# Quantitative Verification of Fairness in Tree Ensembles

arXiv: https://arxiv.org/abs/2512.16386 · HTML: https://mlsanity.com/paper/2512.16386/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 36 references that are in the map.
- First posted on arXiv 18 Dec 2025.

## Builds on (most cited first, 14 of 14)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Towards Evaluating the Robustness of Neural Networks](https://mlsanity.com/paper/1608.04644/) (2016)
- [The Limitations of Deep Learning in Adversarial Settings](https://mlsanity.com/paper/1511.07528/) (2015)
- [Distillation as a Defense to Adversarial Perturbations against Deep Neural Networks](https://mlsanity.com/paper/1511.04508/) (2015)
- [Fairness Testing: Testing Software for Discrimination](https://mlsanity.com/paper/1709.03221/) (2017)
- [Self-Attention Between Datapoints: Going Beyond Individual Input-Output Pairs in Deep Learning](https://mlsanity.com/paper/2106.02584/) (2021)
- [Globally-Robust Neural Networks](https://mlsanity.com/paper/2102.08452/) (2021)
- [Measuring Neural Net Robustness with Constraints](https://mlsanity.com/paper/1605.07262/) (2016)
- [Verifying Individual Fairness in Machine Learning Models](https://mlsanity.com/paper/2006.11737/) (2020)
- [Evasion and Hardening of Tree Ensemble Classifiers](https://mlsanity.com/paper/1509.07892/) (2015)
- [Robustness Verification of Tree-based Models](https://mlsanity.com/paper/1906.03849/) (2019)
- [Improving Neural Network Verification through Spurious Region Guided Refinement](https://mlsanity.com/paper/2010.07722/) (2020)

## Abstract (arXiv)

This work focuses on quantitative verification of fairness in tree ensembles. Unlike traditional verification approaches that merely return a single counterexample when the fairness is violated, quantitative verification estimates the ratio of all counterexamples and characterizes the regions where they occur, which is important information for diagnosing and mitigating bias. To date, quantitative verification has been explored almost exclusively for deep neural networks (DNNs). Representative methods, such as DeepGemini and FairQuant, all build on the core idea of Counterexample-Guided Abstraction Refinement, a generic framework that could be adapted to other model classes. We extended the framework into a model-agnostic form, but discovered two limitations: (i) it can provide only lower bounds, and (ii) its performance scales poorly. Exploiting the discrete structure of tree ensembles, our work proposes an efficient quantification technique that delivers any-time upper and lower bounds. Experiments on five widely used datasets demonstrate its effectiveness and efficiency. When applied to fairness testing, our quantification method significantly outperforms state-of-the-art testing techniques.