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

TL;DR

This paper derives analytical bounds on the local Lipschitz constants of affine-ReLU functions in neural networks, enabling tighter robustness estimates for models like AlexNet and smaller datasets.

Contribution

It introduces a mathematical method to compute tighter local Lipschitz bounds for affine-ReLU layers, improving robustness analysis over standard spectral norm bounds.

Findings

Tighter bounds than spectral norm product for small perturbations

Applicable to large networks like AlexNet and smaller datasets

Efficient computation of bounds for practical networks

Abstract

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.