Efficient Proximal Mapping of the 1-path-norm of Shallow Networks

Fabian Latorre, Paul Rolland, Nadav Hallak, Volkan Cevher
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:5651-5661, 2020.

Abstract

We demonstrate two new important properties of the 1-path-norm of shallow neural networks. First, despite its non-smoothness and non-convexity it allows a closed form proximal operator which can be efficiently computed, allowing the use of stochastic proximal-gradient-type methods for regularized empirical risk minimization. Second, when the activation functions is differentiable, it provides an upper bound on the Lipschitz constant of the network. Such bound is tighter than the trivial layer-wise product of Lipschitz constants, motivating its use for training networks robust to adversarial perturbations. In practical experiments we illustrate the advantages of using the proximal mapping and we compare the robustness-accuracy trade-off induced by the 1-path-norm, L1-norm and layer-wise constraints on the Lipschitz constant (Parseval networks).

Cite this Paper

BibTeX
@InProceedings{pmlr-v119-latorre20a, title = {Efficient Proximal Mapping of the 1-path-norm of Shallow Networks}, author = {Latorre, Fabian and Rolland, Paul and Hallak, Nadav and Cevher, Volkan}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {5651--5661}, year = {2020}, editor = {III, Hal Daumé and Singh, Aarti}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/latorre20a/latorre20a.pdf}, url = {https://proceedings.mlr.press/v119/latorre20a.html}, abstract = {We demonstrate two new important properties of the 1-path-norm of shallow neural networks. First, despite its non-smoothness and non-convexity it allows a closed form proximal operator which can be efficiently computed, allowing the use of stochastic proximal-gradient-type methods for regularized empirical risk minimization. Second, when the activation functions is differentiable, it provides an upper bound on the Lipschitz constant of the network. Such bound is tighter than the trivial layer-wise product of Lipschitz constants, motivating its use for training networks robust to adversarial perturbations. In practical experiments we illustrate the advantages of using the proximal mapping and we compare the robustness-accuracy trade-off induced by the 1-path-norm, L1-norm and layer-wise constraints on the Lipschitz constant (Parseval networks).} }
Endnote
%0 Conference Paper %T Efficient Proximal Mapping of the 1-path-norm of Shallow Networks %A Fabian Latorre %A Paul Rolland %A Nadav Hallak %A Volkan Cevher %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-latorre20a %I PMLR %P 5651--5661 %U https://proceedings.mlr.press/v119/latorre20a.html %V 119 %X We demonstrate two new important properties of the 1-path-norm of shallow neural networks. First, despite its non-smoothness and non-convexity it allows a closed form proximal operator which can be efficiently computed, allowing the use of stochastic proximal-gradient-type methods for regularized empirical risk minimization. Second, when the activation functions is differentiable, it provides an upper bound on the Lipschitz constant of the network. Such bound is tighter than the trivial layer-wise product of Lipschitz constants, motivating its use for training networks robust to adversarial perturbations. In practical experiments we illustrate the advantages of using the proximal mapping and we compare the robustness-accuracy trade-off induced by the 1-path-norm, L1-norm and layer-wise constraints on the Lipschitz constant (Parseval networks).
APA
Latorre, F., Rolland, P., Hallak, N. & Cevher, V.. (2020). Efficient Proximal Mapping of the 1-path-norm of Shallow Networks. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:5651-5661 Available from https://proceedings.mlr.press/v119/latorre20a.html.

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