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How Far Can Fairness Constraints Help Recover From Biased Data?

Mohit Sharma, Amit Deshpande

TL;DR

This work shows that carefully chosen fairness constraints, notably equal opportunity, can recover optimal and fair classifiers on the original distribution even when training data are severely biased. By revealing that several data-bias mechanisms induce a linear fractional transformation of the group-aware regression function, the authors derive threshold-based characterizations that align biased-distribution optima with the Bayes optimal fair classifier under concrete conditions. The contributions include extending recovery to Massart noise, introducing reject-option and ε-robustness frameworks for arbitrary distributions and hypothesis classes, and modeling time-varying bias in multi-stage pipelines with both infinite and finite horizons. These results highlight a principled route to mitigating bias-induced fairness-accuracy trade-offs in practical settings, with implications for robust, fair decision-making in dynamic environments.

Abstract

A general belief in fair classification is that fairness constraints incur a trade-off with accuracy, which biased data may worsen. Contrary to this belief, Blum & Stangl (2019) show that fair classification with equal opportunity constraints even on extremely biased data can recover optimally accurate and fair classifiers on the original data distribution. Their result is interesting because it demonstrates that fairness constraints can implicitly rectify data bias and simultaneously overcome a perceived fairness-accuracy trade-off. Their data bias model simulates under-representation and label bias in underprivileged population, and they show the above result on a stylized data distribution with i.i.d. label noise, under simple conditions on the data distribution and bias parameters. We propose a general approach to extend the result of Blum & Stangl (2019) to different fairness constraints, data bias models, data distributions, and hypothesis classes. We strengthen their result, and extend it to the case when their stylized distribution has labels with Massart noise instead of i.i.d. noise. We prove a similar recovery result for arbitrary data distributions using fair reject option classifiers. We further generalize it to arbitrary data distributions and arbitrary hypothesis classes, i.e., we prove that for any data distribution, if the optimally accurate classifier in a given hypothesis class is fair and robust, then it can be recovered through fair classification with equal opportunity constraints on the biased distribution whenever the bias parameters satisfy certain simple conditions. Finally, we show applications of our technique to time-varying data bias in classification and fair machine learning pipelines.

How Far Can Fairness Constraints Help Recover From Biased Data?

TL;DR

This work shows that carefully chosen fairness constraints, notably equal opportunity, can recover optimal and fair classifiers on the original distribution even when training data are severely biased. By revealing that several data-bias mechanisms induce a linear fractional transformation of the group-aware regression function, the authors derive threshold-based characterizations that align biased-distribution optima with the Bayes optimal fair classifier under concrete conditions. The contributions include extending recovery to Massart noise, introducing reject-option and ε-robustness frameworks for arbitrary distributions and hypothesis classes, and modeling time-varying bias in multi-stage pipelines with both infinite and finite horizons. These results highlight a principled route to mitigating bias-induced fairness-accuracy trade-offs in practical settings, with implications for robust, fair decision-making in dynamic environments.

Abstract

A general belief in fair classification is that fairness constraints incur a trade-off with accuracy, which biased data may worsen. Contrary to this belief, Blum & Stangl (2019) show that fair classification with equal opportunity constraints even on extremely biased data can recover optimally accurate and fair classifiers on the original data distribution. Their result is interesting because it demonstrates that fairness constraints can implicitly rectify data bias and simultaneously overcome a perceived fairness-accuracy trade-off. Their data bias model simulates under-representation and label bias in underprivileged population, and they show the above result on a stylized data distribution with i.i.d. label noise, under simple conditions on the data distribution and bias parameters. We propose a general approach to extend the result of Blum & Stangl (2019) to different fairness constraints, data bias models, data distributions, and hypothesis classes. We strengthen their result, and extend it to the case when their stylized distribution has labels with Massart noise instead of i.i.d. noise. We prove a similar recovery result for arbitrary data distributions using fair reject option classifiers. We further generalize it to arbitrary data distributions and arbitrary hypothesis classes, i.e., we prove that for any data distribution, if the optimally accurate classifier in a given hypothesis class is fair and robust, then it can be recovered through fair classification with equal opportunity constraints on the biased distribution whenever the bias parameters satisfy certain simple conditions. Finally, we show applications of our technique to time-varying data bias in classification and fair machine learning pipelines.
Paper Structure (22 sections, 20 theorems, 82 equations, 2 figures)

This paper contains 22 sections, 20 theorems, 82 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Data Bias Models & Fair Classification
  4. Regression Functions on Biased Data
  5. Fair Classification on Biased Data
  6. Recovering Optimal Classifier from Biased Data for Massart Label Noise
  7. Generalizing Blum & Stangl blum2019recovering Recovery Result for Massart Noise
  8. Proof Sketches
  9. Recovery of Optimal Reject Option Classifiers from Biased Data for Arbitrary Data Distributions
  10. Recovering Robust Hypothesis under Data Bias for Arbitrary Data Distributions and Arbitrary Hypothesis Classes
  11. Recovering from Time-Varying Data Bias
  12. Repeated Data Bias & Infinite Time Horizon
  13. Time-Varying Data Bias Pipeline with Finite Steps
  14. Impact Statement
  15. Proofs
  16. ...and 7 more sections

Key Result

Proposition 1

Suppose $S \geq 0$, $R+S \geq 0$, and $PS - QR \geq 0$, then the transformation $\tilde{\eta}(x, a) = \dfrac{P \eta(x, a) + Q}{R \eta(x, a) + S}$ is order-preserving, i.e., $\eta(x_{1}, a) \leq \eta(x_{2}, a)$ iff $\tilde{\eta}(x_{1}, a) \leq \tilde{\eta}(x_{2}, a)$.

Figures (2)

  • Figure 1: Recovery region for $\beta_{p}, \beta_{n} \in (0, 1]$ given by the constraints $(1-r)(1-2\delta) + r\left((1-\delta) \beta_{p} (1-2\nu) - \delta \beta_{n}\right) > 0$ and $(1-r)(1-2\delta) + r\left((1-\delta) \beta_{n} - \delta \beta_{p} (1-2\nu)\right) > 0$ as in Theorem \ref{['thm:blum_stangl_eo_massart']}, when $r=0.25$, $\nu=0.05$, and $\delta=0.45$. We can recover optimal and fair classifiers for a large range of data biases, including extreme under-representation, i.e., region close to the origin $(0, 0)$, by applying just equal opportunity constraints.
  • Figure 2: Recovery region for $\beta_{p}, \beta_{n} \in (0, 1]$ given by the constraints $r \left((1-\nu)\beta_{p} - (1-\epsilon)\beta_{n}\right) + \epsilon (1-r) \geq 0$ and $r \left((1+\epsilon)\beta_{n} - (1-\nu)\beta_{p}\right) + \epsilon (1-r) \geq 0$ as in Theorem \ref{['thm:eo-robust-recovery']}, when $r=0.2$, $\nu=0.1$, and $\epsilon=0.05$. Even for arbitrary distributions and hypothesis classes, optimal and fair classifiers can be recovered from extreme under-representation and for a large range of data biases using just equal opportunity constraints.

Theorems & Definitions (41)

  • Example 1
  • Example 2
  • Example 3
  • Proposition 1
  • Proposition 2
  • Corollary 3
  • Remark
  • Proposition 4
  • Theorem 5
  • Theorem 6
  • ...and 31 more