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Disparate Conditional Prediction in Multiclass Classifiers

Sivan Sabato, Eran Treister, Elad Yom-Tov

TL;DR

This work extends the Disparate Conditional Prediction (DCP) fairness auditing framework from binary to multiclass classifiers under multiclass equalized odds. It develops two auditing regimes: (i) known confusion matrices per protected sub-population, enabling exact-like bounds via local optimization, and (ii) unknown confusion matrices, where bounds rely on population-level frequencies. The authors formulate a sequential linear programming approach with eta-splitting to handle non-smooth objective terms, and they propose greedy initializations to improve convergence. Empirical results on US Census, Natality, and other datasets demonstrate that the proposed bounds are typically tight, enabling practical identification of classifiers with substantial disparate conditional predictions. The work provides open-source code and lays groundwork for robust, interpretation-friendly fairness auditing in multiclass contexts.

Abstract

We propose methods for auditing multiclass classifiers for fairness under multiclass equalized odds,by estimating the deviation from equalized odds when the classifier is not completely fair. We generalize to multiclass classifiers the measure of Disparate Conditional Prediction (DCP), originally suggested by Sabato & Yom-Tov (2020) for binary classifiers. DCP is defined as the fraction of the population for which the classifier predicts with conditional prediction probabilities that differ from the closest common baseline. We provide new local-optimization methods for estimating the multiclass DCPunder two different regimes,one in which the conditional confusion matrices for each protected sub-population are known, and one in which these cannot be estimated, for instance, because the classifier is inaccessible or because good-quality individual-level data is not available. These methods can be used to detect classifiers that likely treat a significant fraction of the population unfairly. Experiments demonstrate the accuracy of the methods. Code is provided at https://github.com/sivansabato/DCPmulticlass.

Disparate Conditional Prediction in Multiclass Classifiers

TL;DR

This work extends the Disparate Conditional Prediction (DCP) fairness auditing framework from binary to multiclass classifiers under multiclass equalized odds. It develops two auditing regimes: (i) known confusion matrices per protected sub-population, enabling exact-like bounds via local optimization, and (ii) unknown confusion matrices, where bounds rely on population-level frequencies. The authors formulate a sequential linear programming approach with eta-splitting to handle non-smooth objective terms, and they propose greedy initializations to improve convergence. Empirical results on US Census, Natality, and other datasets demonstrate that the proposed bounds are typically tight, enabling practical identification of classifiers with substantial disparate conditional predictions. The work provides open-source code and lays groundwork for robust, interpretation-friendly fairness auditing in multiclass contexts.

Abstract

We propose methods for auditing multiclass classifiers for fairness under multiclass equalized odds,by estimating the deviation from equalized odds when the classifier is not completely fair. We generalize to multiclass classifiers the measure of Disparate Conditional Prediction (DCP), originally suggested by Sabato & Yom-Tov (2020) for binary classifiers. DCP is defined as the fraction of the population for which the classifier predicts with conditional prediction probabilities that differ from the closest common baseline. We provide new local-optimization methods for estimating the multiclass DCPunder two different regimes,one in which the conditional confusion matrices for each protected sub-population are known, and one in which these cannot be estimated, for instance, because the classifier is inaccessible or because good-quality individual-level data is not available. These methods can be used to detect classifiers that likely treat a significant fraction of the population unfairly. Experiments demonstrate the accuracy of the methods. Code is provided at https://github.com/sivansabato/DCPmulticlass.
Paper Structure (16 sections, 1 theorem, 30 equations, 4 figures, 10 tables, 1 algorithm)

This paper contains 16 sections, 1 theorem, 30 equations, 4 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Paper structure.
  3. Related Work
  4. Preliminaries
  5. The $\mathrm{DCP}$ Measure for Multiclass Classifiers
  6. Bounding the $\mathrm{DCP}$ of a Multiclass Classifier
  7. Best-case $\mathrm{DCP}$ without Confusion Matrices
  8. Experiments
  9. Bounding the $\mathrm{DCP}$ with Known Confusion Matrices
  10. Bounding min$\mathrm{DCP}$ without Confusion Matrices
  11. Conclusion
  12. Deferred Experiment Results
  13. An Additional Experiment
  14. The Greedy Initialization Procedure
  15. More Details on the Local Optimization Procedure
  16. ...and 1 more sections

Key Result

Theorem 4.1

For multiclass classification, Eq. (eq:unfbasic) implies where $\boldsymbol{\alpha}_{\textsf{b}}^y := (\alpha_{{\textsf{b}}}^{y\hat{y}})_{\hat{y}\in \mathcal{Y}}$.

Figures (4)

  • Figure 1: The function $\eta(a,b)$.
  • Figure 2: The linear approximation of $\eta(\alpha,b)$ using the split to $\eta_1$ and $\eta_2$ around $\alpha_0=0.35$, for a fixed $b=0.25$.
  • Figure 3: Calculated value $\texttt{min$\mathrm{DCP}$}$ for each election year in the UK elections dataset, where the value is reported for the classifier that attempts to predict this year's election result using the results of the previous election.
  • Figure 4: The convergence of Algorithm \ref{['alg:FairV1']} for different trust region (maximal step size) parameters $\tau$.

Theorems & Definitions (2)

  • Theorem 4.1
  • proof