# Probabilistic Verification of Neural Networks Against Group Fairness

arXiv: https://arxiv.org/abs/2107.08362 · HTML: https://mlsanity.com/paper/2107.08362/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 58 references that are in the map.
- First posted on arXiv 18 Jul 2021.

## Builds on (most cited first, 25 of 25)

- [FaceNet: A Unified Embedding for Face Recognition and Clustering](https://mlsanity.com/paper/1503.03832/) (2015)
- [Man is to Computer Programmer as Woman is to Homemaker? Debiasing Word Embeddings](https://mlsanity.com/paper/1607.06520/) (2016)
- [End to End Learning for Self-Driving Cars](https://mlsanity.com/paper/1604.07316/) (2016)
- [Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks](https://mlsanity.com/paper/1702.01135/) (2017)
- [Certifying and removing disparate impact](https://mlsanity.com/paper/1412.3756/) (2014)
- [A Reductions Approach to Fair Classification](https://mlsanity.com/paper/1803.02453/) (2018)
- [Adversarial Examples for Evaluating Reading Comprehension Systems](https://mlsanity.com/paper/1707.07328/) (2017)
- [Generating Natural Language Adversarial Examples](https://mlsanity.com/paper/1804.07998/) (2018)
- [Preventing Fairness Gerrymandering: Auditing and Learning for Subgroup Fairness](https://mlsanity.com/paper/1711.05144/) (2017)
- [TextBugger: Generating Adversarial Text Against Real-world Applications](https://mlsanity.com/paper/1812.05271/) (2018)
- [Formal Verification of Piece-Wise Linear Feed-Forward Neural Networks](https://mlsanity.com/paper/1705.01320/) (2017)
- [Formal Security Analysis of Neural Networks using Symbolic Intervals](https://mlsanity.com/paper/1804.10829/) (2018)
- [Counterfactual Fairness in Text Classification through Robustness](https://mlsanity.com/paper/1809.10610/) (2018)
- [A Convex Framework for Fair Regression](https://mlsanity.com/paper/1706.02409/) (2017)
- [Branch and Bound for Piecewise Linear Neural Network Verification](https://mlsanity.com/paper/1909.06588/) (2019)
- [Fairness Testing: Testing Software for Discrimination](https://mlsanity.com/paper/1709.03221/) (2017)
- [Automated Directed Fairness Testing](https://mlsanity.com/paper/1807.00468/) (2018)
- [An Abstraction-Based Framework for Neural Network Verification](https://mlsanity.com/paper/1910.14574/) (2019)
- [Probabilistic Verification of Fairness Properties via Concentration](https://mlsanity.com/paper/1812.02573/) (2018)
- [POPQORN: Quantifying Robustness of Recurrent Neural Networks](https://mlsanity.com/paper/1905.07387/) (2019)
- [Verifying Recurrent Neural Networks using Invariant Inference](https://mlsanity.com/paper/2004.02462/) (2020)
- [Automated Test Generation to Detect Individual Discrimination in AI Models](https://mlsanity.com/paper/1809.03260/) (2018)
- [Genetic programming approaches to learning fair classifiers](https://mlsanity.com/paper/2004.13282/) (2020)
- [SOCRATES: Towards a Unified Platform for Neural Network Analysis](https://mlsanity.com/paper/2007.11206/) (2020)
- [Robustness Verification for Classifier Ensembles](https://mlsanity.com/paper/2005.05587/) (2020)

## Built on by (most cited first, 17 of 17)

- [Causality-based Neural Network Repair](https://mlsanity.com/paper/2204.09274/) (2022)
- [Monitoring Algorithmic Fairness](https://mlsanity.com/paper/2305.15979/) (2023)
- [Monitoring Algorithmic Fairness under Partial Observations](https://mlsanity.com/paper/2308.00341/) (2023)
- [Online Safety Analysis for LLMs: a Benchmark, an Assessment, and a Path Forward](https://mlsanity.com/paper/2404.08517/) (2024)
- [Explainable Global Fairness Verification of Tree-Based Classifiers](https://mlsanity.com/paper/2209.13179/) (2022)
- [Certified Continual Learning for Neural Network Regression](https://mlsanity.com/paper/2407.06697/) (2024)
- [Repairing Deep Neural Networks Based on Behavior Imitation](https://mlsanity.com/paper/2305.03365/) (2023)
- [LUNA: A Model-Based Universal Analysis Framework for Large Language Models](https://mlsanity.com/paper/2310.14211/) (2023)
- [Towards Better Fairness-Utility Trade-off: A Comprehensive Measurement-Based Reinforcement Learning Framework](https://mlsanity.com/paper/2307.11379/) (2023)
- [Extending Neural Network Verification to a Larger Family of Piece-wise Linear Activation Functions](https://mlsanity.com/paper/2311.10780/) (2023)
- [Language Models That Walk the Talk: A Framework for Formal Fairness Certificates](https://mlsanity.com/paper/2505.12767/) (2025)
- [Monitoring of Static Fairness](https://mlsanity.com/paper/2507.03048/) (2025)
- [Algorithmic Fairness: A Runtime Perspective](https://mlsanity.com/paper/2507.20711/) (2025)
- [The Fair Game: Auditing & Debiasing AI Algorithms Over Time](https://mlsanity.com/paper/2508.06443/) (2025)
- [Certified and accurate computation of function space norms of deep neural networks](https://mlsanity.com/paper/2603.06431/) (2026)
- [Provable Fairness Repair for Deep Neural Networks](https://mlsanity.com/paper/2605.19549/) (2026)
- [Energy Shields for Fairness](https://mlsanity.com/paper/2605.24926/) (2026)

## Abstract (arXiv)

Fairness is crucial for neural networks which are used in applications with important societal implication. Recently, there have been multiple attempts on improving fairness of neural networks, with a focus on fairness testing (e.g., generating individual discriminatory instances) and fairness training (e.g., enhancing fairness through augmented training). In this work, we propose an approach to formally verify neural networks against fairness, with a focus on independence-based fairness such as group fairness. Our method is built upon an approach for learning Markov Chains from a user-provided neural network (i.e., a feed-forward neural network or a recurrent neural network) which is guaranteed to facilitate sound analysis. The learned Markov Chain not only allows us to verify (with Probably Approximate Correctness guarantee) whether the neural network is fair or not, but also facilities sensitivity analysis which helps to understand why fairness is violated. We demonstrate that with our analysis results, the neural weights can be optimized to improve fairness. Our approach has been evaluated with multiple models trained on benchmark datasets and the experiment results show that our approach is effective and efficient.