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Lipschitz constants are connected to many properties of neural networks, suchas robustness, fairness, and generalization. Existing methods for computingLipschitz constants either produce relatively loose upper bounds or are limitedto small networks. In this paper, we develop an efficient framework forcomputing the $\ell_\infty$ local Lipschitz constant of a neural network bytightly upper bounding the norm of Clarke Jacobian via linear boundpropagation. We formulate the computation of local Lipschitz constants with alinear bound propagation process on a high-order backward graph induced by thechain rule of Clarke Jacobian. To enable linear bound propagation, we derivetight linear relaxations for specific nonlinearities in Clarke Jacobian. Thisformulate unifies existing ad-hoc approaches such as RecurJac, which can beseen as a special case of ours with weaker relaxations. The bound propagationframework also allows us to easily borrow the popular Branch-and-Bound (BaB)approach from neural network verification to further tighten Lipschitzconstants. Experiments show that on tiny models, our method produces comparablebounds compared to exact methods that cannot scale to slightly larger models;on larger models, our method efficiently produces tighter results than existingrelaxed or naive methods, and our method scales to much larger practical modelsthat previous works could not handle. We also demonstrate an application onprovable monotonicity analysis. Code is available athttps://github.com/shizhouxing/Local-Lipschitz-Constants.

Efficiently Computing Local Lipschitz Constants of Neural Networks via Bound Propagation