Deep neural networks (DNNs) are vulnerable to adversarial examples, in which small and often imperceptible perturbations change the class label of an image[33, 9, 23, 24]. Because of the security concerns this raises, there is increasing interest in studying these attacks themselves, and also designing mechanisms to defend against them.
Adversarial examples were originally formed by selecting a single base image, and then sneaking that base image into a different class using a small perturbation [9, 7, 16]. This is done most effectively using (potentially expensive) iterative optimization procedures [8, 16, 2].
Different from per-instance perturbation attacks, Moosavi-Dezfooli et al. [19, 20] show there exists “universal” perturbations that can be added to any image to change its class label (fig. 1). Universal perturbations empower attackers who cannot generate per-instance adversarial examples on the go, or who want to change the identity of an object to be selected later in the field. What’s worse, universal perturbations have good cross-model transferability, which facilitates black-box attacks.
Among various methods for hardening networks to per-instance attacks, adversarial training  is known to dramatically increase robustness . In this process, adversarial examples are produced for each mini-batch during training, and injected into the training data. While effective at increasing robustness, the high cost of this process precludes its use on large and complex datasets. This cost comes from the adversarial example generation process, which frequently requires 5-30 iterations to produce an example. Unfortunately, adversarial training using cheap, non-iterative methods generally does not result in robustness against stronger iterative adversaries .
This paper studies effective methods for producing and deflecting universal adversarial attacks. First, we pose the creation of universal perturbations as an optimization problem that can be effectively solved by stochastic gradient methods. This method dramatically reduces the time needed to produce attacks as compared to . The efficiency of this formulation empowers us to consider universal adversarial training. We formulate the adversarial training problem as a min-max optimization where the minimization is over the network parameters and the maximization is over the universal perturbation. This problem can be solved quickly using alternating stochastic gradient methods with no inner loops, making it far more efficient than per-instance adversarial training with a strong adversary.
Interestingly, universal adversarial training has a number of unexpected and useful side effects. While our models are trained to resist universal perturbations, they are also quite resistant to black-box per-instance attacks – achieving resistance comparable to 7-step PGD adversarial training at a fraction of the cost. Furthermore, per-instance adversarial examples built for attacking our universally hardened model transfer to other (black-box) natural and robust models very well.
2 Related work
We briefly review per-instance perturbation attack techniques that are closely related to our paper and will be used in our experiments. The Fast Gradient Sign Method (FGSM)  is one of the most popular one-step gradient-based approaches for
-bounded attacks. FGSM applies one step of gradient ascent in the direction of the sign of the gradient of the loss function with respect to the input image. When a model is adversarially trained, the gradient of the loss function may be very small near unmodified images. In this case, the R-FGSM method remains effective by first using a random perturbation to step off the image manifold, and then applying FGSM. Projected Gradient Descent (PGD) [15, 16] iteratively applies FGSM multiple times, and is one of the strongest per-instance attacks [16, 2]. The PGD version we use in this paper is adopted from  and applies an initial random perturbation before multiple steps of gradient ascent. Finally, DeepFool  is an iterative method based on a linear approximation of the training loss objective. This method formed the backbone of the original method for producing universal adversarial examples.
Adversarial training, in which adversarial attacks are injected into the dataset during training, is an effective method to learn a robust model resistant to attacks [16, 2, 11, 29, 31]. Robust models adversarially trained with FGSM can resist FGSM attacks , but can be vulnerable to PGD attacks . Madry et al.  suggest strong attacks are important, and they use the iterative PGD method in the inner loop for generating adversarial examples when optimizing the min-max problem. PGD adversarial training is effective but time-consuming. The cost of the inner PGD loop is high, although this can sometimes be replaced with neural models for attack generation [3, 26, 36]. These robust models are adversarially trained to fend off per-instance perturbations and have not been designed for, or tested against, universal perturbations.
Unlike per-instance perturbations, universal perturbations can be directly added to any test image to fool the classifier. In , universal perturbations for image classification are generated by iteratively optimizing the per-instance adversarial loss for training samples using DeepFool . In addition to classification tasks, universal perturbations are also shown to exist for semantic segmentation . Robust universal adversarial examples are generated as a universal targeted adversarial patch in . They are targeted since they cause misclassification of the images to a given target class. Moosavi-Dezfooli et al.  prove the existence of small universal perturbations under certain curvature conditions of decision boundaries. Data-independent universal perturbations are also shown to exist and can be generated by maximizing spurious activations at each layer. These universal perturbations are slightly weaker than the data dependent approaches .
There has been very little work on defending against universal attacks. To the best of our knowledge, the only dedicated study is by Akhtar et al., who propose a perturbation rectifying network that pre-processes input images to remove the universal perturbation . The rectifying network is trained on universal perturbations that are built for the downstream classifier. While other methods of data sanitization exist [28, 17] , it has been shown (at least for per-instance adversarial examples) that this type of defense is easily subverted by an attacker who is aware that a defense network is being used .
A recent preprint  models the problem of defending against universal perturbations as a two-player min-max game. However, unlike us, and similar to per-instance adversarial training, after each gradient descent iteration for updating the DNN parameters, they generate a universal adversarial example in an iterative fashion. Since the generation of universal adversarial perturbations is very time-consuming , this makes their approach very slow in practice and prevents them from training the neural network parameters for many iterations.
3 Optimization for universal perturbation
Given a set of training samples and a network with frozen parameter that maps images onto labels, Moosavi-Dezfooli et al.  propose to find universal perturbations that satisfy,
represents the “fooling ratio,” which is the fraction of images whose perturbed class label differs from the original label The parameter controls the diameter of the bounded perturbation, and
is a small tolerance hyperparameter. Problem (1) is solved by the iterative method in algorithm 1 . This iterative solver relies on an inner loop to apply DeepFool  to each training instance, which makes the solver slow. Moreover, the outer loop of algorithm 1 is not guaranteed to converge.
Different from , we consider the following optimization problem for building universal perturbations,
where represents the loss used for training DNNs. This simple formulation (2) searches for a universal perturbation that maximizes the training loss, and thus forces images into the wrong class.
The naive formulation (2) suffers from a potentially significant drawback; the cross-entropy loss is unbounded from above, and can be arbitrarily large when evaluated on a single image. In the worst-case, a perturbation that causes misclassification of just a single image can maximize (2) by forcing the average loss to infinity. To force the optimizer to find a perturbation that fools many instances, we propose a “clipped” version of the cross entropy loss,
We cap the loss function at to prevent any single image from dominating the objective in (2), and giving us a better surrogate of misclassification accuracy. In section 5.2, we investigate the effect of clipping with different .
We directly solve eq. 2 by a stochastic gradient method described in algorithm 2. Each iteration begins by using gradient ascent to update the universal perturbation to maximize the loss. Then, is projected onto the
-norm ball to prevent it from growing too large. We experiment with various optimizers for this ascent step, including Stochastic Gradient Descent (SGD), Momentum SGD (MSGD), Projected Gradient Descent (PGD), and ADAM.
We test this method by attacking a naturally trained WideResnet CIFAR-10 model from . Stochastic gradient methods that use “normalized” gradients (ADAM and PGD) are less sensitive to learning rate and converge faster, as shown in fig. 2. We visualize the generated universal perturbation from different optimizers in fig. 3. Compared to the noisy perturbation generated by SGD, normalized gradient methods produced stronger attacks with more well-defined geometric structures and checkerboard patterns. The final evaluation accuracies (on test-examples) after adding universal perturbations with were 42.56% for the SGD perturbation, 13.08% for MSGD, 13.30% for ADAM, and 13.79% for PGD. The clean test accuracy of WideResnet is 95.2%.
The proposed method of universal attack using a clipped loss function has several advantages. It is based on a standard stochastic gradient method that comes with convergence guarantees when a decreasing learning rate is used . Also, each iteration is based on a minibatch of samples instead of one instance, which accelerates computation on a GPU. Finally, each iteration requires a simple gradient update instead of the complex DeepFool inner loop; we empirically verify fast convergence and good performance of the proposed method (see section 5).
4 Universal adversarial training
We now consider training robust classifiers that are resistant to universal perturbations. In particular, we consider universal adversarial training, and formulate this problem as a min-max optimization problem,
where represents the neural network weights, represents training samples, represents universal perturbation noise, and is the loss function. We solve eq. 4 by alternating stochastic gradient methods in algorithm 3. Each iteration alternatively updates the neural network weights using gradient descent, and then updates the universal perturbation using ascent.
The standard formulation (5) searches for per-instance perturbed images , while our formulation in (4) maximizes using a universal perturbation . Madry et al.  solve (5) by a stochastic method. In each iteration, an adversarial example is generated for an input instance by the PGD iterative method, and the neural network parameter is updated once . Our formulation (algorithm 3) only maintains one single perturbation that is used and refined in all iterations. For this reason, we need only update and once per step (i.e., there is no expensive inner loop), and these updates accumulate for both and through training.
In fig. 4, we present training curves for the universal adversarial training process on the WideResnet model from  using the CIFAR-10 dataset. We consider different rules for updating during universal adversarial training,
and ADAM . We found that the FGSM update rule was most effective when combined with the SGD optimizer for updating neural network weights .
One way to assess the update rule is to plot the model accuracy before and after the ascent step (i.e., the perturbation update). It is well-known that adversarial training is more effective when stronger attacks are used. In the extreme case of a do-nothing adversary, the adversarial training method degenerates to natural training. In fig. 5, we see a gap between the accuracy curves plotted before and after gradient ascent. We find that the FGSM update rule leads to a larger gap, indicating a stronger adversary. Correspondingly, we find that the FGSM update rule yields networks that are more robust to attacks as compared to SGD update (see fig. 5). Interestingly, although universal training using an FGSM update and ADAM update are both robust to universal perturbation attacks, a robust model with FGSM update is more robust to per-instance attacks than that which uses the ADAM update rule. The accuracy of a universally hardened network against a white-box per-instance PGD attack is 17.21% for FGSM universal training, and only 2.57% for ADAM universal training.
4.1 Attacking hardened models
We evaluate the robustness of different models by applying algorithm 2 to try to find universal perturbations. We attack universally adversarial trained models (produced by eq. 4) using the FGSM universal update rule (uFGSM), or the SGD universal update rule (uSGD). We also consider a robust model from per-instance adversarial training (eq. 5) with adversarial steps of the FGSM and PGD type .
The training curves for the robust WideResnet models on CIFAR-10 are plotted in fig. 5. Robust models adversarially trained with weaker attackers such as uSGD and FGSM are relatively vulnerable to universal perturbations, while robust models from PGD  and uFGSM can resist universal perturbations. We apply PGD (using the sign of the gradient) and ADAM in algorithm 3 to generate universal perturbations for these robust models, and show such perturbations in fig. 6. Comparing fig. 6 (a,b,c,d) with fig. 6 (e,f,g,h), we see that universal perturbations generated by PGD and ADAM are different but have similar patterns. Universal perturbations generated for weaker robust models have more textures, as shown in fig. 6 (a,d,e,h) .
5 Universal perturbations for ImageNet
To validate the performance of our proposed optimization on different architectures and more complex datasets, we apply algorithm 2 to various popular architectures designed for classification on the ImageNet dataset . We compare our method of universal perturbation generation with the current state-of-the-art method, Iterative DeepFool (iDeepFool for short) . We use the authors’ code to run the iDeepFool attack on these classification networks. For fair comparison, we execute both our method and iDeepFool on the exact same 5000 training data points and terminate both methods after 10 epochs. We use for constraint following , use a step-size of 1.0 for our method, and use suggested parameters for iDeepFool. We independently execute iDeepFool since we are interested in the accuracy of the classifier on attacked images – a metric not reported in their paper 111They report “fooling ratio” which is the ratio of examples who’s label prediction changes after applying the universal perturbation. This has become an uncommon metric since the fooling ratio can increase if the universal perturbation causes an example that was originally miss-classified to become correctly classified..
5.1 Benefits of the proposed method
We compare the performance of our stochastic gradient method for eq. 2 and the iDeepFool method for eq. 1 in . We generate universal perturbation for Inception  and VGG  networks trained on ImageNet , and report the top-1 accuracy in table 1. Universal perturbation generated by both iDeepFool and our method can fool networks and degrade the classification accuracy. Universal perturbations generated for the training samples generalize well and cause the accuracy of the validation samples to drop. However, when given a fixed computation budget such as number of passes on the training data (i.e., epochs), our method outperforms iDeepFool by a large margin. Our stochastic gradient method generates the universal perturbations at a much faster pace than iDeepFool. About 20 faster on InceptionV1 and 6 on VGG16 (13 on average).
After verifying the effectiveness and efficiency of our proposed stochastic gradient method222Unless otherwise specified, we use the sign-of-gradient PGD for our stochastic gradient optimizer in algorithm 2., we use our algorithm 2 to generate universal perturbations for more advanced architectures such as ResNet-V1 152  and Inception-V3  (and for other experiments in the remaining sections). Our attacks degrade the validation accuracy of ResNet-V1 152 and Inception-V3 from 76.8% and 78% to 16.4% and 20.1%, respectively. The final universal perturbations used for the results presented in this section are illustrated in fig. 7.
|iDeepFool time (s)||9856||6076|
|our time (s)||482||953|
5.2 The effect of clipping
In this section we analyze the effect of the “clipping” loss parameter in eq. 2. For this purpose, similar to our other ablation experiments, we generate universal perturbation by solving eq. 2 using PGD for Inception-V3 on ImageNet.
Since the results and performance could slightly vary with different random initializations, we run each experiment with 5 random subsets of training data. The accuracy reported is the classification accuracy on the entire validation set of ImageNet after adding the universal perturbation. The results are summarized in fig. 8. The results showcase the value of our proposed loss function for finding universal adversarial perturbations.
5.3 How much training data does the attack need?
As in , we analyze how the number of training points () affects the strength of universal perturbations in fig. 9. In particular, we build using varying amounts of training data. For each experiment, we report the accuracy on the entire validation set after we add the perturbation . We consider four cases for : 500, 1000, 2000, and 4000 333The number of epochs ( in algorithm 2) was 100 epochs for 500 data samples, 40 for 1000 and 2000 samples, and 10 for 4000 samples..
6 Experiment: universal adversarial training
In this section, we analyze our robust models that are universal adversarially trained by solving the min-max problem (section 4) using algorithm 3. We use for the constraint for CIFAR-10 following , and for ImageNet following .
6.1 Defense against white-box attack
We compare our universal adversarilly trained model’s robustness with other hardened models against white-box attacks, where the (robust) models are fully revealed to the attackers. We attack the hardened and natural models using universal perturbations (section 3) and per-instance perturbations (FGSM , R-FGSM , and a 20-step -bounded PGD attack with step-size 2 ). We also report the performance of per-instance adversarially trained models which are trained with per-instance attacks such as FGSM, R-FGSM and PGD . We use universal adversarial training from algorithm 3 to build a robust WideResnet  on CIFAR-10 , and a robust AlexNet  using 5000 training samples of ImageNet . The PGD per-instance adversarial training is done by training on adversarial examples that are built using steps of PGD following , which makes it slower than the non-iterative adversarial training methods such as our universal adversarial training, FGSM, and R-FGSM adversarial training.
We summarize the CIFAR-10 results in table 2. The natural model, as also seen in section 3, is vulnerable to universal and per-instance perturbations. Our robust model achieves best classification (i.e. highest robustness) accuracy against universal perturbation attacks. The 20-step PGD attack fools the natural, FGSM robust, and R-FGSM robust models almost every time. Interestingly, our model is relatively resistant to the PGD attack, though not as robust as the PGD-based robust model. This result is particularly interesting when we consider that our method is hardened using universal perturbations. While the computational cost of our method is similar to that of non-iterative per-instance adversarial training methods (FGSM, and RFGSM), our model is considerably more robust against the PGD attack that is known to be the strongest per-instance attack.
Since our universal adversarial training algorithm is cheap, it scales to large datasets such as ImageNet. As seen in fig. 10 (a), the AlexNet trained using our universal adversarial training algorithm (algorithm 3) is robust against universal attacks generated using both algorithm 1 and algorithm 2. The naturally trained AlexNet is susceptible to universal attacks. The final attacks generated for the robust and natural models are presented in fig. 10 (b,c). The universal perturbation generated for the robust AlexNet model has little structure compared to the universal perturbation built for the naturally trained AlexNet. This is similar to the trend we observed in fig. 3 and fig. 6 for the WideResnet models trained on CIFAR-10.
6.2 Transferability and black-box robustness
We study the transferability of our robust model in the black-box threat setting, in which we generate adversarial examples based on a source model and use them to attack a target model. We study the transferability of the adversarial 20-step PGD per-instance examples between various models with the WideResnet architecture that are trained on CIFAR-10: natural trained model, FGSM trained robust model, R-FGSM trained robust model, PGD trained robust model , and our robust model.
The results are summarized in table 3. By examining rows of table 3, both the PGD-based robust model and our robust model are fairly hardened to black-box attacks made for various source models. By examining columns of table 3, we can compare the transferability of the attacks made for various source models. In this metric, the attacks built for our robust model are the strongest in terms of transferability and can deteriorate the performance of both natural and other robust models. An adversary can enhance her black-box attack by first making her source model universally robust!
6.3 Visualizing attacks on robust models
Tsipras et al.  use several visualization techniques to analyze PGD-based robust models and show some unexpected benefits of adversarial robustness. Similarly, we generate large per-instance adversarial examples using a PGD attack without random initialization. Large perturbations make the perturbations visible. Adversarial examples built in this way for both a natural model and our robust model are illustrated in fig. 11. Many of the adversarial examples of the natural model look similar to the original image and have a lot of “random” noise on the background, while adversarial examples for our robust model produce salient characteristics of another class and align well with human perception. The elimination of structured universal perturbations during universal adversarial training seems to have this interesting side-effect that was only recently shown for PGD adversarial training.
We proposed using stochastic gradient methods and a “clipped” loss function as an effective universal attack that generates universal perturbations much faster than previous methods. To defend against these universal adversaries, we proposed to train robust models by optimizing a min-max problem using alternating stochastic gradient methods. We systematically study the robustness of our robust model under the white-box and black-box threat models. Our experiments suggest that our robust model can resist white-box universal perturbations, and to some extent per-instance perturbations. In the black-box threat model, our robust model is as robust as the much more expensive PGD adversarial training. Moreover, the per-instance adversarial examples generated for our robust model transfer better to attack other models. Due to the relatively cheap computational overhead of our proposed universal adversarial training algorithm, we can easily train robust models for large-scale datasets such as ImageNet.
Acknowledgements: Goldstein and his students were supported by DARPA’s Lifelong Learning Machines and YFA programs, the Office of Naval Research, the AFOSR MURI program, and the Sloan Foundation. Davis and his students were supported by the Office of the Director of National Intelligence (ODNI), and IARPA (2014-14071600012). The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon.”
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