The Lipschitz constant of a network plays an important role in many applications of deep learning, such as robustness certification and Wasserstein Generative Adversarial Network. We introduce a semidefinite programming hierarchy to estimate the global and local Lipschitz constant of a multiple layer deep neural network. The novelty is to combine a polynomial lifting for ReLU functions derivatives with a weak generalization of Putinar's positivity certificate. This idea could also apply to other, nearly sparse, polynomial optimization problems in machine learning. We empirically demonstrate that our method provides a trade-off with respect to state of the art linear programming approach, and in some cases we obtain better bounds in less time.
@article{arxiv.2002.03657,
title = {Semialgebraic Optimization for Lipschitz Constants of ReLU Networks},
author = {Tong Chen and Jean-Bernard Lasserre and Victor Magron and Edouard Pauwels},
journal= {arXiv preprint arXiv:2002.03657},
year = {2020}
}