text: Adversarial examples are a challenging open problem for deep neural networks. We propose in this paper to add a penalization term that forces the decision function to be at in some regions of the input space, such that it becomes, at least locally, less sensitive to attacks. Our proposition is theoretically motivated and shows on a rst set of carefully conducted experiments that it behaves as expected when used alone, and seems promising when coupled with adversarial training.
  @cite_1: Deep learning algorithms have been shown to perform extremely well on manyclassical machine learning problems. However, recent studies have shown thatdeep learning, like other machine learning techniques, is vulnerable to adversarial samples: inputs crafted to force adeep neural network (DNN) to provide adversary-selected outputs. Such attackscan seriously undermine the security of the system supported by the DNN, sometimes with devastating consequences. For example, autonomous vehicles canbe crashed, illicit or illegal content can bypass content filters, or biometricauthentication systems can be manipulated to allow improper access. In thiswork, we introduce a defensive mechanism called defensive distillationto reduce the effectiveness of adversarial samples on DNNs. We analyticallyinvestigate the generalizability and robustness properties granted by the useof defensive distillation when training DNNs. We also empirically study theeffectiveness of our defense mechanisms on two DNNs placed in adversarialsettings. The study shows that defensive distillation can reduce effectivenessof sample creation from 95 to less than 0.5 on a studied DNN. Such dramaticgains can be explained by the fact that distillation leads gradients used inadversarial sample creation to be reduced by a factor of 1030. We alsofind that distillation increases the average minimum number of features thatneed to be modified to create adversarial samples by about 800 on one of theDNNs we tested.
summary: The difference between double backpropagation and our gradient penalization is that we penalize the @math -norm of the gradient of each output with respect to the input while, in backpropagation, the @math -norm of the loss is penalized. Hence, we should not have the problem that occurs when the penalization term is multiplied by a small error vector. Nevertheless, this might not be a problem when using a different loss function. Although empirical evidence show double backpropagation's efficiency to enhance generalization, it is insufficient to defend against adversarial examples. That is what @cite_1 highlights saying limiting sensitivity to infinitesimal perturbation [e.g., using double backpropagation][] drucker1992improving only provides constraints very near training examples, so it does not solve the adversarial perturbation problem . But there are evidence that coupling that gradient penalty with adversarial training increases the robustness.