Efficiently Training Low-Curvature Neural Networks

TL;DR

This paper introduces low-curvature neural networks (LCNNs) that achieve enhanced robustness and gradient stability by minimizing a curvature bound, using novel architectural components, without sacrificing predictive accuracy.

Contribution

The paper proposes a method to train low-curvature neural networks by minimizing a curvature bound and introduces new layers like centered-softplus and Lipschitz batch normalization.

Findings

LCNNs exhibit significantly lower curvature than standard models.

LCNNs demonstrate increased adversarial robustness.

LCNNs maintain comparable predictive performance to traditional networks.

Abstract

The highly non-linear nature of deep neural networks causes them to be susceptible to adversarial examples and have unstable gradients which hinders interpretability. However, existing methods to solve these issues, such as adversarial training, are expensive and often sacrifice predictive accuracy. In this work, we consider curvature, which is a mathematical quantity which encodes the degree of non-linearity. Using this, we demonstrate low-curvature neural networks (LCNNs) that obtain drastically lower curvature than standard models while exhibiting similar predictive performance, which leads to improved robustness and stable gradients, with only a marginally increased training time. To achieve this, we minimize a data-independent upper bound on the curvature of a neural network, which decomposes overall curvature in terms of curvatures and slopes of its constituent layers. To efficiently minimize this bound, we introduce two novel architectural components: first, a non-linearity called centered-softplus that is a stable variant of the softplus non-linearity, and second, a Lipschitz-constrained batch normalization layer. Our experiments show that LCNNs have lower curvature, more stable gradients and increased off-the-shelf adversarial robustness when compared to their standard high-curvature counterparts, all without affecting predictive performance. Our approach is easy to use and can be readily incorporated into existing neural network models.