How Does Noise Help Robustness? Explanation and Exploration under the Neural SDE Framework

CVPR, pp. 279-287, 2020.

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neural stochastic differential equationtest accuracydifferential equationgood generalizationadversarial exampleMore(10+)
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We propose a new continuous neural network framework called Neural Stochastic Differential Equation, which naturally incorporates various commonly used regularization mechanisms based on random noise injection

Abstract:

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 framew...More

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Introduction
  • Despite the superhuman performance in many computer vision tasks, recent findings [2, 8, 28] demonstrate that deep neural networks remain to be more fragile than human or even shallow models
  • Existing work support such phenomenon from different perspectives; for instance, on CIFAR-10 and ImageNet, [25] shows that the test accuracy drops by 5% − 15% if the authors replace the original test set by a new one.
  • It is very interesting to see whether there is a unified way to mitigate all the problems, and whether the authors can find a theoretical explanation for it
Highlights
  • Despite the superhuman performance in many computer vision tasks, recent findings [2, 8, 28] demonstrate that deep neural networks remain to be more fragile than human or even shallow models. Existing work support such phenomenon from different perspectives; for instance, on CIFAR-10 and ImageNet, [25] shows that the test accuracy drops by 5% − 15% if we replace the original test set by a new one
  • Even on the same test set, unnoticeable adversarial perturbations crafted by specific algorithms [19] can make the test accuracy close to zero
  • To study and understand how randomness stabilizes neural networks, we propose a new continuous neural network framework called Neural Stochastic Differential Equation (Neural Stochastic differential equation), which models the continuous limits of ResNet based on the recent proposed Neural ODE model [3] and adds stochastic diffusion and jump terms to cover various commonly used regularization mechanisms based on random noise, including Dropout, stochastic depth and Gaussian smoothing
  • While [30] deals with adversarial robustness, their method is still based on adversarial training
  • We introduce the Neural Stochastic differential equation model, which can stabilize the prediction of Neural ODE by injecting stochastic noise
Conclusion
  • The authors introduce the Neural SDE model, which can stabilize the prediction of Neural ODE by injecting stochastic noise.
  • The authors' model can achieve better generalization and improve the robustness under both adversarial and non-adversarial noises
Summary
  • Introduction:

    Despite the superhuman performance in many computer vision tasks, recent findings [2, 8, 28] demonstrate that deep neural networks remain to be more fragile than human or even shallow models
  • Existing work support such phenomenon from different perspectives; for instance, on CIFAR-10 and ImageNet, [25] shows that the test accuracy drops by 5% − 15% if the authors replace the original test set by a new one.
  • It is very interesting to see whether there is a unified way to mitigate all the problems, and whether the authors can find a theoretical explanation for it
  • Conclusion:

    The authors introduce the Neural SDE model, which can stabilize the prediction of Neural ODE by injecting stochastic noise.
  • The authors' model can achieve better generalization and improve the robustness under both adversarial and non-adversarial noises
Tables
  • Table1: Evaluating the model generalization under different choices of diffusion matrix G(ht, t; v) introduced above. For the three noise types, we search a suitable parameter σt for each of them so that the diffusion matrix G properly regularizes the model. TTN means testing time noise. We observe adding noises can improve the test accuracy over Neural ODE, and furthermore, noise at testing time is beneficial
  • Table2: Testing accuracy results under different levels of non-adversarial perturbations
Download tables as Excel
Related work
  • Our work is inspired by the success of the Neural ODE network, and we seek to improve the generalization and robustness of Neural ODE by adding noise in the dynamic system. Regularization mechanisms such as dropout cannot be easily incorporated in the original Neural ODE due to its deterministic nature.

    Neural ODE The idea of formulating ResNet as a dynamic system was discussed in [5]. A framework was proposed to link existing deep architectures with discretized numerical ODE solvers [18], and was shown to be parameter efficient. These networks adopt layer-wise architecture – each layer is parameterized by different independent weights. The Neural ODE model [3] computes hidden states in a different way: it directly models the dynamics of hidden states by an ODE solver, with the dynamics parameterized by a shared model. A memory efficient approach to compute gradient by adjoint methods was developed, making it possible to train large, multi-scale generative networks [1, 9]. Our work can be regarded as an extension of this framework, with the purpose of incorporating a variety of noise-injection based regularization mechanisms. Stochastic differential equation (SDE) in the context of neural network has been studied recently, focusing either on understanding how dropout shapes the loss landscape [27], or on using SDE as a universal function approximation tool to learn the solution of high dimensional PDEs [23]. Instead, we aim to explain why adding random noise boosts the stability of deep networks, and demonstrates the improved generalization and robustness.
Funding
  • This work is partially supported by NSF under IIS1719097
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