Multivariate Lipschitz Analysis of the Stability of Neural Networks

The stability of neural networks with respect to adversarial perturbations has been extensively studied. One of the main strategies consist of quantifying the Lipschitz regularity of neural networks. In this paper, we introduce a multivariate Lipschitz constant-based stability analysis of fully connected neural networks allowing us to capture the influence of each input or group of inputs on the neural network stability. Our approach relies on a suitable re-normalization of the input space, with the objective to perform a more precise analysis than the one provided by a global Lipschitz constant. We investigate the mathematical properties of the proposed multivariate Lipschitz analysis and show its usefulness in better understanding the sensitivity of the neural network with regard to groups of inputs. We display the results of this analysis by a new representation designed for machine learning practitioners and safety engineers termed as a Lipschitz star. The Lipschitz star is a graphical and practical tool to analyze the sensitivity of a neural network model during its development, with regard to different combinations of inputs. By leveraging this tool, we show that it is possible to build robust-by-design models using spectral normalization techniques for controlling the stability of a neural network, given a safety Lipschitz target. Thanks to our multivariate Lipschitz analysis, we can also measure the efficiency of adversarial training in inference tasks. We perform experiments on various open access tabular datasets, and also on a real Thales Air Mobility industrial application subject to certification requirements.