# LipBaB: Computing exact Lipschitz constant of ReLU networks

arXiv: https://arxiv.org/abs/2105.05495 · HTML: https://mlsanity.com/paper/2105.05495/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 15 references that are in the map.
- First posted on arXiv 12 May 2021.

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

- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [Towards Fast Computation of Certified Robustness for ReLU Networks](https://mlsanity.com/paper/1804.09699/) (2018)
- [Lipschitz-Margin Training: Scalable Certification of Perturbation Invariance for Deep Neural Networks](https://mlsanity.com/paper/1802.04034/) (2018)
- [Formal Security Analysis of Neural Networks using Symbolic Intervals](https://mlsanity.com/paper/1804.10829/) (2018)
- [Evaluating the Robustness of Neural Networks: An Extreme Value Theory Approach](https://mlsanity.com/paper/1801.10578/) (2018)
- [The Lipschitz Constant of Self-Attention](https://mlsanity.com/paper/2006.04710/) (2020)
- [Efficient Formal Safety Analysis of Neural Networks](https://mlsanity.com/paper/1809.08098/) (2018)
- [Robust Large Margin Deep Neural Networks](https://mlsanity.com/paper/1605.08254/) (2016)
- [Exactly Computing the Local Lipschitz Constant of ReLU Networks](https://mlsanity.com/paper/2003.01219/) (2020)
- [Lipschitz constant estimation of Neural Networks via sparse polynomial optimization](https://mlsanity.com/paper/2004.08688/) (2020)

## Built on by (most cited first, 16 of 16)

- [Efficiently Computing Local Lipschitz Constants of Neural Networks via Bound Propagation](https://mlsanity.com/paper/2210.07394/) (2022)
- [Gradient-Regularized Out-of-Distribution Detection](https://mlsanity.com/paper/2404.12368/) (2024)
- [GoTube: Scalable Stochastic Verification of Continuous-Depth Models](https://mlsanity.com/paper/2107.08467/) (2021)
- [Parameterized Hardness of Zonotope Containment and Neural Network Verification](https://mlsanity.com/paper/2509.22849/) (2025)
- [Policy Verification in Stochastic Dynamical Systems Using Logarithmic Neural Certificates](https://mlsanity.com/paper/2406.00826/) (2024)
- [Imitation Learning of MPC with Neural Networks: Error Guarantees and Sparsification](https://mlsanity.com/paper/2501.03671/) (2025)
- [An automatic differentiation system for the age of differential privacy](https://mlsanity.com/paper/2109.10573/) (2021)
- [Adversarial Sample Detection Through Neural Network Transport Dynamics](https://mlsanity.com/paper/2306.04252/) (2023)
- [ECLipsE-Gen-Local: Efficient Compositional Local Lipschitz Estimates for Deep Neural Networks](https://mlsanity.com/paper/2510.05261/) (2025)
- [ExLipBaB: Exact Lipschitz Constant Computation for Piecewise Linear Neural Networks](https://mlsanity.com/paper/2602.15499/) (2026)
- [A Domain-Theoretic Framework for Robustness Analysis of Neural Networks](https://mlsanity.com/paper/2203.00295/) (2022)
- [Verifying Global Neural Network Specifications using Hyperproperties](https://mlsanity.com/paper/2306.12495/) (2023)
- [Robustness Analysis of Continuous-Depth Models with Lagrangian Techniques](https://mlsanity.com/paper/2308.12192/) (2023)
- [Exploration Behavior of Untrained Policies](https://mlsanity.com/paper/2506.22566/) (2025)
- [Demystifying Lipschitz verification: positive matrices, negative results](https://mlsanity.com/paper/2603.28113/) (2026)
- [LiD-GLM: Lipschitz-constrained Deep Generalized Linear Models](https://mlsanity.com/paper/2608.16340/) (2026)

## Abstract (arXiv)

The Lipschitz constant of neural networks plays an important role in several contexts of deep learning ranging from robustness certification and regularization to stability analysis of systems with neural network controllers. Obtaining tight bounds of the Lipschitz constant is therefore important. We introduce LipBaB, a branch and bound framework to compute certified bounds of the local Lipschitz constant of deep neural networks with ReLU activation functions up to any desired precision. We achieve this by bounding the norm of the Jacobians, corresponding to different activation patterns of the network caused within the input domain. Our algorithm can provide provably exact computation of the Lipschitz constant for any p-norm.