text: We provide recovery guarantees for compressible signals that have been corrupted with noise and extend the framework introduced in [1] to defend neural networks against @math -norm and @math -norm attacks. Concretely, for a signal that is approximately sparse in some transform domain and has been perturbed with noise, we provide guarantees for accurately recovering the signal in the transform domain. We can then use the recovered signal to reconstruct the signal in its original domain while largely removing the noise. Our results are general as they can be directly applied to most unitary transforms used in practice and hold for both @math -norm bounded noise and @math -norm bounded noise. In the case of @math -norm bounded noise, we prove recovery guarantees for Iterative Hard Thresholding (IHT) and Basis Pursuit (BP). For the case of @math -norm bounded noise, we provide recovery guarantees for BP. These guarantees theoretically bolster the defense framework introduced in [1] for defending neural networks against adversarial inputs. Finally, we experimentally demonstrate this defense framework using both IHT and BP against the One Pixel Attack [21], Carlini-Wagner @math and @math attacks [3], Jacobian Saliency Based attack [18], and the DeepFool attack [17] on CIFAR-10 [12], MNIST [13], and Fashion-MNIST [27] datasets. This expands beyond the experimental demonstrations of [1].
  @cite_18: Recent work has demonstrated that neural networks are vulnerable to adversarial examples, i.e., inputs that are almost indistinguishable from natural data and yet classified incorrectly by the network. In fact, some of the latest findings suggest that the existence of adversarial attacks may be an inherent weakness of deep learning models. To address this problem, we study the adversarial robustness of neural networks through the lens of robust optimization. This approach provides us with a broad and unifying view on much of the prior work on this topic. Its principled nature also enables us to identify methods for both training and attacking neural networks that are reliable and, in a certain sense, universal. In particular, they specify a concrete security guarantee that would protect against any adversary. These methods let us train networks with significantly improved resistance to a wide range of adversarial attacks. They also suggest the notion of security against a first-order adversary as a natural and broad security guarantee. We believe that robustness against such well-defined classes of adversaries is an important stepping stone towards fully resistant deep learning models.
  @cite_14: Feature squeezing is a recently-introduced framework for mitigating and detecting adversarial examples. In previous work, we showed that it is effective against several earlier methods for generating adversarial examples. In this short note, we report on recent results showing that simple feature squeezing techniques also make deep learning models significantly more robust against the Carlini Wagner attacks, which are the best known adversarial methods discovered to date.
  @cite_22: Abstract: Several machine learning models, including neural networks, consistently misclassify adversarial examples---inputs formed by applying small but intentionally worst-case perturbations to examples from the dataset, such that the perturbed input results in the model outputting an incorrect answer with high confidence. Early attempts at explaining this phenomenon focused on nonlinearity and overfitting. We argue instead that the primary cause of neural networks' vulnerability to adversarial perturbation is their linear nature. This explanation is supported by new quantitative results while giving the first explanation of the most intriguing fact about them: their generalization across architectures and training sets. Moreover, this view yields a simple and fast method of generating adversarial examples. Using this approach to provide examples for adversarial training, we reduce the test set error of a maxout network on the MNIST dataset.
  @cite_28: Adversarial examples are perturbed inputs designed to fool machine learning models. Adversarial training injects such examples into training data to increase robustness. To scale this technique to large datasets, perturbations are crafted using fast single-step methods that maximize a linear approximation of the model's loss. We show that this form of adversarial training converges to a degenerate global minimum, wherein small curvature artifacts near the data points obfuscate a linear approximation of the loss. The model thus learns to generate weak perturbations, rather than defend against strong ones. As a result, we find that adversarial training remains vulnerable to black-box attacks, where we transfer perturbations computed on undefended models, as well as to a powerful novel single-step attack that escapes the non-smooth vicinity of the input data via a small random step. We further introduce Ensemble Adversarial Training, a technique that augments training data with perturbations transferred from other models. On ImageNet, Ensemble Adversarial Training yields models with strong robustness to black-box attacks. In particular, our most robust model won the first round of the NIPS 2017 competition on Defenses against Adversarial Attacks.
  @cite_21: In recent years, deep neural network approaches have been widely adopted for machine learning tasks, including classification. However, they were shown to be vulnerable to adversarial perturbations: carefully crafted small perturbations can cause misclassification of legitimate images. We propose Defense-GAN, a new framework leveraging the expressive capability of generative models to defend deep neural networks against such attacks. Defense-GAN is trained to model the distribution of unperturbed images. At inference time, it finds a close output to a given image which does not contain the adversarial changes. This output is then fed to the classifier. Our proposed method can be used with any classification model and does not modify the classifier structure or training procedure. It can also be used as a defense against any attack as it does not assume knowledge of the process for generating the adversarial examples. We empirically show that Defense-GAN is consistently effective against different attack methods and improves on existing defense strategies. Our code has been made publicly available at this https URL
summary: Most defenses against adversarial inputs do not come with theoretical guarantees. Instead, a large body of research has focused on finding practical ways to improve robustness to adversarial inputs by either augmenting the training data @cite_22 , using adversarial inputs from various networks @cite_28 , or by reducing the dimensionality of the input @cite_14 . For instance, @cite_18 use robust optimization to make the network robust to worst case adversarial perturbations on the training data. However, the effectiveness of their approach is determined by the amount and quality of training data available and its similarity to the distribution of the test data. An approach similar to ours but without any theoretical guarantees is @cite_21 . In this work, the authors use Generative Adversarial Networks (GANs) to estimate the distribution of the training data and during inference, use a GAN to reconstruct an input that is most similar to a given test input and is not adversarial.