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Impacts of Individual Fairness on Group Fairness from the Perspective of Generalized Entropy

Youngmi Jin, Jio Gim, Tae-Jin Lee, Young-Joo Suh

TL;DR

This work investigates how enforcing a target degree of individual fairness, quantified by generalized entropy $I_{\alpha}$, impacts group fairness via the GE between-group term $V$. It defines a fair ERM (FERM-GE) with a GE constraint and employs a Hedge-based algorithm to obtain a randomized classifier that trades off empirical error and GE. The authors extend GE to continuous spaces, derive PAC-style deviation bounds guaranteeing learnability, and empirically show that stronger individual fairness does not always yield larger group fairness gains, while highlighting an intrinsic accuracy–fairness trade-off. Overall, the paper provides a rigorous, algorithmically tractable framework to study and control the interaction between individual and group fairness using GE, with practical implications for fairness-aware learning.

Abstract

This paper investigates how the degree of group fairness changes when the degree of individual fairness is actively controlled. As a metric quantifying individual fairness, we consider generalized entropy (GE) recently introduced into machine learning community. To control the degree of individual fairness, we design a classification algorithm satisfying a given degree of individual fairness through an empirical risk minimization (ERM) with a fairness constraint specified in terms of GE. We show the PAC learnability of the fair ERM problem by proving that the true fairness degree does not deviate much from an empirical one with high probability for finite VC dimension if the sample size is big enough. Our experiments show that strengthening individual fairness degree does not always lead to enhancement of group fairness.

Impacts of Individual Fairness on Group Fairness from the Perspective of Generalized Entropy

TL;DR

This work investigates how enforcing a target degree of individual fairness, quantified by generalized entropy , impacts group fairness via the GE between-group term . It defines a fair ERM (FERM-GE) with a GE constraint and employs a Hedge-based algorithm to obtain a randomized classifier that trades off empirical error and GE. The authors extend GE to continuous spaces, derive PAC-style deviation bounds guaranteeing learnability, and empirically show that stronger individual fairness does not always yield larger group fairness gains, while highlighting an intrinsic accuracy–fairness trade-off. Overall, the paper provides a rigorous, algorithmically tractable framework to study and control the interaction between individual and group fairness using GE, with practical implications for fairness-aware learning.

Abstract

This paper investigates how the degree of group fairness changes when the degree of individual fairness is actively controlled. As a metric quantifying individual fairness, we consider generalized entropy (GE) recently introduced into machine learning community. To control the degree of individual fairness, we design a classification algorithm satisfying a given degree of individual fairness through an empirical risk minimization (ERM) with a fairness constraint specified in terms of GE. We show the PAC learnability of the fair ERM problem by proving that the true fairness degree does not deviate much from an empirical one with high probability for finite VC dimension if the sample size is big enough. Our experiments show that strengthening individual fairness degree does not always lead to enhancement of group fairness.
Paper Structure (31 sections, 16 theorems, 111 equations, 9 figures, 3 tables, 1 algorithm)

This paper contains 31 sections, 16 theorems, 111 equations, 9 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Definition of Generalized Entropy Index
  4. Applying GE to Algorithmic Fairness
  5. GE as an Individual Fairness
  6. Problem Formulation
  7. Deviation Bounds of Empirical Fairness
  8. FERM-GE
  9. Experiments
  10. Individual Fairness $I_{\alpha}$ vs. Group Fairness $V$
  11. Trade-off between Fairness and Accuracy
  12. Summary
  13. Detail Explanation of Algorithm \ref{['alg:fair']}: A Learning Algorithm Achieving FERM
  14. Experimental Setup and Supplementary Experiments
  15. Data and Implementation
  16. ...and 16 more sections

Key Result

Theorem 1

(Standard VC Dimension Bound) For any distribution $P$ over $\mathcal{X} \times \mathcal{Y}$, let $S =\{(\bm{x}_i, y_i)\}_{i=1}^n$ be a sample data set i.i.d according to $P$. For any $0 < \delta < \frac{1}{2}$ and any $h \in \mathcal{H}$, with probability at least $1-\delta$, it holds that where $d_{\mathcal{H}}$ is the VC dimension of $\mathcal{H}$.

Figures (9)

  • Figure 1: $I_{\alpha}$ and $V$ when $\alpha =1$ ($x$ axis is $\gamma$, left $y$ axis $I_{\alpha}$, and right $y$ axis $V$)
  • Figure 2: Adult income: test error when $a=5$ ($x$ axis is $\gamma$)
  • Figure 4: Comparison with existing algorithms ($y$ axis is error)
  • Figure 5: COMPAS: Averaged test error when $a=5$ ($x$ axis is $\gamma$)
  • Figure 6: COMPAS: Averaged test $I_{\alpha}$ when $a=5$ ($x$ axis is $\gamma$)
  • ...and 4 more figures

Theorems & Definitions (24)

  • Example 1
  • Definition 1
  • Theorem 1
  • Definition 2
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7: Repetition of Theorem \ref{['thm:compare-opt']}
  • ...and 14 more