# Branch and Bound for Piecewise Linear Neural Network Verification

arXiv: https://arxiv.org/abs/1909.06588 · HTML: https://mlsanity.com/paper/1909.06588/

## Key facts

- Cited by 138 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Sep 2019.

## Built on by (most cited first, 30 of 138)

- [Beta-CROWN: Efficient Bound Propagation with Per-neuron Split Constraints for Complete and Incomplete Neural Network Robustness Verification](https://mlsanity.com/paper/2103.06624/) (2021)
- [General Cutting Planes for Bound-Propagation-Based Neural Network Verification](https://mlsanity.com/paper/2208.05740/) (2022)
- [Complete Verification via Multi-Neuron Relaxation Guided Branch-and-Bound](https://mlsanity.com/paper/2205.00263/) (2022)
- [PRIMA: General and Precise Neural Network Certification via Scalable Convex Hull Approximations](https://mlsanity.com/paper/2103.03638/) (2021)
- [The Second International Verification of Neural Networks Competition (VNN-COMP 2021): Summary and Results](https://mlsanity.com/paper/2109.00498/) (2021)
- [First Three Years of the International Verification of Neural Networks Competition (VNN-COMP)](https://mlsanity.com/paper/2301.05815/) (2023)
- [Certified Training: Small Boxes are All You Need](https://mlsanity.com/paper/2210.04871/) (2022)
- [Causality-based Neural Network Repair](https://mlsanity.com/paper/2204.09274/) (2022)
- [Neural Network Verification with Branch-and-Bound for General Nonlinearities](https://mlsanity.com/paper/2405.21063/) (2024)
- [Efficiently Computing Local Lipschitz Constants of Neural Networks via Bound Propagation](https://mlsanity.com/paper/2210.07394/) (2022)
- [Improved Branch and Bound for Neural Network Verification via Lagrangian Decomposition](https://mlsanity.com/paper/2104.06718/) (2021)
- [How to Certify Machine Learning Based Safety-critical Systems? A Systematic Literature Review](https://mlsanity.com/paper/2107.12045/) (2021)
- [When Deep Learning Meets Polyhedral Theory: A Survey](https://mlsanity.com/paper/2305.00241/) (2023)
- [Scalable Neural Network Verification with Branch-and-bound Inferred Cutting Planes](https://mlsanity.com/paper/2501.00200/) (2024)
- [VeriX: Towards Verified Explainability of Deep Neural Networks](https://mlsanity.com/paper/2212.01051/) (2022)
- [Provably Bounding Neural Network Preimages](https://mlsanity.com/paper/2302.01404/) (2023)
- [Understanding Certified Training with Interval Bound Propagation](https://mlsanity.com/paper/2306.10426/) (2023)
- [Efficient Neural Network Analysis with Sum-of-Infeasibilities](https://mlsanity.com/paper/2203.11201/) (2022)
- [Open- and Closed-Loop Neural Network Verification using Polynomial Zonotopes](https://mlsanity.com/paper/2207.02715/) (2022)
- [A DPLL(T) Framework for Verifying Deep Neural Networks](https://mlsanity.com/paper/2307.10266/) (2023)
- [Expressive Losses for Verified Robustness via Convex Combinations](https://mlsanity.com/paper/2305.13991/) (2023)
- [IBP Regularization for Verified Adversarial Robustness via Branch-and-Bound](https://mlsanity.com/paper/2206.14772/) (2022)
- [Scalable Verification of GNN-based Job Schedulers](https://mlsanity.com/paper/2203.03153/) (2022)
- [Convex Bounds on the Softmax Function with Applications to Robustness Verification](https://mlsanity.com/paper/2303.01713/) (2023)
- [BDD4BNN: A BDD-based Quantitative Analysis Framework for Binarized Neural Networks](https://mlsanity.com/paper/2103.07224/) (2021)
- [Probabilistic Verification of Neural Networks Against Group Fairness](https://mlsanity.com/paper/2107.08362/) (2021)
- [CertiFair: A Framework for Certified Global Fairness of Neural Networks](https://mlsanity.com/paper/2205.09927/) (2022)
- [RvLLM: LLM Runtime Verification with Domain Knowledge](https://mlsanity.com/paper/2505.18585/) (2025)
- [Toward Certified Robustness Against Real-World Distribution Shifts](https://mlsanity.com/paper/2206.03669/) (2022)
- [NLP Verification: Towards a General Methodology for Certifying Robustness](https://mlsanity.com/paper/2403.10144/) (2024)

## Abstract (arXiv)

The success of Deep Learning and its potential use in many safety-critical applications has motivated research on formal verification of Neural Network (NN) models. In this context, verification involves proving or disproving that an NN model satisfies certain input-output properties. Despite the reputation of learned NN models as black boxes, and the theoretical hardness of proving useful properties about them, researchers have been successful in verifying some classes of models by exploiting their piecewise linear structure and taking insights from formal methods such as Satisifiability Modulo Theory. However, these methods are still far from scaling to realistic neural networks. To facilitate progress on this crucial area, we exploit the Mixed Integer Linear Programming (MIP) formulation of verification to propose a family of algorithms based on Branch-and-Bound (BaB). We show that our family contains previous verification methods as special cases. With the help of the BaB framework, we make three key contributions. Firstly, we identify new methods that combine the strengths of multiple existing approaches, accomplishing significant performance improvements over previous state of the art. Secondly, we introduce an effective branching strategy on ReLU non-linearities. This branching strategy allows us to efficiently and successfully deal with high input dimensional problems with convolutional network architecture, on which previous methods fail frequently. Finally, we propose comprehensive test data sets and benchmarks which includes a collection of previously released testcases. We use the data sets to conduct a thorough experimental comparison of existing and new algorithms and to provide an inclusive analysis of the factors impacting the hardness of verification problems.