How Does Noise Help Robustness? Explanation and Exploration under the Neural SDE Framework
Xuanqing Liu,  Tesi Xiao,  Si Si,  Qin Cao,  Sanjiv Kumar,  Cho-Jui Hsieh

Neural Ordinary Differential Equation (Neural ODE) has been proposed as a continuous approximation to the ResNet architecture. Some commonly used regularization mechanisms in discrete neural networks (e.g., dropout, Gaussian noise) are missing in current Neural ODE networks. In this paper, we propose a new continuous neural network framework called Neural Stochastic Differential Equation (Neural SDE), which naturally incorporates various commonly used regularization mechanisms based on random noise injection. For regularization purposes, our framework includes multiple types of noise patterns, such as dropout, additive, and multiplicative noise, which are common in plain neural networks. We provide some theoretical analyses explaining the improved robustness of our models against input perturbations. Furthermore, we demonstrate that the Neural SDE network can achieve better generalization than the Neural ODE and is more resistant to adversarial and non-adversarial input perturbations.