# Analytical bounds on the local Lipschitz constants of affine-ReLU functions

arXiv: https://arxiv.org/abs/2008.06141 · HTML: https://mlsanity.com/paper/2008.06141/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Aug 2020.

## Built on by (most cited first, 2 of 2)

- [Certified Robustness via Dynamic Margin Maximization and Improved Lipschitz Regularization](https://mlsanity.com/paper/2310.00116/) (2023)
- [A Neurosymbolic Approach to the Verification of Temporal Logic Properties of Learning enabled Control Systems](https://mlsanity.com/paper/2303.05394/) (2023)

## Abstract (arXiv)

In this paper, we determine analytical bounds on the local Lipschitz constants of of affine functions composed with rectified linear units (ReLUs). Affine-ReLU functions represent a widely used layer in deep neural networks, due to the fact that convolution, fully-connected, and normalization functions are all affine, and are often followed by a ReLU activation function. Using an analytical approach, we mathematically determine upper bounds on the local Lipschitz constant of an affine-ReLU function, show how these bounds can be combined to determine a bound on an entire network, and discuss how the bounds can be efficiently computed, even for larger layers and networks. We show several examples by applying our results to AlexNet, as well as several smaller networks based on the MNIST and CIFAR-10 datasets. The results show that our method produces tighter bounds than the standard conservative bound (i.e. the product of the spectral norms of the layers' linear matrices), especially for small perturbations.