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

TL;DR

This paper introduces a closed-form proximal operator for the 1-path-norm in shallow neural networks, enabling efficient regularization and providing tighter Lipschitz bounds for robustness, with practical benefits demonstrated in experiments.

Contribution

It presents a novel closed-form proximal operator for the 1-path-norm and shows its effectiveness in regularization and robustness training of shallow networks.

Findings

Proximal operator for 1-path-norm can be computed efficiently.

Using 1-path-norm improves robustness-accuracy trade-offs.

Compared to L1 and layer-wise constraints, 1-path-norm offers tighter Lipschitz bounds.

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).