While neural networks have achieved high performance in different learning tasks, their accuracy drops significantly in the presence of small adversarial perturbations to inputs. Defenses based on regularization and adversarial training are often followed by new attacks to defeat them. In this paper, we propose attack-agnostic robustness certificates for a multi-label classification problem using a deep ReLU network. Although computing the exact distance of a given input sample to the classification decision boundary requires solving a non-convex optimization, we characterize two lower bounds for such distances, namely the simplex certificate and the decision boundary certificate. These robustness certificates leverage the piece-wise linear structure of ReLU networks and use the fact that in a polyhedron around a given sample, the prediction function is linear. In particular, the proposed simplex certificate has a closed-form, is differentiable and is an order of magnitude faster to compute than the existing methods even for deep networks. In addition to theoretical bounds, we provide numerical results for our certificates over MNIST and compare them with some existing upper bounds.
@cite_18: 
@cite_37: Many machine learning models are vulnerable to adversarial examples: inputs that are specially crafted to cause a machine learning model to produce an incorrect output. Adversarial examples that affect one model often affect another model, even if the two models have different architectures or were trained on different training sets, so long as both models were trained to perform the same task. An attacker may therefore train their own substitute model, craft adversarial examples against the substitute, and transfer them to a victim model, with very little information about the victim. Recent work has further developed a technique that uses the victim model as an oracle to label a synthetic training set for the substitute, so the attacker need not even collect a training set to mount the attack. We extend these recent techniques using reservoir sampling to greatly enhance the efficiency of the training procedure for the substitute model. We introduce new transferability attacks between previously unexplored (substitute, victim) pairs of machine learning model classes, most notably SVMs and decision trees. We demonstrate our attacks on two commercial machine learning classification systems from Amazon (96.19 misclassification rate) and Google (88.94 ) using only 800 queries of the victim model, thereby showing that existing machine learning approaches are in general vulnerable to systematic black-box attacks regardless of their structure.
@cite_26: We show that defensive distillation is not secure: it is no more resistant to targeted misclassification attacks than unprotected neural networks.
@cite_28: 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.
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@cite_53: 
@cite_21: Following the recent adoption of deep neural networks (DNN) accross a wide range of applications, adversarial attacks against these models have proven to be an indisputable threat. Adversarial samples are crafted with a deliberate intention of undermining a system. In the case of DNNs, the lack of better understanding of their working has prevented the development of efficient defenses. In this paper, we propose a new defense method based on practical observations which is easy to integrate into models and performs better than state-of-the-art defenses. Our proposed solution is meant to reinforce the structure of a DNN, making its prediction more stable and less likely to be fooled by adversarial samples. We conduct an extensive experimental study proving the efficiency of our method against multiple attacks, comparing it to numerous defenses, both in white-box and black-box setups. Additionally, the implementation of our method brings almost no overhead to the training procedure, while maintaining the prediction performance of the original model on clean samples.
@cite_17: It is well known that it is possible to construct "adversarial examples" for neural networks: inputs which are misclassified by the network yet indistinguishable from true data. We propose a simple modification to standard neural network architectures, thermometer encoding, which significantly increases the robustness of the network to adversarial examples. We demonstrate this robustness with experiments on the MNIST, CIFAR-10, CIFAR-100, and SVHN datasets, and show that models with thermometer-encoded inputs consistently have higher accuracy on adversarial examples, without decreasing generalization. State-of-the-art accuracy under the strongest known white-box attack was increased from 93.20 to 94.30 on MNIST and 50.00 to 79.16 on CIFAR-10. We explore the properties of these networks, providing evidence that thermometer encodings help neural networks to find more-non-linear decision boundaries.
@cite_52: Adversarial examples are maliciously perturbed inputs designed to mislead machine learning (ML) models at test-time. They often transfer: the same adversarial example fools more than one model. In this work, we propose novel methods for estimating the previously unknown dimensionality of the space of adversarial inputs. We find that adversarial examples span a contiguous subspace of large ( 25) dimensionality. Adversarial subspaces with higher dimensionality are more likely to intersect. We find that for two different models, a significant fraction of their subspaces is shared, thus enabling transferability. In the first quantitative analysis of the similarity of different models' decision boundaries, we show that these boundaries are actually close in arbitrary directions, whether adversarial or benign. We conclude by formally studying the limits of transferability. We derive (1) sufficient conditions on the data distribution that imply transferability for simple model classes and (2) examples of scenarios in which transfer does not occur. These findings indicate that it may be possible to design defenses against transfer-based attacks, even for models that are vulnerable to direct attacks.
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@cite_24: 
@cite_55: Neural networks provide state-of-the-art results for most machine learning tasks. Unfortunately, neural networks are vulnerable to adversarial examples: given an input @math and any target classification @math , it is possible to find a new input @math that is similar to @math but classified as @math . This makes it difficult to apply neural networks in security-critical areas. Defensive distillation is a recently proposed approach that can take an arbitrary neural network, and increase its robustness, reducing the success rate of current attacks' ability to find adversarial examples from @math to @math . In this paper, we demonstrate that defensive distillation does not significantly increase the robustness of neural networks by introducing three new attack algorithms that are successful on both distilled and undistilled neural networks with @math probability. Our attacks are tailored to three distance metrics used previously in the literature, and when compared to previous adversarial example generation algorithms, our attacks are often much more effective (and never worse). Furthermore, we propose using high-confidence adversarial examples in a simple transferability test we show can also be used to break defensive distillation. We hope our attacks will be used as a benchmark in future defense attempts to create neural networks that resist adversarial examples.
@cite_49: We identify obfuscated gradients as a phenomenon that leads to a false sense of security in defenses against adversarial examples. While defenses that cause obfuscated gradients appear to defeat optimization-based attacks, we find defenses relying on this effect can be circumvented. For each of the three types of obfuscated gradients we discover, we describe indicators of defenses exhibiting this effect and develop attack techniques to overcome it. In a case study, examining all defenses accepted to ICLR 2018, we find obfuscated gradients are a common occurrence, with 7 of 8 defenses relying on obfuscated gradients. Using our new attack techniques, we successfully circumvent all 7 of them.
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@cite_11: Standard methods for generating adversarial examples for neural networks do not consistently fool neural network classifiers in the physical world due to a combination of viewpoint shifts, camera noise, and other natural transformations, limiting their relevance to real-world systems. We demonstrate the existence of robust 3D adversarial objects, and we present the first algorithm for synthesizing examples that are adversarial over a chosen distribution of transformations. We synthesize two-dimensional adversarial images that are robust to noise, distortion, and affine transformation. We apply our algorithm to complex three-dimensional objects, using 3D-printing to manufacture the first physical adversarial objects. Our results demonstrate the existence of 3D adversarial objects in the physical world.
The adversarial attacks and defenses have received significant attention from the machine learning community in the last couple of years @cite_9 @cite_18 @cite_53 @cite_49 @cite_11 @cite_17 @cite_56 @cite_28 @cite_21 @cite_37 @cite_52 @cite_26 @cite_47 @cite_13 @cite_24 @cite_12 . A wide range of defenses have been proposed to harden neural networks against adversarial attacks. However, a pattern has emerged in which the majority of adversarial defenses are broken by new attacks. For example, @cite_55 , @cite_49 , @cite_11 , @cite_18 broke several of these proposed defenses.