Chordal Sparsity for Lipschitz Constant Estimation of Deep Neural Networks

TL;DR

This paper introduces Chordal-LipSDP, a scalable semidefinite programming method leveraging chordal sparsity to efficiently estimate Lipschitz constants of deep neural networks, enhancing robustness guarantees.

Contribution

It develops a chordally sparse formulation of LipSDP, significantly improving scalability while maintaining zero accuracy loss in Lipschitz constant estimation.

Findings

Outperforms LipSDP in deep networks due to decomposition of large constraints

Enables tighter Lipschitz estimates with a tunable sparsity parameter

Demonstrates scalability through extensive numerical experiments

Abstract

Lipschitz constants of neural networks allow for guarantees of robustness in image classification, safety in controller design, and generalizability beyond the training data. As calculating Lipschitz constants 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 semidefinite constraint, is now 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.