Albeit displaying remarkable performance across a range of tasks, Deep Neural Networks (DNNs) are highly vulnerable to adversarial examples, which are carefully crafted examples generated by adding a certain degree of noise (a.k.a. perturbations) to the corresponding original images, typically appearing quasi-imperceptible to humans (Papernot, McDaniel). Importantly, these adversarial examples are transferable from one network to another, even when the other network fashions a different architecture and possibly trained on a different subset of training data (Liu, Chen; Florian, Papernot). Transferability permits an adversarial attack, without knowing the internals of the target network, posing serious security concerns on the practical deployment of these models.
Adversarial perturbations are either instance-specific or instance-agnostic. The instance-specific attacks iteratively optimize a perturbation pattern specific to an input sample (e.g., Fawzi (Fawzi, Dezfooli); Goodfellow (Shlens); Kurakin (Goodfellow); Dezfooli (Fawzi); Nguyen (Yosinski); Dong (Pang); Li (Bai)). In comparison, the instance-agnostic attacks learn a universal perturbation or a function that finds adversarial patterns on a data distribution instead of a single sample. For example, Dezfooli (Fawzi) proposed universal adversarial perturbations that can fool a model on the majority of the source dataset images. To reduce dependency on the input data samples, Mopuri (Garg) maximizes layer activations of the source network while Mopuri (Uppala) extracts deluding perturbations using class impressions relying on the source label space. To enhance the transferability of instance-agnostic approaches, recent generative models attempt to directly craft perturbations using an adversarially trained function Baluja (Fischer); Poursaeed (Katsman).
We observe that most prior works on crafting adversarial attacks suffer from two pivotal limitations that restrict their transferability to real-world scenarios. (a) Existing attacks rely directly or indirectly on the source (training) data, which hampers their transferability to other domains. From a practical standpoint, source domain can be unknown, or the domain-specific data may be unavailable to the attacker. Therefore, a true "black-box" attack must be able to fool learned models across different target domains without ever being explicitly trained on those data domains. (b) Instance-agnostic attacks, compared with their counterparts, are far more scalable to large datasets as they avoid expensive per-instance iterative optimization. However, they demonstrate weaker transferability rates than the instance-specific attacks. Consequently, the design of highly transferable instance-agnostic attacks that also generalize across unseen domains is a largely unsolved problem.
In this work, we introduce ‘domain-agnostic’ generation of adversarial examples, with the aim of relaxing the source data reliance assumption. In particular, we propose a flexible framework capable of launching vastly transferable adversarial attacks, e.g., perturbations found on paintings, comics or medical images are shown to trick natural image classifiers trained on ImageNet dataset with high fooling rates. A distinguishing feature of our approach is the introduction of relativistic loss that explicitly enforces learning of domain-invariant adversarial patterns. Our attack algorithm is highly scalable to large-scale datasets since it learns a universal adversarial function that avoids expensive iterative optimization from instance-specific attacks. While enjoying the efficient inference time of instance-agnostic methods, our algorithm outperforms all existing attack methods (both instance-specific and agnostic) by a significant margin ( average increase in fooling rate from naturally trained Inception-v3 to adversarially trained models in comparison to state-of-the-art Dong (Pang)) and sets the new state-of-the-art under both white-box and black-box settings. Figure 1 provides an overview of our approach.
2 Related Work
Image-dependent Perturbations: Several approaches target creation of image-dependent perturbations. Szegedy (Zaremba) noticed that despite exhibiting impressive performance, neural networks can be fooled through maliciously crafted perturbations that appear quasi-imperceptible to humans. Following this finding, many approaches Fawzi (Fawzi, Dezfooli); Goodfellow (Shlens); Kurakin (Goodfellow); Dezfooli (Fawzi); Nguyen (Yosinski) investigate the existence of these perturbations. They either apply gradient ascent in the pixel space or solve complex optimizations. Recently, a few methods Xie (Zhang); Dong (Pang) propose input or gradient transformation modules to improve the transferability of adversarial examples. A common characteristic of the aforementioned approaches is their data-dependence; the perturbations are computed for each data-point separately in a mutually exclusive way. Further, these approaches render inefficiently at inference time since they iterate on the input multiple times. In contrast, we resort to a data-independent approach based on a generator, demonstrating improved inference-time efficiency along with high transferability rates.
Universal Adversarial Perturbation: Seminal work of Dezfooli (Fawzi)
introduces the existence of Universal Adversarial Perturbation (UAP). It is a single noise vector which when added to a data-point can fool a pretrained model.Dezfooli (Fawzi) crafts UAP in an iterative fashion utilizing target data-points that is capable of flipping their labels. Though it can generate image-agnostic UAP, the success ratio of their attack is proportional to the number of training samples used for crafting UAP. Mopuri (Garg) proposes a so-called data-independent algorithm by maximizing the product of mean activations at multiple layers given a universal perturbation as input. This method crafts a so-called data-independent perturbation, however, the attack success ratio is not comparable to Dezfooli (Fawzi). Instead, we propose a fully distribution-agnostic approach that crafts adversarial examples directly from a learned generator, as opposed to first generating perturbations followed by their addition to images.
Generator-oriented Perturbations: Another branch of attacks leverage generative models to craft adversaries. Baluja (Fischer) learns a generator network to perturb images, however, the unbounded perturbation magnitude in their case might render perceptible perturbations at test time. Xiao (Li) apply generative adversarial networks to craft visually realistic perturbations and build distilled network to perform black-box attack. Similarly, Poursaeed (Katsman); Mopuri (Uppala) train generators to create adversaries to launch attacks; the former uses target data directly and the latter relies on class impressions.
A common trait of prior work is that they either rely directly (or indirectly) upon the data distribution and/or entail access to its label space for creating adversarial examples (Table 1). In contrast, we propose a flexible, distribution-agnostic approach - inculcating relativistic loss - to craft adversarial examples that achieves state-of-the-art results both under white-box and black-box attack settings.
|FFF (Mopuri, Garg)||Pretrained-net/data||Low||✓||✗|
|AAA (Mopuri, Uppala)||Class Impressions||Medium||✗||✗|
|UAP (Dezfooli, Fawzi)||ImageNet||Low||✗||✗|
|GAP (Poursaeed, Katsman)||ImageNet||Medium||✗||✗|
|RHP (Li, Bai)||ImageNet||Medium||✗||✗|
|Ours||Arbitrary (Paintings, Comics, Medical scans etc.)||High||✓||✓|
3 Cross-Domain Transferable Perturbations
Our proposed approach is based on a generative model that is trained using an adversarial mechanism. Assume we have an input image belonging to a source domain . We aim to train a universal function that learns to add a perturbation pattern on the source domain which can successfully fool a network trained on source as well as any target domain when fed with perturbed inputs . Importantly, our training is only performed on the unlabelled source domain dataset with samples: and the target domain is not used at all during training. For brevity, in the following discussion, we will only refer the input and perturbed images using and respectively and the domain will be clear from the context.
The proposed framework consists of a generator and a discriminator parameterized by and . In our case, we initialize discriminator with a pretrained network and the parameters are remained fixed while the is learned. The output of is scaled to have a fixed norm and it lies within a bound; . The perturbed images as well as the real images
are passed through the discriminator. The output of the discriminator denotes the class probabilities, where
is the number of classes. This is different from the traditional GAN framework where a discriminator only estimate whether an input is real or fake. For an adversarial attack, the goal is to fool a network on most examples by making minor changes to its inputs, i.e.,
where, is the fooling ratio, is the ground-truth label for the example and the predictions on clean images are given by, . Note that we do not necessarily require the ground-truth labels of source domain images to craft a successful attack. In the case of adversarial attacks based on a traditional GAN framework, the following objective is maximized for the generator to achieve the maximal fooling rate:
is the one-hot encoded label vector for an input example. The above objective seeks to maximize the discriminator error on the perturbed images that are output from the generator network.
We argue that the objective given by Eq. 2 does not directly enforce transferability for the generated perturbations . This is primarily due to the reason that the discriminator’s response for clean examples is totally ignored in the conventional generative attacks. Here, inspired by the generative adversarial network in Jolicoeur (Martineau), we propose a relativistic adversarial perturbation (RAP) generation approach that explicitly takes in to account the discriminator’s predictions on clean images. Alongside reducing the classifier’s confidence on perturbed images, the attack algorithm also forces the discriminator to maintain a high confidence scores for the clean samples. The proposed objective is given by:
The cross entropy loss would be higher when the perturbed image is scored significantly lower than the clean image response for the ground-truth class i.e., . The discriminator basically seeks to increase the ‘fooling gap’ () between the true and perturbed samples. Through such relative discrimination, we not only report better transferability rates across networks trained on the same domain, but most importantly show excellent cross-domain transfer rates for the instance-agnostic perturbations. We attribute this behaviour to the fact that once a perturbation pattern is optimized using the proposed loss on a source distribution (e.g., paintings, cartoon images), the generator learns a "contrastive" signal that is agnostic to the underlying distribution. As a result, when the same perturbation pattern is applied to networks trained on totally different domain (e.g., natural images), it still achieves the state-of-the-art attack transferability rates. Table 2 shows the gain in transferability when using relativistic cross-entropy (Eq. 3) in comparison to simple cross-entropy loss (Eq. 2).
For an untargeted attack, the above mentioned objective in Eq. 2 and 3 suffices, however, for a targeted adversarial attack, the prediction for the perturbed image must match a given target class i.e.,
. For such a case, we employ the following loss function:
The overall training scheme for the generative network is given in Algorithm 1.
|Cross Entropy (CE)||79.21||78.96||69.32||66.45|
4.1 Rules of the Game
We report results using following three different attack settings in our experiments: (a) White-box. Attacker has access to the original model (both architecture and parameters) and the training data distribution. (b) Black-box. Attacker has access to a pretrained model on the same distribution but without any knowledge of the target architecture and target data distribution. (c) Cross-domain Black-box. Attacker has neither access to (any) pretrained model, nor to its label space and its training data distribution. It then has to seek a transferable adversarial function that is learned from a model pretrained on a possibly different distribution than the original. Hence, this setting is relatively far more challenging than the plain black-box setting.
|Fool Rate ()||Top-1 ()||Fool Rate ()||Top-1 ()||Fool Rate ()||Top-1 ()|
4.2 Experimental Settings
Generator Architecture. We chose ResNet architecture introduced in (Johnson, Alahi) as the generator network ; it consists of downsampling, residual and upsampling blocks. For training, we used Adam optimizer (Kingma, Ba)
with a learning rate of 1e-4 and values of exponential decay rate for first and second moments set to 0.5 and 0.999, respectively. Generators are learned against the four pretrained ImageNet models including VGG-16, VGG-19(Simonyan, Zisserman), Inception (Inc-v3) (Szegedy, Vanhoucke,Ioffe,Shlens,Wojna), ResNet-152 (He, Zhang,Ren,Sun) and ChexNet (which is a Dense-121 Huang (Liu) network trained to diagnose pneumonia) (Rajpurkar, Irvin).
Datasets. We consider the following datasets for generator training namely Paintings (Painter, by), Comics (Cenk, BircanoÄŸlu), ImageNet and a subset of ChestX-ray (ChestX) (Rajpurkar, Irvin). There are approximately 80k samples in Paintings, 50k in Comics, 1.2 million in ImageNet training set and 10k in ChestX.
Inference: Inference is performed on ImageNet validation set (val-set) (50k samples), a subset (5k samples) of ImageNet proposed by (Li, Bai) and ImageNet-NeurIPS NeurIPS (Attacks) (1k samples) dataset.
Evaluation Metrics: We use the fooling rate (percentage of input samples for which predicted label is flipped after adding adversarial perturbations), top-1 accuracy and % increase in error rate (the difference between error rate of adversarial and clean images) to evaluate our proposed approach.
Table 3 shows the cross-domain black-box setting results, where attacker have no access to model architecture, parameters, its training distribution or label space. Note that ChestX (Rajpurkar, Irvin) does not have much texture, an important feature to deceive ImageNet models (Geirhos, Rubisch), yet the transferability rate of perturbations learned against ChexNet is much better than the Gaussian noise.
Tables 5 and 5 show the comparison of our method against different universal methods on both naturally and adversarially trained models Florian (Kurakin) (Inc-v3, Inc-v4 and IncRes-v2). Our attack success rate is much higher both in white-box and black-box settings. Notably, for the case of adversarially trained models, Gaussian smoothing on top of our approach leads to significant increase in transferability. We provide further comparison with GAP Poursaeed (Katsman) in the supplementary material. Figures 3 and 4 show the model’s output and attention shift on example adversaries.
4.2.2 Comparison with State-of-the-Art
Finally, we compare our method with recently proposed instance-specific attack method (Dong, Pang) that exhibits high transferability to adversarially trained models. For the very first time in literature, we showed that a universal function like ours can attain much higher transferability rate, outperforming the state-of-the-art instance-specific translation invariant method (Dong, Pang) by a large average absolute gain of 46.6% and 86.5% (in fooling rates) on both naturally and adversarially trained models, respectively, as reported in Table 6. The naturally trained models are Inception-v3 (Inc-v3) Szegedy (Vanhoucke,Ioffe,Shlens,Wojna), Inception-v4 (Inc-v4), Inception Resnet v2 (IncRes-v2) Szegedy (Ioffe) and Resnet v2-152 (Res-152) He (Zhang)). The adversarially trained models are from Florian (Kurakin).
|Attack||Naturally Trained||Adversarially Trained|
4.3 Transferability: Naturally Trained vs. Adversarially Trained
Furthermore, we study the impact of training iterations and Gaussian smoothing Dong (Pang) on the transferability of our generative adversarial examples. We report results using naturally and adversarially trained IncRes-v2 model (Szegedy, Ioffe) as other models exhibit similar behaviour. Figure 5 displays the transferability (in %age fool rate) as a function of the number of training epochs (a-b) and various kernel sizes for Gaussian smoothing (c-d).
Firstly, we observe a gradual increase in the transferability of generator against the naturally trained model as the training epochs advance. In contrast the transferability deteriorates against the adversarially trained model. Therefore, when targeting naturally trained models, we train for ten epochs on Paintings, Comics, and ChestX datasets (although we anticipate better performance for higher epochs). When targeting adversarially trained models, we deploy an early stopping criterion to obtain the best trained generator since the performance drops on such models as epochs are increased. This fundamentally shows the reliance of naturally and adversarially trained models on different set of features.
Our results clearly demonstrate that the adversarial solution space is shared across different architectures and even across distinct data domains. Since we train our generator against naturally trained models only, therefore it converges to a solution space on which an adversarially trained model has already been trained. As a result, our perturbations gradually become weaker against adversarially trained models as the training progress. A visual demonstration is provided in supplementary material.
Secondly, the application of Gaussian smoothing reveals different results on naturally trained and adversarially trained models. After applying smoothing, adversaries become stronger for adversarially trained models and get weaker for naturally trained models. We achieve optimal results with the kernel size of 3 and for adversarially trained models and use these settings consistently in our experiments. We apply Gaussian kernel on the unrestricted generator’s output, therefore as the kernel size is increased, generator’s output becomes very smooth and after projection within valid range, adversaries become weaker.
Adversarial examples have been shown to be transferable across different models trained on the same domain. For the first time in literature, we show that the cross-domain transferable adversaries exists that can fool the target domain networks with high success rates. We propose a novel generative framework that learns to generate strong adversaries using a relativistic discriminator. Surprisingly, our proposed universal adversarial function can beat the instance-specific attack methods that were previously found to be much stronger compared to the universal perturbations. Our generative attack model trained on Chest X-ray and Comics images, can fool VGG-16, ResNet50 and Dense-121 models with a success rate of and , respectively, without having any knowledge of data distribution or label space.
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Supplementary: Cross Domain Transferability of Adversarial Perturbations
We further compare our method with GAP  in Sec. 1 to demonstrate superiority of our approach. In Sec. 2, we visually demonstrate the effect of training time and Gaussian kernel size of the generated adversaries. Finally, in Sec. 3, we show adversaries produced by different generators as well as demonstrate attention shift on adversarial examples.
1 Comparison with GAP
|Fool Rate ()||Top-1 ()||Fool Rate ()||Top-1 ()||Fool Rate ()||Top-1 ()|
2 Effect of Training Time and Gaussian Kernel Size
Figures 1 and 2 show the evolution of generative adversaries as the number of epochs increases. At initial epochs, adversaries are more smoother and more transferable against adversarially trained models. On the other hand, as training progress, generator converges to a solution with locally strong patterns that are more transferable to naturally trained models.
Figure 5 demonstrates the attention shift on generative adversarial examples produced by our method. Figures 6, 7, 8 and 9 show examples of different clean images and their corresponding adversaries produced by different generators.