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].
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.
@cite_18 @cite_14 @cite_22 @cite_28 @cite_21
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Neural networks are vulnerable to adversarial examples and researchers have proposed many heuristic attack and defense mechanisms. We take the principled view of distributionally robust optimization, which guarantees performance under adversarial input perturbations. By considering a Lagrangian penalty formulation of perturbation of the underlying data distribution in a Wasserstein ball, we provide a training procedure that augments model parameter updates with worst-case perturbations of training data. For smooth losses, our procedure provably achieves moderate levels of robustness with little computational or statistical cost relative to empirical risk minimization. Furthermore, our statistical guarantees allow us to efficiently certify robustness for the population loss. For imperceptible perturbations, our method matches or outperforms heuristic approaches. Neural networks are vulnerable to adversarial examples and researchers have proposed many heuristic attack and defense mechanisms. We address this problem through the principled lens of distributionally robust optimization, which guarantees performance under adversarial input perturbations. By considering a Lagrangian penalty formulation of perturbing the underlying data distribution in a Wasserstein ball, we provide a training procedure that augments model parameter updates with worst-case perturbations of training data. For smooth losses, our procedure provably achieves moderate levels of robustness with little computational or statistical cost relative to empirical risk minimization. Furthermore, our statistical guarantees allow us to efficiently certify robustness for the population loss. For imperceptible perturbations, our method matches or outperforms heuristic approaches. Machine learning and deep learning in particular has advanced tremendously on perceptual tasks in recent years. However, it remains vulnerable against adversarial perturbations of the input that have been crafted specifically to fool the system while being quasi-imperceptible to a human. In this work, we propose to augment deep neural networks with a small detector'' subnetwork which is trained on the binary classification task of distinguishing genuine data from data containing adversarial perturbations. Our method is orthogonal to prior work on addressing adversarial perturbations, which has mostly focused on making the classification network itself more robust. We show empirically that adversarial perturbations can be detected surprisingly well even though they are quasi-imperceptible to humans. Moreover, while the detectors have been trained to detect only a specific adversary, they generalize to similar and weaker adversaries. In addition, we propose an adversarial attack that fools both the classifier and the detector and a novel training procedure for the detector that counteracts this attack. Machine learning and deep learning in particular has advanced tremendously on perceptual tasks in recent years. However, it remains vulnerable against adversarial perturbations of the input that have been crafted specifically to fool the system while being quasi-imperceptible to a human. In this work, we propose to augment deep neural networks with a small "detector" subnetwork which is trained on the binary classification task of distinguishing genuine data from data containing adversarial perturbations. Our method is orthogonal to prior work on addressing adversarial perturbations, which has mostly focused on making the classification network itself more robust. We show empirically that adversarial perturbations can be detected surprisingly well even though they are quasi-imperceptible to humans. Moreover, while the detectors have been trained to detect only a specific adversary, they generalize to similar and weaker adversaries. In addition, we propose an adversarial attack that fools both the classifier and the detector and a novel training procedure for the detector that counteracts this attack.