Chordal Sparsity for Lipschitz Constant Estimation of Deep Neural Networks

Computing Lipschitz constants of neural networks allows for robustness guarantees in image classification, safety in controller design, and generalization beyond the training data. As calculating Lipschitz constants of neural networks is NP-hard, techniques for estimating Lipschitz constants must navigate the trade-off between scalability and accuracy. In this work, we significantly push the scalability frontier of a semidefinite programming technique known as LipSDP while achieving zero accuracy loss. We first show that LipSDP has chordal sparsity, which allows us to derive a chordally sparse formulation that we call Chordal-LipSDP. The key benefit is that the main computational bottleneck of LipSDP, a large linear matrix inequality, can be decomposed into an equivalent collection of smaller ones — allowing Chordal-LipSDP to outperform LipSDP particularly as the network depth grows. Moreover, our formulation uses a tunable sparsity parameter that enables one to gain tighter estimates without incurring a significant computational cost. We illustrate the scalability of our approach through extensive numerical experiments.