Computable Lipschitz Bounds for Deep Neural Networks
This work addresses the need for formal robustness guarantees in neural-network models, though it is incremental as it builds on existing bounds by extending them to different norms and network types.
The authors tackled the problem of deriving sharp and computable Lipschitz constant bounds for deep neural networks to guarantee robustness, proposing two novel bounds for feed-forward and convolutional networks that showed better accuracy in numerical tests, including one bound being optimal in a simple analytical case.
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.