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

arXiv: https://arxiv.org/abs/2007.01003 · HTML: https://mlsanity.com/paper/2007.01003/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jul 2020.

## Built on by (most cited first, 2 of 2)

- [PathProx: A Proximal Gradient Algorithm for Weight Decay Regularized Deep Neural Networks](https://mlsanity.com/paper/2210.03069/) (2022)
- [Hidden Synergy: L1 Weight Normalization and 1-Path-Norm Regularization](https://mlsanity.com/paper/2404.19112/) (2024)

## Abstract (arXiv)

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).