[Paper Review] Verifying Individual Fairness in Machine Learning Models

Philips George John, Deepak Vijaykeerthy+1 more|arXiv (Cornell University)|Jun 21, 2020

Adversarial Robustness in Machine LearningComputer Science33 references17 citations

TL;DR

This paper proposes the first framework for verifying individual fairness in machine learning models by checking global robustness: whether similar inputs (according to a flexible, attribute-specific metric) receive similar predictions. It introduces sound but incomplete verification algorithms for linear and kernelized classifiers (polynomial/RBF), using semidefinite programming relaxations to bound prediction differences, with experimental validation on public datasets showing effective detection of individual bias.

ABSTRACT

We consider the problem of whether a given decision model, working with structured data, has individual fairness. Following the work of Dwork, a model is individually biased (or unfair) if there is a pair of valid inputs which are close to each other (according to an appropriate metric) but are treated differently by the model (different class label, or large difference in output), and it is unbiased (or fair) if no such pair exists. Our objective is to construct verifiers for proving individual fairness of a given model, and we do so by considering appropriate relaxations of the problem. We construct verifiers which are sound but not complete for linear classifiers, and kernelized polynomial/radial basis function classifiers. We also report the experimental results of evaluating our proposed algorithms on publicly available datasets.

Motivation & Objective

• To address the lack of verification techniques for individual fairness, which requires global robustness across all inputs rather than group-level fairness or local adversarial robustness.
• To develop a formal, sound verification method that detects individual bias by checking whether close inputs (under a flexible metric) yield significantly different model outputs.
• To extend existing verification approaches—previously limited to local robustness or group fairness—to the global, individual fairness setting.
• To provide a practical framework for verifying fairness in white-box models using mathematical optimization, particularly semidefinite programming.
• To evaluate the method empirically on public datasets, demonstrating its ability to detect individual-level unfairness in real-world models.

Proposed method

• The paper defines individual fairness via a flexible metric that partitions input attributes into subsets, each with a threshold for allowable perturbation, ensuring inputs are close if their differences per attribute are within the respective threshold.
• For linear classifiers, the verification problem is reduced to a linear program, which is exact but potentially exponential in time; for kernelized models, it is relaxed into a semidefinite program (SDP) using sum-of-squares (SOS) techniques.
• The core method uses semidefinite programming relaxations to compute lower bounds on the maximum output difference between close inputs, enabling sound verification of fairness.
• For RBF kernel models, the algorithm iterates over feasible pairs of input perturbations in the sensitive attribute space and solves a constrained optimization to find the minimal prediction difference.
• The method leverages Putinar’s Positivstellensatz and SOS hierarchies to construct lower bounds on polynomial optimization problems over semi-algebraic sets, ensuring soundness.
• The framework outputs a lower bound on the maximum output difference between close inputs, with a counterexample if the difference exceeds a fairness threshold.

Experimental results

Research questions

• RQ1 Can we formally verify individual fairness in machine learning models by checking global robustness across all input pairs?
• RQ2 How can we adapt existing verification techniques—originally designed for local adversarial robustness or group fairness—to the global, individual fairness setting?
• RQ3 What mathematical relaxation techniques can be used to soundly verify individual fairness in non-linear models like kernelized classifiers?
• RQ4 To what extent can semidefinite programming and sum-of-squares relaxations be used to bound prediction differences under input perturbations?
• RQ5 How effective is the proposed framework in detecting individual-level bias in real-world datasets?

Key findings

• The proposed verification framework is sound but incomplete for linear and kernelized classifiers, with exact verification possible in exponential time for linear models.
• For kernelized models, the method uses semidefinite programming relaxations based on sum-of-squares representations to compute lower bounds on the maximum output difference between close inputs.
• The framework successfully detects individual bias in RBF kernel models by identifying input pairs with large output differences despite small input perturbations.
• Experimental evaluation on public datasets confirms the method's ability to identify unfair predictions, with the output lower bound reflecting the worst-case fairness violation.
• The method generalizes prior definitions of fairness by subsuming both group fairness and counterfactual fairness under a unified, flexible metric-based framework.
• The use of Putinar’s Positivstellensatz enables constrained optimization over semi-algebraic sets, allowing fairness verification even with complex input constraints.