Computable Lipschitz Bounds for Deep Neural Networks

TL;DR

This paper introduces new computable bounds for the Lipschitz constants of deep neural networks, emphasizing the importance of different norms, and demonstrates their effectiveness through theoretical analysis and numerical experiments.

Contribution

The paper proposes two novel bounds for Lipschitz constants of neural networks under $l^1$ and $l^8f$ norms, improving over existing bounds and addressing convolutional networks.

Findings

New bounds are more accurate than existing ones.

One bound is exact for simple analytical cases.

Bounds are validated on MNIST and random matrix tests.

Abstract

Deriving sharp and computable upper bounds of the Lipschitz constant of deep neural networks is crucial to formally guarantee the robustness of neural-network based models. We analyse three existing upper bounds written for the $l^2$ norm. We highlight the importance of working with the $l^1$ and $l^\infty$ norms and we propose two novel bounds for both feed-forward fully-connected neural networks and convolutional neural networks. We treat the technical difficulties related to convolutional neural networks with two different methods, called explicit and implicit. Several numerical tests empirically confirm the theoretical results, help to quantify the relationship between the presented bounds and establish the better accuracy of the new bounds. Four numerical tests are studied: two where the output is derived from an analytical closed form are proposed; another one with random matrices; and the last one for convolutional neural networks trained on the MNIST dataset. We observe that one of our bound is optimal in the sense that it is exact for the first test with the simplest analytical form and it is better than other bounds for the other tests.