# Formalising the Robustness of Counterfactual Explanations for Neural Networks

arXiv: https://arxiv.org/abs/2208.14878 · HTML: https://mlsanity.com/paper/2208.14878/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 47 references that are in the map.
- First posted on arXiv 31 Aug 2022.

## Builds on (most cited first, 27 of 27)

- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Towards Evaluating the Robustness of Neural Networks](https://mlsanity.com/paper/1608.04644/) (2016)
- [Explanation in Artificial Intelligence: Insights from the Social Sciences](https://mlsanity.com/paper/1706.07269/) (2017)
- [Counterfactual Explanations without Opening the Black Box: Automated Decisions and the GDPR](https://mlsanity.com/paper/1711.00399/) (2017)
- [Explaining Machine Learning Classifiers through Diverse Counterfactual Explanations](https://mlsanity.com/paper/1905.07697/) (2019)
- [OpenML: networked science in machine learning](https://mlsanity.com/paper/1407.7722/) (2014)
- [Towards Robust Interpretability with Self-Explaining Neural Networks](https://mlsanity.com/paper/1806.07538/) (2018)
- [Actionable Recourse in Linear Classification](https://mlsanity.com/paper/1809.06514/) (2018)
- [Algorithmic Recourse: from Counterfactual Explanations to Interventions](https://mlsanity.com/paper/2002.06278/) (2020)
- [FACE: Feasible and Actionable Counterfactual Explanations](https://mlsanity.com/paper/1909.09369/) (2019)
- [Interpretable Counterfactual Explanations Guided by Prototypes](https://mlsanity.com/paper/1907.02584/) (2019)
- [Model-Agnostic Counterfactual Explanations for Consequential Decisions](https://mlsanity.com/paper/1905.11190/) (2019)
- [Evaluating the Robustness of Neural Networks: An Extreme Value Theory Approach](https://mlsanity.com/paper/1801.10578/) (2018)
- [Multi-Objective Counterfactual Explanations](https://mlsanity.com/paper/2004.11165/) (2020)
- [Preserving Causal Constraints in Counterfactual Explanations for Machine Learning Classifiers](https://mlsanity.com/paper/1912.03277/) (2019)
- [The Hidden Assumptions Behind Counterfactual Explanations and Principal Reasons](https://mlsanity.com/paper/1912.04930/) (2019)
- [Counterfactual Explanations Can Be Manipulated](https://mlsanity.com/paper/2106.02666/) (2021)
- [Towards Robust and Reliable Algorithmic Recourse](https://mlsanity.com/paper/2102.13620/) (2021)
- [Interpretable Predictions of Tree-based Ensembles via Actionable Feature Tweaking](https://mlsanity.com/paper/1706.06691/) (2017)
- [On Generating Plausible Counterfactual and Semi-Factual Explanations for Deep Learning](https://mlsanity.com/paper/2009.06399/) (2020)
- [Exploring Counterfactual Explanations Through the Lens of Adversarial Examples: A Theoretical and Empirical Analysis](https://mlsanity.com/paper/2106.09992/) (2021)
- [Robust Counterfactual Explanations for Tree-Based Ensembles](https://mlsanity.com/paper/2207.02739/) (2022)
- [The Intriguing Relation Between Counterfactual Explanations and Adversarial Examples](https://mlsanity.com/paper/2009.05487/) (2020)
- [Abstraction based Output Range Analysis for Neural Networks](https://mlsanity.com/paper/2007.09527/) (2020)
- [BRPO: Batch Residual Policy Optimization](https://mlsanity.com/paper/2002.05522/) (2020)
- [Scaling Guarantees for Nearest Counterfactual Explanations](https://mlsanity.com/paper/2010.04965/) (2020)
- [Formalizing Generalization and Robustness of Neural Networks to Weight Perturbations](https://mlsanity.com/paper/2103.02200/) (2021)

## Built on by (most cited first, 24 of 24)

- [Robust Counterfactual Explanations for Neural Networks With Probabilistic Guarantees](https://mlsanity.com/paper/2305.11997/) (2023)
- [Towards Data-and Knowledge-Driven Artificial Intelligence: A Survey on Neuro-Symbolic Computing](https://mlsanity.com/paper/2210.15889/) (2022)
- [Robust Counterfactual Explanations in Machine Learning: A Survey](https://mlsanity.com/paper/2402.01928/) (2024)
- [Provably Robust and Plausible Counterfactual Explanations for Neural Networks via Robust Optimisation](https://mlsanity.com/paper/2309.12545/) (2023)
- [Promoting Counterfactual Robustness through Diversity](https://mlsanity.com/paper/2312.06564/) (2023)
- [Recourse under Model Multiplicity via Argumentative Ensembling (Technical Report)](https://mlsanity.com/paper/2312.15097/) (2023)
- [Model Reconstruction Using Counterfactual Explanations: A Perspective From Polytope Theory](https://mlsanity.com/paper/2405.05369/) (2024)
- [Contestable AI needs Computational Argumentation](https://mlsanity.com/paper/2405.10729/) (2024)
- [Counterfactual Explanations with Probabilistic Guarantees on their Robustness to Model Change](https://mlsanity.com/paper/2408.04842/) (2024)
- [Interval Abstractions for Robust Counterfactual Explanations](https://mlsanity.com/paper/2404.13736/) (2024)
- [Generating Likely Counterfactuals Using Sum-Product Networks](https://mlsanity.com/paper/2401.14086/) (2024)
- [Rigorous Probabilistic Guarantees for Robust Counterfactual Explanations](https://mlsanity.com/paper/2407.07482/) (2024)
- [RobustX: Robust Counterfactual Explanations Made Easy](https://mlsanity.com/paper/2502.13751/) (2025)
- [Exploring Energy Landscapes for Minimal Counterfactual Explanations: Applications in Cybersecurity and Beyond](https://mlsanity.com/paper/2503.18185/) (2025)
- [Counterfactual Explanations for Model Ensembles Using Entropic Risk Measures](https://mlsanity.com/paper/2503.07934/) (2025)
- [Verified Training for Counterfactual Explanation Robustness under Data Shift](https://mlsanity.com/paper/2403.03773/) (2024)
- [ElliCE: Efficient and Provably Robust Algorithmic Recourse via the Rashomon Sets](https://mlsanity.com/paper/2602.07674/) (2026)
- [AR-Pro: Counterfactual Explanations for Anomaly Repair with Formal Properties](https://mlsanity.com/paper/2410.24178/) (2024)
- [Argumentative Ensembling for Robust Recourse under Model Multiplicity](https://mlsanity.com/paper/2506.20260/) (2025)
- [Synthesising Counterfactual Explanations via Label-Conditional Gaussian Mixture Variational Autoencoders](https://mlsanity.com/paper/2510.04855/) (2025)
- [Fixed Point Explainability](https://mlsanity.com/paper/2505.12421/) (2025)
- [Evaluating Counterfactual Explanation Methods on Incomplete Inputs](https://mlsanity.com/paper/2604.08004/) (2026)
- [Towards Verified and Targeted Explanations through Formal Methods](https://mlsanity.com/paper/2604.14209/) (2026)
- [Target-confidence Recourse Using tSeTlin machines: TRUST](https://mlsanity.com/paper/2606.18832/) (2026)

## Abstract (arXiv)

The use of counterfactual explanations (CFXs) is an increasingly popular explanation strategy for machine learning models. However, recent studies have shown that these explanations may not be robust to changes in the underlying model (e.g., following retraining), which raises questions about their reliability in real-world applications. Existing attempts towards solving this problem are heuristic, and the robustness to model changes of the resulting CFXs is evaluated with only a small number of retrained models, failing to provide exhaustive guarantees. To remedy this, we propose Delta-robustness, the first notion to formally and deterministically assess the robustness (to model changes) of CFXs for neural networks. We introduce an abstraction framework based on interval neural networks to verify the Delta-robustness of CFXs against a possibly infinite set of changes to the model parameters, i.e., weights and biases. We then demonstrate the utility of this approach in two distinct ways. First, we analyse the Delta-robustness of a number of CFX generation methods from the literature and show that they unanimously host significant deficiencies in this regard. Second, we demonstrate how embedding Delta-robustness within existing methods can provide CFXs which are provably robust.