Efficient Proximal Mapping of the 1-path-norm of Shallow Networks
This work addresses the challenge of improving adversarial robustness in neural networks, offering a novel regularization approach that is more efficient and tighter than existing methods, though it is incremental as it builds on prior norms and constraints.
The paper tackles the problem of training shallow neural networks with robustness to adversarial perturbations by introducing an efficient proximal operator for the 1-path-norm, which allows for regularized empirical risk minimization and provides a tighter upper bound on the network's Lipschitz constant compared to layer-wise methods.
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).