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Poster
Rethinking Lipschitz Neural Networks and Certified Robustness: A Boolean Function Perspective
Bohang Zhang · Du Jiang · Di He · Liwei Wang

Wed Nov 30 09:00 AM -- 11:00 AM (PST) @ Hall J #923
in Poster Session 3 »
Designing neural networks with bounded Lipschitz constant is a promising way to obtain certifiably robust classifiers against adversarial examples. However, the relevant progress for the important $\ell_\infty$ perturbation setting is rather limited, and a principled understanding of how to design expressive $\ell_\infty$ Lipschitz networks is still lacking. In this paper, we bridge the gap by studying certified $\ell_\infty$ robustness from a novel perspective of representing Boolean functions. We derive two fundamental impossibility results that hold for any standard Lipschitz network: one for robust classification on finite datasets, and the other for Lipschitz function approximation. These results identify that networks built upon norm-bounded affine layers and Lipschitz activations intrinsically lose expressive power even in the two-dimensional case, and shed light on how recently proposed Lipschitz networks (e.g., GroupSort and $\ell_\infty$-distance nets) bypass these impossibilities by leveraging order statistic functions. Finally, based on these insights, we develop a unified Lipschitz network that generalizes prior works, and design a practical version that can be efficiently trained (making certified robust training free). Extensive experiments show that our approach is scalable, efficient, and consistently yields better certified robustness across multiple datasets and perturbation radii than prior Lipschitz networks.
Keywords: Adversarial Robustness · Expressive Power · certified defense · Lipschitz Neural Network

Author Information

Bohang Zhang (Peking University)
Du Jiang (Peking University)
Di He (Peking University)
Liwei Wang (Peking University)

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