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Fairness Testing through Extreme Value Theory

Verya Monjezi, Ashutosh Trivedi, Vladik Kreinovich, Saeid Tizpaz-Niari

arXiv:2501.11597v1cs.SEcs.AIcs.CYcs.LG

TL;DR

Average-focused fairness can miss consequential discrimination in extreme outcomes. This paper defines ECD using GEV tail modeling and finds that its ECD-Fair mitigator outperforms MiniMax-Fairness and reduces tail discrimination in 90% of cases.

  • Problem

    Average-focused fairness can overlook socially influential edge cases, while the literature has given limited attention to fairness in extreme outcomes.

    Source: §I · p. 1

  • Method

    The paper defines ECD as the maximum counterfactual outcome change under a protected-attribute flip and uses GEV distributions to quantify tail fairness.

    Source: §I · p. 2

  • Results

    Average-based mitigators increased worst-case discrimination in 35% of cases while preserving or improving average fairness in 63%; ECD-Fair reduced tail discrimination in 90% of cases.

    Source: §I · p. 2

  • Takeaways & Limitations

    EVT-based analysis models worst-case counterfactual bias with statistical guarantees and reveals limitations of a state-of-the-art bias-reduction algorithm.

    Source: §VIII · p. 11

Abstract

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Data-driven software is increasingly being used as a critical component of automated decision-support systems. Since this class of software learns its logic from historical data, it can encode or amplify discriminatory practices. Previous research on algorithmic fairness has focused on improving average-case fairness. On the other hand, fairness at the extreme ends of the spectrum, which often signifies lasting and impactful shifts in societal attitudes, has received significantly less emphasis. Leveraging the statistics of extreme value theory (EVT), we propose a novel fairness criterion called extreme counterfactual discrimination (ECD). This criterion estimates the worst-case amounts of disadvantage in outcomes for individuals solely based on their memberships in a protected group. Utilizing tools from search-based software engineering and generative AI, we present a randomized algorithm that samples a statistically significant set of points from the tail of ML outcome distributions even if the input dataset lacks a sufficient number of relevant samples. We conducted several experiments on four ML models (deep neural networks, logistic regression, and random forests) over 10 socially relevant tasks from the literature on algorithmic fairness. First, we evaluate the generative AI methods and find that they generate sufficient samples to infer valid EVT distribution in 95% of cases. Remarkably, we found that the prevalent bias mitigators reduce the average-case discrimination but increase the worst-case discrimination significantly in 5% of cases. We also observed that even the tail-aware mitigation algorithm -- MiniMax-Fairness -- increased the worst-case discrimination in 30% of cases. We propose a novel ECD-based mitigator that improves fairness in the tail in 90% of cases with no degradation of the average-case discrimination.

I. INTRODUCTION

The introduction argues that average-focused fairness can overlook consequential discrimination in extreme outcomes, and presents EVT-based ECD and tail-oriented testing and mitigation.

I. INTRODUCTION

  • Data-driven systems learn from historical data and can encode or amplify its biases, motivating tools to assess and mitigate tail harms.Source: §I · p. 1
  • Average-focused fairness can miss socially influential edge cases, despite parity in averages and continuing disparities in high-paying jobs and leadership.Source: §I · p. 1
  • Extreme counterfactual discrimination (ECD) measures the maximum outcome change when a protected attribute is flipped, complementing average causal discrimination.Source: §I · p. 2
  • GEV modeling directly represents the tail, enabling checks of tail validity and statistical guarantees for extreme-discrimination estimates, unlike CVaR.Source: §I · p. 2
  • The randomized testing algorithm uses an exponentiality test and realistic GAN- or VAE-generated samples to address scarce extreme-tail data.Source: §I · p. 2
  • ECD-Fair reduces tail discrimination in 90% of cases and improves average fairness in 45%, outperforming MiniMax-Fairness.Source: §I · p. 2

II. OVERVIEW

The overview introduces threshold-based EVT to estimate extreme counterfactual discrimination (ECD) and illustrates it on an Adult Census DNN. In that example, average-based mitigation improves average fairness while increasing tail discrimination.

Extreme Value Theory

  • The overview uses the EVT threshold approach, fitting a GEV distribution to values above a selected high threshold.The threshold must balance excluding non-tail values against retaining enough tail samples for confidence.Source: §II · p. 2

Dataset and ML Model

  • The example uses 48,842 Adult Census records to predict income above 50K and compares white and black individuals.The DNN has 84% test accuracy and an average odd difference of 0.05.Source: §II · p. 3

Test-case Generations

  • The search algorithm samples 25,658 white and 3,076 black cases, with mean CD -0.04 for white and 0.03 for black.Source: §II · p. 3

Fairness Measures through Extreme Value Distributions

  • In the original DNN, the average CDs for white and black groups differ by 0.07 in favor of white individuals, while expected extreme CDs differ by 0.13.Source: §II · p. 3
  • EG lowers AOD to 0.02 with 2% accuracy loss, but raises the white-group 2,000-interaction return level from 0.16 to 0.95.For the black group, the expected return level rises from 0.47 to 0.81.Source: §II · p. 3

Tail-Aware Bias Reduction Algorithms

  • MiniMax-Fairness reduces ECD to -0.27 while degrading average-based fairness by 0.04; the paper's in-process mitigator reduces ECD to 0.02 with 0.01 degradation.Source: §II · p. 3

III. EXTREME COUNTERFACTUAL DISCRIMINATION

This section defines counterfactual discrimination as an outcome difference between an individual and a counterfactual differing in a protected attribute. ECD focuses on worst-case group values and statistically significant differences between groups.

Definition of CD

  • Counterfactual discrimination is the difference between an individual's model outcome and the outcome after changing the protected attribute while holding other protected attributes fixed.Positive values indicate disadvantage for the individual's group; negative values indicate advantage.Source: §III · p. 4

Average Causal Discrimination

  • Average causal discrimination summarizes CD by group using its expected value for privileged and unprivileged individuals.Source: §III · p. 4

Related Work

  • THEMIS uses z-score testing under a normal-distribution assumption, while the paper argues fairness should be assessed both on average and in the tail.Source: §III · p. 4

Definition of ECD

  • ECD models each group's maximum counterfactual discrimination, tests whether the group difference is statistically significant, provides worst-case guarantees, and supports tail-bias mitigation.Source: §III · p. 4

IV. APPROACH

The approach learns demographic-specific data distributions, searches for statistically adequate tail samples, fits group-wise GEV distributions, and compares their expected worst-case values. It also adapts hyperparameter optimization to target ECD while retaining accuracy constraints.

Approach Overview

  • The approach has three stages: learn target-group distributions, collect tail samples with statistical guarantees, then fit and compare group-wise GEV distributions.Source: §IV · p. 4

Learning the Underlying Distributions

  • GANs and VAEs model data distributions, while CTGAN and TVAE generate target-group tabular samples that reflect the group's underlying distribution.Generation is used to address scarce samples in some protected groups and avoid unrealistic variable combinations from uniform sampling.Source: §IV · pp. 4–5

Collecting Tail Samples

  • The search evaluates the coefficient of variation across the k highest CD values and accepts tail samples when each value is below 1.0 + 1/(4k).If the test fails, CTGAN or TVAE generates samples and the search repeats until the test passes or times out.Source: §IV · p. 5

Inferring Tail Distributions

  • For each group, the method fits a GEV using the k_max largest measurements and treats it as valid when the shape is statistically consistent with ξ <= 0.It compares expected worst-case values and scale; the group difference is µu − µp.Source: §IV · p. 5

Tail-Aware Bias Mitigation

  • The ECD-based mitigator extends PARFAIT-ML by optimizing ECD rather than AOD while keeping the accuracy constraints unchanged.Source: §IV · p. 5

V. EXPERIMENTS

The experiments assess realistic test generation, EVT's ability to quantify tail discrimination, and average-based and tail-aware mitigation. They span socially critical datasets and several model and mitigation families.

Research Questions

  • The study asks how an EVT-based mitigator compares with existing tail-aware bias-reduction methods.Source: §V · p. 6

Datasets and Models

  • The evaluation uses nine fairness datasets and four model families: DNN, logistic regression, SVM, and random forest.The DNN uses a six-layer architecture; the other models use scikit-learn defaults.Source: §V · p. 6

Mitigation Methods

  • Average-based mitigators include EG, Fair-SMOTE, MAAT, and STEALTH, while the tail-aware comparison includes MiniMax-Fairness and an ECD-objective PARFAIT-ML approach.Source: §V · p. 6

Implementation and Statistical Tests

  • The study repeats each experiment 20 times and ranks methods against the vanilla model using Scott-Knott with Cliff's Delta and bootstrap testing.Source: §V · p. 6

A. Evaluating Synthetic Data Generation (RQ1)

The evaluation compares synthetic data generators using statistical similarity and downstream-model performance. CTGAN and TVAE can closely replicate real datasets, but effectiveness varies by dataset.

A. Evaluating Synthetic Data Generation (RQ1)

  • The evaluation compares CTGAN, TVAE, and random generation using statistical similarity and downstream-model performance criteria.The criteria include distribution statistics and the performance of a model trained on generated data versus the actual dataset.Source: A. Evaluating Synthetic Data… · p. 6
  • For Compas, the MDS comparison particularly highlights TVAE-generated data aligning with the actual dataset’s distribution.Figure 3 visualizes the methods’ replication of the real dataset’s characteristics.Source: A. Evaluating Synthetic Data… · p. 7, Figure 3 · p. 7
  • KL-D reached 0.98 and inception distance fell to 0.008, but CTGAN and TVAE effectiveness was dataset-dependent.These are the reported best values across the experiments.Source: A. Evaluating Synthetic Data… · p. 7, Table III · p. 7

B. Feasibility, Usefulness, and Guarantee of EVT (RQ2)

The study tests whether EVT can model counterfactual-discrimination tails and support worst-case guarantees. ECD identifies tail discrimination that average and prevalent tail metrics may miss.

B. Feasibility, Usefulness, and Guarantee of EVT (RQ2)

  • A DNN on Census data showed ACD of 0.05, CVaR of 0.08, and ECD of 0.21, with reported tail discrimination toward women.The reported subgroup values were µM = 0.03 and µF = 0.24.Source: B. Feasibility, Usefulness, and… · p. 7
  • 62 of 80 scenarios (78%) had a negative GEV shape indicating a finite tail; 14 (18%) had a near-zero shape indicating an infinite, exponentially decaying tail.The reported distribution types support extrapolation over unlimited queries for the finite-tail cases and bounded queries for the near-zero cases.Source: B. Feasibility, Usefulness, and… · pp. 7, 9
  • 95% of cases supported EVT-based worst-case guarantees, with feasible extrapolation in 76 of 80 scenarios.The table reports test-case counts, ACD, CVaR, ECD, GEV characteristics, and extrapolation feasibility.Source: B. Feasibility, Usefulness, and… · pp. 7, 9, Table IV · p. 8
  • ECD-based discrimination occurred in 19 of 40 cases (48%), versus ACD discrimination in 10 of 40 (25%).ECD was significantly greater than ACD in 13 cases (33%); in 18 of 40 experiments, ECD found tail discrimination against the unprivileged group missed by CVaR.Source: B. Feasibility, Usefulness, and… · p. 9

C. Validation of Prevalent Bias Mitigation Algorithms (RQ3)

Average-based mitigators often improve average fairness without improving tail fairness. Among the four methods, STEALTH performs best on ECD while preserving or reducing AOD bias in many cases.

C. Validation of Prevalent Bias Mitigation Algorithms (RQ3)

  • EG won on AOD in 19 cases, ahead of STEALTH (14), MAAT (6), and Fair-SMOTE (4).Average AOD was 0.03 for EG, 0.07 for STEALTH, 0.07 for MAAT, and 0.08 for Fair-SMOTE.Source: C. Validation of Prevalent Bias… · p. 9
  • STEALTH won on ECD in 31 of 40 cases, compared with 15 for EG, 12 for MAAT, and 9 for Fair-SMOTE.Average ECD values were 0.04 for STEALTH, 0.20 for EG, 0.08 for MAAT, and 0.12 for Fair-SMOTE.Source: C. Validation of Prevalent Bias… · p. 9
  • 35% of cases showed increased ECD unfairness under average-based mitigation, while average fairness was preserved or improved in 63%.The comparison covers EG, Fair-SMOTE, MAAT, and STEALTH.Source: C. Validation of Prevalent Bias… · p. 9

D. Tail-aware Mitigation Algorithms (RQ4)

ECD-Fair is compared with MiniMax-Fairness and STEALTH on DNN and logistic-regression benchmarks. It outperforms MiniMax-Fairness and slightly outperforms STEALTH overall on tail and average fairness.

D. Tail-aware Mitigation Algorithms (RQ4)

  • ECD-Fair won on ECD in 18 of 20 cases, versus 10 for MiniMax-Fairness.ECD-Fair had average AOD and ECD of 0.04 and 0.03, respectively, across the benchmarks.Source: D. Tail-aware Mitigation Algorithms… · p. 9
  • ECD-Fair did not increase tail unfairness in any benchmark, while STEALTH degraded it in 2 benchmarks.ECD-Fair and STEALTH won on ECD in 18 and 16 cases, respectively, and ECD-Fair slightly outperformed STEALTH overall on tail and average metrics.Source: D. Tail-aware Mitigation Algorithms… · p. 9
  • ECD-Fair improved tail fairness in 90% of cases, compared with 80% for STEALTH, and reduced AOD bias in 45% versus 40%.ECD-Fair significantly outperformed MiniMax-Fairness, the state-of-the-art tail-aware baseline.Source: D. Tail-aware Mitigation Algorithms… · p. 10

VI. DISCUSSIONS

The discussion identifies limits in estimating tail fairness: ground truth is unavailable, and the approach depends on protected attributes and valid counterfactual representations. Its conclusions also depend on representative in-distribution samples and on whether the tested algorithms, hyperparameters, and datasets cover challenging fairness scenarios.

  • The tail lacks ground truth, so maximum individual discrimination in the validation set serves only as a lower bound.Source: §VI · p. 10
  • The approach requires protected attributes at inference, so it cannot assess fairness through unawareness, which removes those attributes.Source: §VI · p. 10
  • The approach relies on representative samples from the training distribution and may not be valid for out-of-distribution queries.Source: §VI · p. 11
  • The approach assumes that flipping sensitive values yields valid representations; whether tested algorithms, hyperparameters, and datasets cover challenging scenarios remains open.Source: §VI · p. 11

VII. RELATED WORK

The related work contrasts tail-based discrimination testing with average-focused fairness measures and other worst-case approaches. It also distinguishes this paper’s use of EVT on discrimination tails from work applying EVT to income inequality and recommendation systems.

Fairness Testing of Data-Driven Software

  • THEMIS measures average causal discrimination with counterfactual queries and normal-distribution guarantees, whereas this work targets extreme discrimination with extreme-value guarantees and generative tail cases.Source: §VII · p. 11

Fairness in the Tail

  • CVaR-based fairness targets expected and upper-quantile group loss, while Diana et al. constrain group loss; the authors say CVaR may miss tail discrimination and its shape.Source: §VII · p. 11

Intersectional Fairness

  • Intersectional fairness,, summarizes subgroup fairness using average metrics, unlike this paper’s EVT comparison of outcome-distribution tails across groups.Source: §VII · p. 11

Other Application of EVT for Fairness

  • Prior EVT work studies income inequality,, and recommendation-system Matthew effects, while this paper examines discrimination in classification-based decision systems.Source: §VII · p. 11

VIII. CONCLUSION AND FUTURE WORK

The paper concludes that EVT can model worst-case counterfactual bias with statistical guarantees and reveals limitations of a state-of-the-art bias-reduction algorithm in the worst case. It identifies applying EVT to AI harms against vulnerable communities as a future direction.

  • The approach fits worst-case counterfactual bias with statistical guarantees and reveals limitations of a state-of-the-art bias-reduction algorithm in the worst case.Source: §VIII · p. 11
  • A future direction is using EVT to assess whether automated decision-support software systematically harms a vulnerable community.Source: §VIII · p. 11

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