# Robustness of Neural Networks: A Probabilistic and Practical Approach

arXiv: https://arxiv.org/abs/1902.05983 · HTML: https://mlsanity.com/paper/1902.05983/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Feb 2019.

## Built on by (most cited first, 30 of 38)

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- [RoMA: a Method for Neural Network Robustness Measurement and Assessment](https://mlsanity.com/paper/2110.11088/) (2021)
- [Verification of Neural Networks' Global Robustness](https://mlsanity.com/paper/2402.19322/) (2024)
- [A.I. Robustness: a Human-Centered Perspective on Technological Challenges and Opportunities](https://mlsanity.com/paper/2210.08906/) (2022)
- [TrajPAC: Towards Robustness Verification of Pedestrian Trajectory Prediction Models](https://mlsanity.com/paper/2308.05985/) (2023)
- [Robustness Assessment of a Runway Object Classifier for Safe Aircraft Taxiing](https://mlsanity.com/paper/2402.00035/) (2024)
- [PREMAP: A Unifying PREiMage APproximation Framework for Neural Networks](https://mlsanity.com/paper/2408.09262/) (2024)
- [Risk Verification of Stochastic Systems with Neural Network Controllers](https://mlsanity.com/paper/2209.09881/) (2022)
- [Probabilistic Robustness in Deep Learning: A Concise yet Comprehensive Guide](https://mlsanity.com/paper/2502.14833/) (2025)
- [Revisiting Outage for Edge Inference Systems](https://mlsanity.com/paper/2504.03686/) (2025)
- [A Novel Noise Injection-based Training Scheme for Better Model Robustness](https://mlsanity.com/paper/2302.10802/) (2023)
- [A Survey of Neural Network Robustness Assessment in Image Recognition](https://mlsanity.com/paper/2404.08285/) (2024)
- [Monitoring Robustness and Individual Fairness](https://mlsanity.com/paper/2506.00496/) (2025)
- [Incremental Satisfiability Modulo Theory for Verification of Deep Neural Networks](https://mlsanity.com/paper/2302.06455/) (2023)
- [DEM: A Method for Certifying Deep Neural Network Classifier Outputs in Aerospace](https://mlsanity.com/paper/2401.02283/) (2024)
- [Improving Data-aware and Parameter-aware Robustness for Continual Learning](https://mlsanity.com/paper/2405.17054/) (2024)
- [Probabilistically Tightened Linear Relaxation-based Perturbation Analysis for Neural Network Verification](https://mlsanity.com/paper/2507.05405/) (2025)
- [Is the Last Layer Sufficient for Uncertainty Quantification?](https://mlsanity.com/paper/2605.30741/) (2026)
- [An Overview and Prospective Outlook on Robust Training and Certification of Machine Learning Models](https://mlsanity.com/paper/2208.07464/) (2022)
- [gRoMA: a Tool for Measuring the Global Robustness of Deep Neural Networks](https://mlsanity.com/paper/2301.02288/) (2023)
- [Adversarial Examples in Environment Perception for Automated Driving (Review)](https://mlsanity.com/paper/2504.08414/) (2025)
- [Extracting Robust Register Automata from Neural Networks over Data Sequences](https://mlsanity.com/paper/2511.19100/) (2025)
- [SimCert: Probabilistic Certification for Behavioral Similarity in Deep Neural Network Compression](https://mlsanity.com/paper/2603.14818/) (2026)
- [Models of Computational Profiles to Study the Likelihood of DNN Metamorphic Test Cases](https://mlsanity.com/paper/2107.13491/) (2021)
- [Quantifying probabilistic robustness of tree-based classifiers against natural distortions](https://mlsanity.com/paper/2208.10354/) (2022)
- [Artificial Intelligence Nomenclature Identified From Delphi Study on Key Issues Related to Trust and Barriers to Adoption for Autonomous Systems](https://mlsanity.com/paper/2210.09086/) (2022)
- [Boosting-based Construction of BDDs for Linear Threshold Functions and Its Application to Verification of Neural Networks](https://mlsanity.com/paper/2306.05211/) (2023)

## Abstract (arXiv)

Neural networks are becoming increasingly prevalent in software, and it is therefore important to be able to verify their behavior. Because verifying the correctness of neural networks is extremely challenging, it is common to focus on the verification of other properties of these systems. One important property, in particular, is robustness. Most existing definitions of robustness, however, focus on the worst-case scenario where the inputs are adversarial. Such notions of robustness are too strong, and unlikely to be satisfied by-and verifiable for-practical neural networks. Observing that real-world inputs to neural networks are drawn from non-adversarial probability distributions, we propose a novel notion of robustness: probabilistic robustness, which requires the neural network to be robust with at least (1 - ε) probability with respect to the input distribution. This probabilistic approach is practical and provides a principled way of estimating the robustness of a neural network. We also present an algorithm, based on abstract interpretation and importance sampling, for checking whether a neural network is probabilistically robust. Our algorithm uses abstract interpretation to approximate the behavior of a neural network and compute an overapproximation of the input regions that violate robustness. It then uses importance sampling to counter the effect of such overapproximation and compute an accurate estimate of the probability that the neural network violates the robustness property.