Semialgebraic Optimization for Lipschitz Constants of ReLU Networks

Part of Advances in Neural Information Processing Systems 33 (NeurIPS 2020)

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Tong Chen, Jean B. Lasserre, Victor Magron, Edouard Pauwels


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.