise an approximation algorithm named "Approximation of distance between sets (ApproxDist)" to facilitate accurate estimation of distances, and we further demonstrate its algorithmic effectiveness under certain reasonable assumptions. Empirical results indicate that the proposed fairness measure HFM is valid and that the proposed ApproxDist is effective and efficient.
Paper Structure (23 sections, 2 theorems, 21 equations, 8 figures, 5 tables, 2 algorithms)

This paper contains 23 sections, 2 theorems, 21 equations, 8 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Sources of biases
  4. Mechanisms to enhance fairness
  5. Types of fairness measures
  6. Methodology
  7. Model fairness assessment from a manifold perspective
  8. A prompt approximation of distances between sets for Euclidean spaces
  9. Algorithmic effectiveness analysis of ApproxDist
  10. Empirical Results
  11. Experimental setups
  12. Datasets
  13. Evaluation metrics
  14. Baseline fairness measures
  15. Implementation details
  16. ...and 8 more sections

Key Result

Lemma 1

Let $\bm{v}_1$ (resp. $\bm{v}_2$) be a vector in the $n$-dimensional Euclidean space $\mathbb{R}^n$ with length $r_1$ (resp. $r_2$) such that $r_1\leqslant r_2$. Let $\bm{w}\subset \mathbb{R}^n$ be a unit vector. We define $\mathbb{P}(\bm{v}_1,\bm{v}_2)$ as the probability that $|\langle \bm{w},\bm{ here $\phi$ represents the angle between $\bm{v}_1$ and $\bm{v}_2$.

Figures (8)

  • Figure 1: Comparison of baseline fairness measures and the proposed HFM, evaluated on test data. (a,d) Scatter plots cruz2022fairgbm showing fairness and accuracy (resp. $\mathrm{f}_1$ score) of each fairness measure; (b,e) Scatter plots showing fairness and the variation of test accuracy (resp. $\mathrm{f}_1$ score) performance between the raw and disturbed data, where in the disturbed data, only sensitive attributes are changed if there are any. \ref{['subfig:nondelta,e']} Plot cruz2022fairgbm of trade-offs between fairness and error rates per fairness measure, and \ref{['subfig:nondelta,f']} Plot of trade-offs between fairness and $(1-\mathrm{f}_1\text{ score})$ per fairness measure; Lines in \ref{['subfig:nondelta,e']} and \ref{['subfig:nondelta,f']} show the mean value, and shades show 95% confidence intervals; Also note that in \ref{['subfig:nondelta,e']} and \ref{['subfig:nondelta,f']}, the smaller the better.
  • Figure 2: Comparison of baseline fairness measures and the proposed HFM. (a) Friedman test chart (non-overlapping means significant difference) on the test accuracy, which rejects the null hypothesis that "all fairness-relevant rules have the same evaluation performance" at the significance level of 5%, and where CD means the critical difference of average rank difference, calculated by Nemenyi post-hoc test zhou2021machine; (b) The aggregated rank of each fairness-relevant rule (the smaller the better) qian2015pareto on the test accuracy; (d--e) Friedman test chart and the aggregated rank of each fairness-relevant rule on the $\mathrm{f}_1$ score, evaluated on test data; (c) and (f) The aggregated rank of each fairness-relevant rule on the discriminative risk (DR), where the ranking rule in (a--c) is $0.05\times\text{error rate}+0.95\times\text{fairness}$, and that in (d--f) is $0.1\times(1-\mathrm{f}_1\text{ score})+0.9\times\text{fairness}$.
  • Figure 3: Comparison of approximation distances between sets with precise distances that are calculated directly by definition, evaluated on test data. (a) Scatter plot showing approximated values and precise values of distances between sets, and correlation here refers to the Pearson correlation coefficient; (b) Relative difference comparison of ApproxDist with direct computation concerning distance values. (c--d) Comparison of time cost (second) between ApproxDist and direct computation based on Eq. \ref{['eq:5']}.
  • Figure 4: Effects of hyperparameters $m_1$ and $m_2$ in ApproxDist. (a--b) The effect of the hyperparameter $m_2$ on the distance value; \ref{['subfig:pm,c']} The effect of the hyperparameter $m_2$ on the time cost, where $m_1$ is set to 20. (d--e) The effect of the hyperparameter $m_1$ on the distance value; \ref{['subfig:pm,f']} The effect of the hyperparameter $m_1$ on the time cost, where $m_2$ is set to $\lceil2\lg~n\rceil$ in terms of $n$---the size of the corresponding dataset.
  • Figure 5: Illustration of Lemma \ref{['projection 1']}. The dashed lines represent the hyperplanes that are perpendicular to $\bm{v}_1-\bm{v}_2$ and $\bm{v}_1+\bm{v}_2$ respectively, and $\theta$ denotes the acute angle between the hyperplanes.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Lemma 1
  • Proposition 2
  • proof : Proof of Lemma \ref{['projection 1']}
  • proof : Proof of Proposition \ref{['density']}