%0 Journal Article %1 anil2018sorting %A Anil, Cem %A Lucas, James %A Grosse, Roger %D 2018 %K convex deep-learning optimization %T Sorting out Lipschitz function approximation %U http://arxiv.org/abs/1811.05381 %X Training neural networks under a strict Lipschitz constraint is useful for provable adversarial robustness, generalization bounds, interpretable gradients, and Wasserstein distance estimation. By the composition property of Lipschitz functions, it suffices to ensure that each individual affine transformation or nonlinear activation is 1-Lipschitz. The challenge is to do this while maintaining the expressive power. We identify a necessary property for such an architecture: each of the layers must preserve the gradient norm during backpropagation. Based on this, we propose to combine a gradient norm preserving activation function, GroupSort, with norm-constrained weight matrices. We show that norm-constrained GroupSort architectures are universal Lipschitz function approximators. Empirically, we show that norm-constrained GroupSort networks achieve tighter estimates of Wasserstein distance than their ReLU counterparts and can achieve provable adversarial robustness guarantees with little cost to accuracy.

@article{anil2018sorting, abstract = {Training neural networks under a strict Lipschitz constraint is useful for provable adversarial robustness, generalization bounds, interpretable gradients, and Wasserstein distance estimation. By the composition property of Lipschitz functions, it suffices to ensure that each individual affine transformation or nonlinear activation is 1-Lipschitz. The challenge is to do this while maintaining the expressive power. We identify a necessary property for such an architecture: each of the layers must preserve the gradient norm during backpropagation. Based on this, we propose to combine a gradient norm preserving activation function, GroupSort, with norm-constrained weight matrices. We show that norm-constrained GroupSort architectures are universal Lipschitz function approximators. Empirically, we show that norm-constrained GroupSort networks achieve tighter estimates of Wasserstein distance than their ReLU counterparts and can achieve provable adversarial robustness guarantees with little cost to accuracy.}, added-at = {2019-10-16T16:53:11.000+0200}, author = {Anil, Cem and Lucas, James and Grosse, Roger}, biburl = {https://www.bibsonomy.org/bibtex/24791d401246e76c128ca45618784251d/kirk86}, description = {[1811.05381] Sorting out Lipschitz function approximation}, interhash = {b84d673adad94dbe99a4c9b793b230d6}, intrahash = {4791d401246e76c128ca45618784251d}, keywords = {convex deep-learning optimization}, note = {cite arxiv:1811.05381Comment: 8 main pages, 21 pages total, 17 figures. Accepted at ICML 2019}, timestamp = {2019-10-16T16:53:11.000+0200}, title = {Sorting out Lipschitz function approximation}, url = {http://arxiv.org/abs/1811.05381}, year = 2018 }