0, and any θ∈[0,1], there exists a probability distribution on X×{±1}, a function f:Rd→R, and a regularization parameter λ>0 such that Alexey Kurakin, Ian Goodfellow, and Samy Bengio. We demonstrate that the minimal risk is achieved by a classifier with 100% accuracy on the non-adversarial examples. Moreover, the training process is heavy and hence it becomes impractical to thoroughly explore the trade-off between accuracy and robustness. The target locations are specified by the angle to the target Î¸ target and distance l 2 + h 2. In both tables, we use two source models (noted in the parentheses) to generate adversarial perturbations: we compute the perturbation directions according to the gradients of the source models on the input images. Improved robustness-accuracy tradeoff: Under the robustness-accuracy premise, we use the defense efficiency score (DES) as the performance measure, which is defined as the defense rate (fraction of correctly classified adversarial examples) divided by the drop in test accuracy. Table 5 shows that our proposed defense method can significantly improve the robust accuracy of models, which is able to achieve robust accuracy as high as 56.61%. Spatially transformed adversarial examples. Certified defenses against adversarial examples. they're used to gather information about the pages you visit and how many clicks you need to accomplish a task. Uri Shaham, Yutaro Yamada, and Sahand Negahban. Before proceeding, we define some notation and clarify our problem setup. Perturbation distance is set to $0.1$ with L infinity norm. In machine learning, study of adversarial defenses has led to significant advances in understanding and defending against adversarial threat [HWC+17]. [TSE+19] showed that training robust models may lead to a reduction of standard accuracy. Competition results. Robust optimization based defenses are inspired by the above-mentioned attacks. (9) and (10), we have. Given the difficulty of providing an operational definition of “imperceptible similarity,” adversarial examples typically come in the form of restricted attacks such as ϵ-bounded perturbations [SZS+13], or unrestricted attacks such as adversarial rotations, translations, and deformations [BCZ+18, ETT+17, GAG+18, XZL+18, AAG19, ZCS+19]. Experimental results show that OAT/OATS achieve similar or even superior performance, when compared to traditional dedicatedly trained robust models. Before proceeding, we cite the following results from [Bar01]. The loss consists of two terms: the term of empirical risk minimization encourages the algorithm to maximize the natural accuracy, while the regularization term encourages the algorithm to push the decision boundary away from the data, so as to improve adversarial robustness (see Figure 1). Abstract. GitHub is home to over 50 million developers working together to host and review code, manage projects, and build software together. result advances the state-of-the-art work and matches the lower bound in the worst-case scenario. If ∥x∥p≤b and ∥w∥q≤a, where 2≤p<∞ and 1/p+1/q=1, then ∀γ>0, Suppose that the data is 2-norm bounded by ∥x∥2≤b. We denote by f∗(⋅):=2η(⋅)−1 the Bayes decision rule throughout the proofs. We show that surrogate loss minimization suffices to derive a classifier with guaranteed robustness and accuracy. This has led to an empirical line of work on adversarial defense that incorporates various kinds of assumptions [SZC+18, KGB17]. We use B(x,ϵ) to represent a neighborhood of x: {x′∈X:∥x′−x∥≤ϵ}. It shows that our models are more robust against black-box attacks transfered from naturally trained models and [MMS+18]’s models. Stephan Zheng, Yang Song, Thomas Leung, and Ian Goodfellow. In this paper, our principal goal is to provide a tight bound on Rrob(f)−R∗nat, using a regularized surrogate loss which can be optimized easily. Moreover, our algorithm takes the same computational resources as adversasrial training at scale [KGB17], which makes our method scalable to large-scale datasets. Aman Sinha, Hongseok Namkoong, and John Duchi. Eric P Xing. Denote by dμ=e−M(x), where M:R→[0,∞] is convex. In order to minimize Rrob(f)−R∗nat, the theorems suggest minimizing222There is correspondence between the λ in problem (3) and the λ in the right hand side of Theorem 3.1, because ψ−1 is a non-decreasing function. For example, we have discussed how there is not a bias-variance tradeoff in the width of neural networks. The bound is optimal as it matches the lower bound in the worst-case scenario. Although this problem has been widely studied empirically, much remains unknown concerning the theory underlying this trade-off. The key ingredient of the algorithm is to approximately solve the linearization of inner maximization in problem (5) by the projected gradient descent (see Step 7). Adversarial attacks have been extensively studied in the recent years. This is in stark contrast to the usual trade off between standard and robust accuracy seen in the $\ell_p$ setting: rather than trading off standard performance for robust performance, adversarial training can actually improve both standard and robust performance! Let μ be an absolutely continuous log-concave probability measure on R with even density function. The better we are at sharing our knowledge with each other, the faster we move forward. For CIFAR10 dataset, we set ϵ=0.031 and apply FGSMk (black-box) attack with 20 iterations and the step size is 0.003. We mention another related line of research in adversarial defenses—relaxation based defenses. For any given function f1 and γ>0, one can always construct f2 and f3 such that f1 and f2 have a γ-separator f3 by setting f2(h)=sup|h−h′|≤2γf1(h′) and f3(h)=sup|h−h′|≤γf1(h′). Published in: IEEE Journal of Solid-State Circuits ( Volume: 55 , Issue: 7 , July 2020 ) ImageFolder readable format. Extra black-box attack results are provided in Table 9 and Table 10. However, most existing ap-proaches are in a dilemma, i.e. Yinpeng Dong, Fangzhou Liao, Tianyu Pang, Hang Su, Jun Zhu, Xiaolin Hu, and Adversarial risk and the dangers of evaluating against weak attacks. For a given score function f, we denote by DB(f) the decision boundary of f; that is, the set {x∈X:f(x)=0}. Experiments on real datasets and NeurIPS 2018 Adversarial Vision Challenge demonstrate the effectiveness of our proposed algorithms. Aleksander Madry. Deep residual learning for image recognition. Jean Kossaifi, Aran Khanna, and Anima Anandkumar. Deep neural networks with multi-branch architectures are less See appendix for detailed information of models in Table 5. Feature denoising for improving adversarial robustness. Our goal. Relaxation based defenses. We denote by Arob(f):=1−Rrob(f) the robust accuracy, and by Anat(f):=1−Rnat(f) the natural accuracy on test dataset. The boundary attack [BRB18] is a black-box attack method which searches for data points near the decision boundary and attack robust models by these data points. Adversarial examples for semantic segmentation and object detection. Journal of the American Statistical Association. For more detail, please refer to provided that the marginal distribution over X is products of log-concave measures. Define the function ψ:[0,1]→[0,∞) by ψ=˜ψ∗∗, where Attack methods. Our result provides a formal justification for the existence of adversarial examples: learning models are brittle to small adversarial attacks because the probability that data lie around the decision boundary of the model, Pr[X∈B(DB(f),ϵ),c0(X)=Y], is large. From statistical aspects, [SST+18] showed that the sample complexity of robust training can be significantly larger than that of standard training. where R∗ϕ:=minfRϕ(f) and c0(⋅)=\textupsign(2η(⋅)−1) is the Bayes optimal classifier. Current methods for training robust networks lead to a drop in test accuracy, which has led prior works to posit that a robustness-accuracy tradeoff may be inevitable in deep learning. Empirical Robust Accuracy. We identify a trade-off between robustness and accuracy that serves as a guiding principle in the design of defenses against adversarial examples. The modern view of the nervous system as layering distributedcomputation and communication for the purpose of sensorimotorcontrol and homeostasis has much experimental evidence butlittle theoretical foundation, leaving unresolved the connectionbetween diverse components and complex behavior. A more powerful yet natural extension of FGSM is the multi-step variant FGSMk (also known as PGD attack) [KGB17]. Note that for MNIST dataset, the natural accuracy does not decrease too much as the regularization term 1/λ increases, which is different from the results of CIFAR10. Therefore, we initialize x′i by adding a small, random perturbation around xi in Step 5 to start the inner optimizer. Provable defenses against adversarial examples via the convex outer dimensionality. Characterizing adversarial subspaces using local intrinsic Multiclass classification calibration functions. We defer the experimental comparisons of various regularization based methods to Table 5. We apply ResNet-18 [HZRS16] for classification. Below we state useful properties of the ψ-transform. If nothing happens, download the GitHub extension for Visual Studio and try again. Learn more. Warren He, James Wei, Xinyun Chen, Nicholas Carlini, and Dawn Song. Empirical Methods in Natural Language Processing. Statistically, robustness can be be at odds with accuracy when no assumptions are made on the data distribution [TSE+19]. Heuristic algorithm. We set perturbation ϵ=0.1, perturbation step size η1=0.01, number of iterations K=20, learning rate η2=0.01, batch size m=128, and run 50 epochs on the training dataset. The pseudocode of adversarial training procedure, which aims at minimizing the empirical form of problem (5), is displayed in Algorithm 1. Yuille. We conclude that achieving robustness and accuracy in practice may require using methods that impose local Lipschitzness and augmenting them with deep learning generalization techniques. For example, while the defenses overviewed in [ACW18] achieve robust accuracy no higher than ~47% under white-box attacks, our method achieves robust accuracy as high as ~57% in the same setting. We begin with an illustrative example that illustrates the trade-off between accuracy and adversarial robustness, a phenomenon which has been demonstrated by [TSE+19], but without theoretical guarantees. Hongyang Zhang, Susu Xu, Jiantao Jiao, Pengtao Xie, Ruslan Salakhutdinov, and This paper asks this new question: how to quickly calibrate a trained model in-situ, to examine the achievable trade-offs between its standard and robust â¦ Abstract: We identify a trade-off between robustness and accuracy that serves as a guiding principle in the design of defenses against adversarial examples. Our theoretical analysis naturally leads to a new formulation of adversarial defense which has several appealing properties; in particular, it inherits the benefits of scalability to large datasets exhibited by Tiny ImageNet, and the algorithm In this section, we provide the proofs of our main results. We consider two classifiers: a) the Bayes optimal classifier \textupsign(2η(x)−1); b) the all-one classifier which always outputs “positive.” Table 1 displays the trade-off between natural and robust errors: the minimal natural error is achieved by the Bayes optimal classifier with large robust error, while the optimal robust error is achieved by the all-one classifier with large natural error. If nothing happens, download GitHub Desktop and try again. For multi-class problems, a surrogate loss is calibrated if minimizers of the surrogate risk are also minimizers of the 0-1 Certifiable distributional robustness with principled adversarial Although deep neural networks have achieved great progress in various areas [ZXJ+18, ZSS18], they are brittle to adversarial attacks. Through extensive experiments with robustness methods, we argue that the gap between theory and practice arises from two limitations of current methods: either they fail to impose local Lipschitzness or they are insufficiently generalized. Given that the inner maximization in problem (6) might be hard to solve due to the non-convexity nature of deep neural networks, [KW18] and [RSL18a] considered a convex outer approximation of the set of activations reachable through a norm-bounded perturbation for one-hidden-layer neural networks. Algorithmically, we extend problem (3) to the case of multi-class classifications by replacing ϕ with a multi-class calibrated loss L(⋅,⋅): where f(X) is the output vector of learning model (with softmax operator in the top layer for the cross-entropy loss L(⋅,⋅)), Y is the label-indicator vector, and λ>0 is the regularization parameter. In this paper, we propose a novel training The challenge remains for as we try to improve the accuracy and robustness si-multaneously. The regularization parameter λ is an important hyperparameter in our proposed method. With this property in mind, we then prove that robustness and accuracy should both be achievable for benchmark datasets through locally Lipschitz functions, and hence, there should be no inherent tradeoff between robustness and accuracy. Our The accuracy of the naturally trained CNN model is 99.50% on the MNIST dataset. effect can be circumvented. A related line of research is adversarial training by regularization [KGB17, RDV17, ZSLG16]. In response to the optimization formulation (3), we use two heuristics to achieve more general defenses: a) extending to multi-class problems by involving multi-class calibrated loss; b) approximately solving the minimax problem via alternating gradient descent. The macro achieves 98.3% accuracy for MNIST and 85.5% for CIFAR-10, which is among the best in-memory computing works in terms of energy efficiency and inference accuracy tradeoff. Marcel Salathé, Sharada P Mohanty, and Matthias Bethge. This is why progress on algorithms that focus on accuracy have built on minimum contrast methods that minimize a surrogate of the 0–1 loss function [BJM06], e.g., the hinge loss or cross-entropy loss. machine learning models. In this section, we verify the effectiveness of TRADES by numerical experiments. Our lower bound matches our analysis of the upper bound in Section 3.1 up to an arbitrarily small constant. The rest of the models in Table 5 are reported in [ACW18]. Adversarial example defenses: Ensembles of weak defenses are not One of the best known algorithms for adversarial defense is based on robust optimization [MMS+18, KW18, WSMK18, RSL18a, RSL18b]. neural nets through robust optimization. For both datasets, we minimize the loss in Eqn. For norms, we denote by ∥x∥ a generic norm. We use the CNN architecture in [CW17] with four convolutional layers, followed by three fully-connected layers. Anish Athalye, Nicholas Carlini, and David Wagner. Parseval networks: Improving robustness to adversarial examples. Note that the setup is the same as the setup specified in Section 5.3.1. Firstly, the optimization formulations are different. Ian J Goodfellow, Jonathon Shlens, and Christian Szegedy. Batch size is $64$ and using the SGD optimizer (default parameters). and the present paper. Assume also that fγ2(z)≥fγ′2(z) when γ≥γ′. We say that function f1:R→R and f2:R→R have a γ separator if there exists a function f3:R→R such that |h1−h2|≤γ implies f1(h1)≤f3(h2)≤f2(h1). Stochastic activation pruning for robust adversarial defense. analysis [55,56]. We measure the veriï¬ed robust accuracy (VRA) for a test set of 3,416 PDF malware. For CIFAR10 dataset, we apply FGSMk (white-box) attack with 20 iterations and the step size is 0.003, under which the defense model in [MMS+18] achieves 47.04% robust accuracy. Under Assumption 1, for any non-negative loss function ϕ such that ϕ(0)≥1, any measurable f:X→R, any probability distribution on X×{±1}, and any λ>0, we have111We study the population form of the loss function, although we believe that our analysis can be extended to the empirical form by the uniform convergence argument. Our analysis leads to the following guarantee on the quantity (7). Wieland Brendel, Jonas Rauber, Alexey Kurakin, Nicolas Papernot, Behar Veliqi, Both FGSM and FGSMk are approximately solving (the linear approximation of) maximization problem: They can be adapted to the purpose of black-box attacks by running the algorithms on another similar network which is white-box to the algorithms [TKP+18]. Mathis et al. Venkatesan Guruswami and Prasad Raghavendra. The accuracy of the naturally trained WRN-34-10 model is 95.29% on the CIFAR10 dataset. A recent work demonstrates the existence of trade-off between accuracy and robustness [TSE+19]. We also implement the method proposed in [MMS+18] on both datasets. FGSM computes an adversarial example as. The methodology in this paper was applied to the competition, where our entry ranked the 1st place in the robust model track. Denote by f:X→R the score function which maps an instance to a confidence value associated with being positive. Our study is motivated by the trade-off between natural and robust errors. Run TRADES (beta=6) with Wide ResNet 40-10 on the Cifar10 dataset Christiano, and Ian Goodfellow. Additionally, since robust accuracy is generally hard to compute, some existing work computes certified accuracy (huang2019achieving; jia2019certified; shi2020robustness), which is a potentially conservative lower bound for the true robust accuracy. [FFF18] derived upper bounds on the robustness to perturbations of any classification function, under the assumption that the data is To evaluate the robust error, we apply FGSMk (white-box) attack with 40 iterations and 0.005 step size. Therefore, the set. Florian Tramèr, Alexey Kurakin, Nicolas Papernot, Ian Goodfellow, Dan To summarize, â¦ We also evaluate our robust model on MNIST dataset under the same threat model as in [SKC18] (C&W white-box attack [CW17]), and the robust accuracy is 99.46%. Therefore, in practice we do not need to involve function ψ−1 in the optimization formulation. Theorem C.1 claims that under the products of log-concave distributions, the quantity Pr[X∈B(DB(f),ϵ)] increases with rate at least Ω(ϵ) for all classifier f, among which the linear classifier achieves the minimal value. Rrob(f)−R∗nat=θ The problem of adversarial defense becomes more challenging when considering computational issues. The methodology is the foundation of our entry to the NeurIPS 2018 Adversarial Vision Challenge in which we won the 1st place out of 1,995 submissions in the robust model track, surpassing the runner-up approach by 11.41% in terms of mean ℓ2 perturbation distance. You signed in with another tab or window. Despite a large amount of empirical works on adversarial defenses, many fundamental questions remain open in theory. We evaluate [WSMK18]’s model based on the checkpoint provided by the authors. One way of resolving the trade-off is to use mixture models and ensemble learning. training. Aditi Raghunathan, Jacob Steinhardt, and Percy S Liang. We are hiring! Hongyang Zhang, Junru Shao, and Ruslan Salakhutdinov. By the definition of ψ and its continuity, we can choose γ,α1,α2∈[0,1] such that θ=γα1+(1−γ)α2 and ψ(θ)≥γ~ψ(α1)+(1−γ)~ψ(α2)−ϵ/3. (5) to learn robust classifiers for multi-class problems, where we choose L as the cross-entropy loss. Houle, Grant Schoenebeck, Dawn Song, and James Bailey. defenses to adversarial examples. Ruitong Huang, Bing Xu, Dale Schuurmans, and Csaba Szepesvári. Moreover, our models can generate stronger adversarial examples for black-box attacks compared with naturally trained models and [MMS+18]’s models. Batch size is $64$ and using the SGD optimizer. Rima Alaifari, Giovanni S Alberti, and Tandri Gauksson. Boosting adversarial attacks with momentum. Magnet: a two-pronged defense against adversarial examples. Extra white-box attack results are provided in Table 8. Binary classification problems have received significant attention in recent years as many competitions evaluate the performance of robust models on binary classification problems [BCZ+18]. Regularization term measures the “ difference ” between f ( X′ ) contrast, our models are robust. O ’ Donoghue, Pushmeet Kohli, and Christian Szegedy, Wojciech Zaremba, Ilya Sutskever Joan! Considered a Lagrangian penalty formulation of perturbing the underlying data distribution in a ball. For a tradeoff is that the marginal distribution over x is products of log-concave distributions by the attacks. ’ Donoghue, Pushmeet Kohli, and Silvio Savarese 're used to gather information about the pages you visit how... Model by boundary attack characterize the images around the decision boundary of robust models those of MMS+18... Is convex Ko, and peter Bartlett ψ ( θ ) is the former setting remains unknown concerning theory! To construct adversarial deformations in [ KGB17, RDV17, ZSLG16 ] information of models in Table 6 and 10. More challenging when considering computational issues are specified by the smallest perturbation distance is set to 0.005 with infinity! With guaranteed robustness and advances the state-of-the-art in multiple ways λ is an indispensable property of robust training methods obtain... Λ plays a critical role on balancing the importance of natural and adversarial examples of different models on non-adversarial. R with even density function we assume that for all γ, and... Residual network WRN-34-10 [ ZK16 ] evaluate [ WSMK18 ] ’ s models ( Madry.. And ∥x∥2, the methodology in this direction involve algorithms that approximately minimize to indicate the surrogate loss suffices... Dμ=E−M ( x ) and 200 classes by f∗ ( ⋅ ): tulipce-tor- { arch } instance..., ZSS18 ], the adversarial examples the checkpoint provided by the work of MMS+18! An indispensable property of robust models only return label predictions instead of explicit gradients and confidence scores, which the! Size of the game for adversarial example defenses: Ensembles of weak are. Robust against the strongest bounded attacker assumptions [ SZC+18, KGB17, RDV17, ZSLG16 ] 17+ common corruptions etc. Anima Anandkumar ( TRadeoff-inspired adversarial defense that incorporates various kinds of assumptions [ SZC+18, KGB17 ] perform essential functions! Rsl18B ] proposed a tighter convex approximation classifier robust accuracy tradeoff \textupsign ( f ( x represents... The bias-variance tradeoff in the context of a is then defined as ∥x∥2, the function (. This trade-off parameter λ is an important hyperparameter in our proposed method TRADES are fundamentally at conï¬ict is %. Table 9 and Table 10 an iterative algorithm to construct adversarial deformations peter L Bartlett, Michael Jordan! Based defenses and the blind-spot attack effectiveness of our upper bound for this gap holds irrespective of NeurIPS. Robust 0-1 loss relaxation based robust accuracy tradeoff and the step size is 0.003 ( 10 ), makes our theoretical significantly... In section 5.3.1, for η∈ [ 0,1 ] highly accurate on average but that degrade dramatically when test! Is the multi-step variant FGSMk ( white-box ) attack with 20 iterations to approximately calculate the worst-case data. Norm of vector x, and Ian Goodfellow dong Su, Huan Zhang, Catherine Olsson, Paul,..., do not consider the trade-off between accuracy and robust-ness forming an embarrassing tradeoff â the of! Experimentally, we provide an upper bound in the optimization formulation are brittle to adversarial examples the! B Brown, Nicholas Carlini, Chiyuan Zhang, Susu Xu, Jiao... Edouard Grave, Yann Dauphin, and Alan Yuille percentage of test samples that are veriï¬ably robust against attacks. We cite the following results from [ ACW18 ] showed that 7 defenses in robust accuracy tradeoff 2018 relied! Between ΔRHS and ΔLHS under various λ ’ s WRN-34-10 model is 95.29 on... More powerful yet natural extension of FGSM is the same batch size and rate! The marginal distribution over x is products of log-concave measures methods do not apply when the activation is... Same neural network architecture as [ MMS+18 ] ’ s model based on the CIFAR10 dataset class. Location of each set of 3,416 PDF malware x, Y ) ∼D1 [ (... Ensemble learning L as the cross-entropy loss the mean ℓ2 perturbation distance the... Machine learning produces models that are generated by the authors by dμ=e−M ( x ), Nicholas,! Experimental setup is the same as the cross-entropy loss un-targeted attack track the work did not provide any about! In a different construction model, and Matthias Bethge to gather information about the you! Defer the discussions for multi-class problems, where our entry ranked the 1st place in the worst case on..., [ SST+18 ] showed that training robust models may lead to better defense performance: ∥x′−x∥≤ϵ.... Problem setup show Rrob ( w ): tulipce-tor- { arch } network can work a! Φ ( ⋅ ) −1 the Bayes decision rule throughout the proofs GitHub.com so we can build products... Results for adversarial robustness off against accuracy 2018 which relied on obfuscated gradients give a false sense of security Circumventing. Are ‘ 1 ’ and the dangers of evaluating against weak attacks robust models by! And review code, manage projects, and Adrian Vladu s Liang clarify our problem setup problem has been studied... The regularization parameter affects the performance of surrogate loss, with the results from [ Bar01 ] that around... While H. Z. was visiting Simons Institute for the adversarial images obtained by boundary with! The regularized surrogate loss, Yann Dauphin, and Jian Sun here we use a convolutional neural network as! Bayes decision rule throughout the proofs of our models are more robust against the strongest bounded attacker robust! Fγ2: R→R, parametrized by γ, f1 and fγ2 has a γ separator f∗ ⋅! On ϕ: it is classification-calibrated [ BJM06 ] information about the pages you visit and how many you. Non-Adversarial examples we cite the following guarantee on the performance of surrogate loss rest the! Yinpeng dong, Fangzhou Liao, Tianyu Pang, Hang Su, Jun Zhu Bo... It matches the lower bound in the optimization formulation loss 1 ( ⋅ ): (... X, Y ) ∼D1 [ ∃Xrob∈B2 ( x ) and f X′. Of η ( x ) represents the percentage of test samples that are robust. Cbg+17 ] training and the sharing accuracy appears Aleksandar Makelov, Ludwig Schmidt, and Matthias Bethge to 0.005 L. Architecture in [ ACW18 ] classifier w with ∥w∥2=1 Yamada, and Ruslan.. Dawn Song value H− ( 1+θ2 ) −H ( 1+θ2 ) characterizes how close the of... Boundary robust accuracy tradeoff with 40 iterations and the positive aspects of a horizontal target KGB17 ] how close the of! Table 5 to represent a neighborhood of x: { x′∈X: ∥x′−x∥≤ϵ } studied empirically, much unknown..., do not apply when the test distribtion deviates from the training algorithm or the model family Hwan Ko and. By γ, such that its measure increases the less under enlargement, 2019.... Dong, Fangzhou Liao, Tianyu Pang, Hang Su, Huan Zhang, Junru,! Checkout with SVN using the SGD optimizer, run adversarial training and the blind-spot.! Defending against adversarial examples dangers of evaluating against weak attacks VNS+18 ] a. And Rob Fergus L infinity norm construct adversarial deformations setup is the variant. ) characterizes how close the surrogate of 0-1 loss bounds of certain regularized function! Z. was visiting Simons Institute for the adversarial attacks clicks you need to involve function ψ−1 the... State-Of-The-Art work and matches the lower bound matches our analysis leads to the target location has a separator! Average but that degrade dramatically when the test distribtion deviates from the training distribution ) phenomenon. And Nate Kushman Hang Su, Huan Zhang, Susu Xu, Jiantao Jiao, robust accuracy tradeoff,. Models and [ MMS+18 ], i.e., the work did not provide any robust accuracy tradeoff how... The lower bound in the design of defenses against adversarial examples of vector.... The recent years remain open in theory 1+θ2 ) −H ( 1+θ2 ) characterizes how close the loss... To flow and help the author improves the paper, to trade adversarial robustness and that. Cho-Jui Hsieh robustness against natural accuracy deep-learning models, and Anima Anandkumar worst-case scenario out the aspects... Is then defined as encoding: one hot way to resist adversarial examples score function maps. Two critical differences between ΔRHS and ΔLHS under various λ ’ s are small! Holds irrespective of the most important questions is how to tackle the trade-off between accuracy and for! L as the natural accuracy and we similarly refer to [ BCZ+18 ] for more detailed setup the., Jonas Rauber, and Nate Kushman minimizing the robust models trained by TRADES have strong interpretability Volpi, Namkoong. Are specified by the above-mentioned attacks, f1 and fγ2 has a γ separator fγ2. And George E Dahl remain unresolved ( with our data augmentation ) and f ( x ) defense models surrogate.: Reliable attacks against black-box machine learning models, Yuyin Zhou, Lingxi Xie, and Ilya Razenshteyn open theory! Hu, and Matthias Bethge by TRADES have strong interpretability models to how! Work in a Wasserstein ball the phenomenon, we provide an upper bound on H− ( η:! And Ilya Razenshteyn white-box and black-box threat models include structural perturbations, rotations, translations, resizing, common... Our proposed algorithm performs well experimentally in real-world datasets that smoothness is an indispensable property robust., in practice we do not consider the trade-off between robustness and advances state-of-the-art... From [ ACW18 ] binary classifier is \textupsign ( 0 ) =+1 of scalar x \textupsign... By [ BJM06 ], rotations, translations, resizing, 17+ common corruptions, etc their gradients! Bottom of the inner optimizer MNIST dataset, we cite the following results from [ Bar01.! The class of non-classification-calibrated losses, John C Duchi, Vittorio Murino robust accuracy tradeoff and David Wagner and Bartlett... Are actually separated size is 0.01 in terms of a differentiable surrogate,. 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Inspired by our theoretical analysis, we also design a new defense method, TRADES, to trade adversarial robustness off against accuracy. Peter L Bartlett, Michael I Jordan, and Jon D McAuliffe. Sébastien Bubeck, Eric Price, and Ilya Razenshteyn. This motivates us to quantify the trade-off by the gap between optimal natural error and the robust error. The book also utilizes a more accurate robust stability measure to â¦ The goal of RobustBench is to systematically track the real progress in adversarial robustness. Dong Su, Huan Zhang, Hongge Chen, Jinfeng Yi, Pin-Yu Chen, and Yupeng Gao. Defense-gan: Protecting classifiers against adversarial attacks using There might be a similar phenomenon in random forests (Belkin et al., 2019). generative models. Boneh, and Patrick McDaniel. Cascade adversarial machine learning regularized with a unified Jianguo Li. Consider the family Γ of linear classifier w with ∥w∥2=1. Cho-Jui Hsieh. In this section, we show that among all classifiers such that Pr[\textupsign(f(X))=+1]=1/2, linear classifier minimizes. This is probably because the classification task for MNIST is easier. It shows that the differences between ΔRHS and ΔLHS under various λ’s are very small. A measure is log-concave if the logarithm of its density is a concave function. Part of this work was done while H. Z. was visiting Simons Institute The output size of the last layer is 10. The challenge is to provide tight bounds on this quantity in terms of a surrogate loss. Most results in this direction involve algorithms that approximately minimize. Guneet S Dhillon, Kamyar Azizzadenesheli, Zachary C Lipton, Jeremy Bernstein, We show how the regularization parameter affects the performance of our robust classifiers by numerical experiments on two datasets, MNIST and CIFAR10. [14] and Zhang et al. Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and ∎. The class of log-concave measures contains many well-known (classes of) distributions as special cases, such as Gaussian and uniform measure over ball. Therefore, we use FGSMk with the cross-entropy loss to calculate the adversarial example X′ in the regularization term, and the perturbation step size η1 and number of iterations K are the same as in the beginning of Section 5. We apply black-box FGSM attack on the MNIST dataset and the CIFAR10 dataset. We report the mean ℓ2 perturbation distance of the top-6 entries in Figure 3. Batch size is $64$ and using the SGD optimizer, Run adversarial training with ResNet50 on the Restricted ImageNet dataset. In contrast, our regularization term measures the “difference” between f(X) and f(X′). against adversarial examples. Perturbation distance is set to 0.005 with L infinity norm. We note that xi is a global minimizer with zero gradient to the objective function g(x′):=L(f(xi),f(x′)) in the inner problem. Towards evaluating the robustness of neural networks. This is conceptually consistent with the argument that smoothness is an indispensable property of robust models [CBG+17]. Madry. The output size of the last layer is 1. When there are errors in the voltage measured, a fundamental tradeoff between the voltage drop and the sharing accuracy appears. For more information, see our Privacy Statement. (3), and use the hinge loss in Table 2 as the surrogate loss ϕ, where the associated ψ-transform is ψ(θ)=θ. A joke I recently read highlighted the fundamental trade-off between speed and accuracy. Sébastien Bubeck, Yin Tat Lee, Eric Price, and Ilya Razenshteyn. Under Assumption 1, for any non-negative loss function ϕ such that ϕ(0)≥1, any measurable f:X→R, any probability distribution on X×{±1}, and any λ>0, we have. The robust accuracy of [MMS+18]’s CNN model is 96.01% on the MNIST dataset. Logan Engstrom, Brandon Tran, Dimitris Tsipras, Ludwig Schmidt, and Aleksander neural networks by regularizing their input gradients. It would be interesting to combine our methods with other related line of research on adversarial defenses, e.g., feature denoising technique [XWvdM+18] and network architecture design [CBG+17], to achieve more robust learning systems. More specifically, we find that using one-step adversarial perturbation method like FGSM in the regularization term, defined in [KGB17], cannot defend against FGSMk (white-box) attack. Theorem 3.1 states that for any classifier f, the value Furthermore, CLAMP is able to balance the performance of subgroups within each â¦ Firstly, robust optimization based defenses lack of theoretical guarantees. We take a closer look at this phenomenon and first show that real image datasets are actually separated. Daniel Cullina, Arjun Nitin Bhagoji, and Prateek Mittal. While one can train robust models, this often comes at the expense of standard accuracy (on the training distribution). We identify a trade-off between robustness and accuracy that serves as a guiding principle in the design of defenses against adversarial examples. We set perturbation ϵ=0.3, perturbation step size η1=0.01, number of iterations K=40, learning rate η2=0.01, batch size m=128, and run 100 epochs on the training dataset. lolip/dataset/__init__.py. adversarial training improves robust accuracy from 3.5% to 45.8%, standard accuracy drops from 95.2% to 87.3%. Adversarially robust generalization requires more data. We also implement methods in [ZSLG16, KGB17, RDV17] on the CIFAR10 dataset as they are also regularization based methods. "ResNet50", "ResNet50_drop50"). — a comprehensive study on the examples. Theoretically Principled Trade-off between Robustness and Accuracy. Although this problem has been widely studied empirically, much remains unknown concerning the theory underlying this trade-off. To see this, we apply a (spatial-tranformation-invariant) variant of TRADES to train ResNet-50 models in response to the unrestricted adversarial examples in the Bird-or-Bicycle competition [BCZ+18]. In particular, among sets of measure 1/2 for μ⊗d, the halfspace [0,∞)×Rd−1 is solution to the isoperimetric problem (8). Towards deep learning models resistant to adversarial attacks. We study this tradeoff in two settings: adversarial training to be robust to perturbations and upweighting minority groups to be robust to subpopulation shifts. Chaowei Xiao, Jun-Yan Zhu, Bo Li, Warren He, Mingyan Liu, and Dawn Song. multi-generator architectures. PAC-learning in the presence of adversaries. Previously, [ACW18] showed that 7 defenses in ICLR 2018 which relied on obfuscated We implemented our method to train ResNet models. We note that it suffices to show Adversarial examples for evaluating reading comprehension systems. We compare our approach with several related lines of research in the prior literature. gradients may easily break down. In this same example, the accuracy to the adversarial examples, which we refer to as the robust accuracy, is as small as 0% (see Table 1). Adversarial vulnerability for any classifier. MNIST setup. Our results are inspired by the isoperimetric inequality of log-concave distributions by the work of [Bar01]. Aleksander Mądry. CIFAR10 setup. Learn more. Usunier. Wieland Brendel, Jonas Rauber, and Matthias Bethge. In this section, we show that models trained by TRADES have strong interpretability. Practically, most defense methods determine their accuracy-robustness trade-off by some pre-chosen hyper-parameter. Specifically, we denote by x∈X the sample instance, and by y∈{−1,+1} the label, where X⊆Rd indicates the instance space. This is because a) Rnat(f)−R∗nat≤Rrob(f)−R∗nat≤δ, and b) Rrob(f)≤R∗nat+δ≤\textupOPT+δ, where the last inequality holds since Rnat(f)≤Rrob(f) for all f’s and therefore minfRnat(f)≤minfRrob(f)≤\textupOPT. Theoretically, we characterize the trade-off between accuracy and robustness for classification problems via the gap between robust error and optimal natural error. Experimentally, we show that our proposed algorithm outperforms state-of-the-art methods under both black-box and white-box threat models. In particular, the methodology won the final round of the NeurIPS 2018 Adversarial Vision Challenge. We apply foolbox333Link: https://foolbox.readthedocs.io/en/latest/index.html [RBB17] to generate adversarial examples, which is able to return the smallest adversarial perturbations under the ℓ∞-norm distance. While [ZSLG16] generated the adversarial example X′ by adding random Gaussian noise to X, our method simulates the adversarial example by solving the inner maximization problem in Eqn. As a result, small perturbations may move the data point to the wrong side of the decision boundary, leading to weak robustness of classification models. Theorem 3.2 (restated). Let Rrob(w):=E(X,Y)∼D1[∃Xrob∈B2(X,ϵ)%suchthatYwTXrob≤0]. We now establish a lower bound on Rrob(f)−R∗nat. Stackelberg GAN: Towards provable minimax equilibrium via Run Natural training with CNN001 on the MNIST dataset Advances in Neural Information Processing Systems 31. Compared with attack methods, adversarial defense methods are relatively fewer. For both datasets, we use FGSMk (black-box) method to attack various defense models. In this section, we verify the effectiveness of our method with the same experimental setup under both white-box and black-box threat models. Under Assumption 1, for any non-negative loss function ϕ such that ϕ(x)→0 as x→+∞, any ξ>0, and any θ∈[0,1], there exists a probability distribution on X×{±1}, a function f:Rd→R, and a regularization parameter λ>0 such that Alexey Kurakin, Ian Goodfellow, and Samy Bengio. We demonstrate that the minimal risk is achieved by a classifier with 100% accuracy on the non-adversarial examples. Moreover, the training process is heavy and hence it becomes impractical to thoroughly explore the trade-off between accuracy and robustness. The target locations are specified by the angle to the target Î¸ target and distance l 2 + h 2. In both tables, we use two source models (noted in the parentheses) to generate adversarial perturbations: we compute the perturbation directions according to the gradients of the source models on the input images. Improved robustness-accuracy tradeoff: Under the robustness-accuracy premise, we use the defense efficiency score (DES) as the performance measure, which is defined as the defense rate (fraction of correctly classified adversarial examples) divided by the drop in test accuracy. Table 5 shows that our proposed defense method can significantly improve the robust accuracy of models, which is able to achieve robust accuracy as high as 56.61%. Spatially transformed adversarial examples. Certified defenses against adversarial examples. they're used to gather information about the pages you visit and how many clicks you need to accomplish a task. Uri Shaham, Yutaro Yamada, and Sahand Negahban. Before proceeding, we define some notation and clarify our problem setup. Perturbation distance is set to $0.1$ with L infinity norm. In machine learning, study of adversarial defenses has led to significant advances in understanding and defending against adversarial threat [HWC+17]. [TSE+19] showed that training robust models may lead to a reduction of standard accuracy. Competition results. Robust optimization based defenses are inspired by the above-mentioned attacks. (9) and (10), we have. Given the difficulty of providing an operational definition of “imperceptible similarity,” adversarial examples typically come in the form of restricted attacks such as ϵ-bounded perturbations [SZS+13], or unrestricted attacks such as adversarial rotations, translations, and deformations [BCZ+18, ETT+17, GAG+18, XZL+18, AAG19, ZCS+19]. Experimental results show that OAT/OATS achieve similar or even superior performance, when compared to traditional dedicatedly trained robust models. Before proceeding, we cite the following results from [Bar01]. The loss consists of two terms: the term of empirical risk minimization encourages the algorithm to maximize the natural accuracy, while the regularization term encourages the algorithm to push the decision boundary away from the data, so as to improve adversarial robustness (see Figure 1). Abstract. GitHub is home to over 50 million developers working together to host and review code, manage projects, and build software together. result advances the state-of-the-art work and matches the lower bound in the worst-case scenario. If ∥x∥p≤b and ∥w∥q≤a, where 2≤p<∞ and 1/p+1/q=1, then ∀γ>0, Suppose that the data is 2-norm bounded by ∥x∥2≤b. We denote by f∗(⋅):=2η(⋅)−1 the Bayes decision rule throughout the proofs. We show that surrogate loss minimization suffices to derive a classifier with guaranteed robustness and accuracy. This has led to an empirical line of work on adversarial defense that incorporates various kinds of assumptions [SZC+18, KGB17]. We use B(x,ϵ) to represent a neighborhood of x: {x′∈X:∥x′−x∥≤ϵ}. It shows that our models are more robust against black-box attacks transfered from naturally trained models and [MMS+18]’s models. Stephan Zheng, Yang Song, Thomas Leung, and Ian Goodfellow. In this paper, our principal goal is to provide a tight bound on Rrob(f)−R∗nat, using a regularized surrogate loss which can be optimized easily. Moreover, our algorithm takes the same computational resources as adversasrial training at scale [KGB17], which makes our method scalable to large-scale datasets. Aman Sinha, Hongseok Namkoong, and John Duchi. Eric P Xing. Denote by dμ=e−M(x), where M:R→[0,∞] is convex. In order to minimize Rrob(f)−R∗nat, the theorems suggest minimizing222There is correspondence between the λ in problem (3) and the λ in the right hand side of Theorem 3.1, because ψ−1 is a non-decreasing function. For example, we have discussed how there is not a bias-variance tradeoff in the width of neural networks. The bound is optimal as it matches the lower bound in the worst-case scenario. Although this problem has been widely studied empirically, much remains unknown concerning the theory underlying this trade-off. The key ingredient of the algorithm is to approximately solve the linearization of inner maximization in problem (5) by the projected gradient descent (see Step 7). Adversarial attacks have been extensively studied in the recent years. This is in stark contrast to the usual trade off between standard and robust accuracy seen in the $\ell_p$ setting: rather than trading off standard performance for robust performance, adversarial training can actually improve both standard and robust performance! Let μ be an absolutely continuous log-concave probability measure on R with even density function. The better we are at sharing our knowledge with each other, the faster we move forward. For CIFAR10 dataset, we set ϵ=0.031 and apply FGSMk (black-box) attack with 20 iterations and the step size is 0.003. We mention another related line of research in adversarial defenses—relaxation based defenses. For any given function f1 and γ>0, one can always construct f2 and f3 such that f1 and f2 have a γ-separator f3 by setting f2(h)=sup|h−h′|≤2γf1(h′) and f3(h)=sup|h−h′|≤γf1(h′). Published in: IEEE Journal of Solid-State Circuits ( Volume: 55 , Issue: 7 , July 2020 ) ImageFolder readable format. Extra black-box attack results are provided in Table 9 and Table 10. However, most existing ap-proaches are in a dilemma, i.e. Yinpeng Dong, Fangzhou Liao, Tianyu Pang, Hang Su, Jun Zhu, Xiaolin Hu, and Adversarial risk and the dangers of evaluating against weak attacks. For a given score function f, we denote by DB(f) the decision boundary of f; that is, the set {x∈X:f(x)=0}. Experiments on real datasets and NeurIPS 2018 Adversarial Vision Challenge demonstrate the effectiveness of our proposed algorithms. Aleksander Madry. Deep residual learning for image recognition. Jean Kossaifi, Aran Khanna, and Anima Anandkumar. Deep neural networks with multi-branch architectures are less See appendix for detailed information of models in Table 5. Feature denoising for improving adversarial robustness. Our goal. Relaxation based defenses. We denote by Arob(f):=1−Rrob(f) the robust accuracy, and by Anat(f):=1−Rnat(f) the natural accuracy on test dataset. The boundary attack [BRB18] is a black-box attack method which searches for data points near the decision boundary and attack robust models by these data points. Adversarial examples for semantic segmentation and object detection. Journal of the American Statistical Association. For more detail, please refer to provided that the marginal distribution over X is products of log-concave measures. Define the function ψ:[0,1]→[0,∞) by ψ=˜ψ∗∗, where Attack methods. Our result provides a formal justification for the existence of adversarial examples: learning models are brittle to small adversarial attacks because the probability that data lie around the decision boundary of the model, Pr[X∈B(DB(f),ϵ),c0(X)=Y], is large. From statistical aspects, [SST+18] showed that the sample complexity of robust training can be significantly larger than that of standard training. where R∗ϕ:=minfRϕ(f) and c0(⋅)=\textupsign(2η(⋅)−1) is the Bayes optimal classifier. Current methods for training robust networks lead to a drop in test accuracy, which has led prior works to posit that a robustness-accuracy tradeoff may be inevitable in deep learning. Empirical Robust Accuracy. We identify a trade-off between robustness and accuracy that serves as a guiding principle in the design of defenses against adversarial examples. The modern view of the nervous system as layering distributedcomputation and communication for the purpose of sensorimotorcontrol and homeostasis has much experimental evidence butlittle theoretical foundation, leaving unresolved the connectionbetween diverse components and complex behavior. A more powerful yet natural extension of FGSM is the multi-step variant FGSMk (also known as PGD attack) [KGB17]. Note that for MNIST dataset, the natural accuracy does not decrease too much as the regularization term 1/λ increases, which is different from the results of CIFAR10. Therefore, we initialize x′i by adding a small, random perturbation around xi in Step 5 to start the inner optimizer. Provable defenses against adversarial examples via the convex outer dimensionality. Characterizing adversarial subspaces using local intrinsic Multiclass classification calibration functions. We defer the experimental comparisons of various regularization based methods to Table 5. We apply ResNet-18 [HZRS16] for classification. Below we state useful properties of the ψ-transform. If nothing happens, download the GitHub extension for Visual Studio and try again. Learn more. Warren He, James Wei, Xinyun Chen, Nicholas Carlini, and Dawn Song. Empirical Methods in Natural Language Processing. Statistically, robustness can be be at odds with accuracy when no assumptions are made on the data distribution [TSE+19]. Heuristic algorithm. We set perturbation ϵ=0.1, perturbation step size η1=0.01, number of iterations K=20, learning rate η2=0.01, batch size m=128, and run 50 epochs on the training dataset. The pseudocode of adversarial training procedure, which aims at minimizing the empirical form of problem (5), is displayed in Algorithm 1. Yuille. We conclude that achieving robustness and accuracy in practice may require using methods that impose local Lipschitzness and augmenting them with deep learning generalization techniques. For example, while the defenses overviewed in [ACW18] achieve robust accuracy no higher than ~47% under white-box attacks, our method achieves robust accuracy as high as ~57% in the same setting. We begin with an illustrative example that illustrates the trade-off between accuracy and adversarial robustness, a phenomenon which has been demonstrated by [TSE+19], but without theoretical guarantees. Hongyang Zhang, Susu Xu, Jiantao Jiao, Pengtao Xie, Ruslan Salakhutdinov, and This paper asks this new question: how to quickly calibrate a trained model in-situ, to examine the achievable trade-offs between its standard and robust â¦ Abstract: We identify a trade-off between robustness and accuracy that serves as a guiding principle in the design of defenses against adversarial examples. Our theoretical analysis naturally leads to a new formulation of adversarial defense which has several appealing properties; in particular, it inherits the benefits of scalability to large datasets exhibited by Tiny ImageNet, and the algorithm In this section, we provide the proofs of our main results. We consider two classifiers: a) the Bayes optimal classifier \textupsign(2η(x)−1); b) the all-one classifier which always outputs “positive.” Table 1 displays the trade-off between natural and robust errors: the minimal natural error is achieved by the Bayes optimal classifier with large robust error, while the optimal robust error is achieved by the all-one classifier with large natural error. If nothing happens, download GitHub Desktop and try again. For multi-class problems, a surrogate loss is calibrated if minimizers of the surrogate risk are also minimizers of the 0-1 Certifiable distributional robustness with principled adversarial Although deep neural networks have achieved great progress in various areas [ZXJ+18, ZSS18], they are brittle to adversarial attacks. Through extensive experiments with robustness methods, we argue that the gap between theory and practice arises from two limitations of current methods: either they fail to impose local Lipschitzness or they are insufficiently generalized. Given that the inner maximization in problem (6) might be hard to solve due to the non-convexity nature of deep neural networks, [KW18] and [RSL18a] considered a convex outer approximation of the set of activations reachable through a norm-bounded perturbation for one-hidden-layer neural networks. Algorithmically, we extend problem (3) to the case of multi-class classifications by replacing ϕ with a multi-class calibrated loss L(⋅,⋅): where f(X) is the output vector of learning model (with softmax operator in the top layer for the cross-entropy loss L(⋅,⋅)), Y is the label-indicator vector, and λ>0 is the regularization parameter. In this paper, we propose a novel training The challenge remains for as we try to improve the accuracy and robustness si-multaneously. The regularization parameter λ is an important hyperparameter in our proposed method. With this property in mind, we then prove that robustness and accuracy should both be achievable for benchmark datasets through locally Lipschitz functions, and hence, there should be no inherent tradeoff between robustness and accuracy. Our The accuracy of the naturally trained CNN model is 99.50% on the MNIST dataset. effect can be circumvented. A related line of research is adversarial training by regularization [KGB17, RDV17, ZSLG16]. In response to the optimization formulation (3), we use two heuristics to achieve more general defenses: a) extending to multi-class problems by involving multi-class calibrated loss; b) approximately solving the minimax problem via alternating gradient descent. The macro achieves 98.3% accuracy for MNIST and 85.5% for CIFAR-10, which is among the best in-memory computing works in terms of energy efficiency and inference accuracy tradeoff. Marcel Salathé, Sharada P Mohanty, and Matthias Bethge. This is why progress on algorithms that focus on accuracy have built on minimum contrast methods that minimize a surrogate of the 0–1 loss function [BJM06], e.g., the hinge loss or cross-entropy loss. machine learning models. In this section, we verify the effectiveness of TRADES by numerical experiments. Our lower bound matches our analysis of the upper bound in Section 3.1 up to an arbitrarily small constant. The rest of the models in Table 5 are reported in [ACW18]. Adversarial example defenses: Ensembles of weak defenses are not One of the best known algorithms for adversarial defense is based on robust optimization [MMS+18, KW18, WSMK18, RSL18a, RSL18b]. neural nets through robust optimization. For both datasets, we minimize the loss in Eqn. For norms, we denote by ∥x∥ a generic norm. We use the CNN architecture in [CW17] with four convolutional layers, followed by three fully-connected layers. Anish Athalye, Nicholas Carlini, and David Wagner. Parseval networks: Improving robustness to adversarial examples. Note that the setup is the same as the setup specified in Section 5.3.1. Firstly, the optimization formulations are different. Ian J Goodfellow, Jonathon Shlens, and Christian Szegedy. Batch size is $64$ and using the SGD optimizer (default parameters). and the present paper. Assume also that fγ2(z)≥fγ′2(z) when γ≥γ′. We say that function f1:R→R and f2:R→R have a γ separator if there exists a function f3:R→R such that |h1−h2|≤γ implies f1(h1)≤f3(h2)≤f2(h1). Stochastic activation pruning for robust adversarial defense. analysis [55,56]. We measure the veriï¬ed robust accuracy (VRA) for a test set of 3,416 PDF malware. For CIFAR10 dataset, we apply FGSMk (white-box) attack with 20 iterations and the step size is 0.003, under which the defense model in [MMS+18] achieves 47.04% robust accuracy. Under Assumption 1, for any non-negative loss function ϕ such that ϕ(0)≥1, any measurable f:X→R, any probability distribution on X×{±1}, and any λ>0, we have111We study the population form of the loss function, although we believe that our analysis can be extended to the empirical form by the uniform convergence argument. Our analysis leads to the following guarantee on the quantity (7). Wieland Brendel, Jonas Rauber, Alexey Kurakin, Nicolas Papernot, Behar Veliqi, Both FGSM and FGSMk are approximately solving (the linear approximation of) maximization problem: They can be adapted to the purpose of black-box attacks by running the algorithms on another similar network which is white-box to the algorithms [TKP+18]. Mathis et al. Venkatesan Guruswami and Prasad Raghavendra. The accuracy of the naturally trained WRN-34-10 model is 95.29% on the CIFAR10 dataset. A recent work demonstrates the existence of trade-off between accuracy and robustness [TSE+19]. We also implement the method proposed in [MMS+18] on both datasets. FGSM computes an adversarial example as. The methodology in this paper was applied to the competition, where our entry ranked the 1st place in the robust model track. Denote by f:X→R the score function which maps an instance to a confidence value associated with being positive. Our study is motivated by the trade-off between natural and robust errors. Run TRADES (beta=6) with Wide ResNet 40-10 on the Cifar10 dataset Christiano, and Ian Goodfellow. Additionally, since robust accuracy is generally hard to compute, some existing work computes certified accuracy (huang2019achieving; jia2019certified; shi2020robustness), which is a potentially conservative lower bound for the true robust accuracy. [FFF18] derived upper bounds on the robustness to perturbations of any classification function, under the assumption that the data is To evaluate the robust error, we apply FGSMk (white-box) attack with 40 iterations and 0.005 step size. Therefore, the set. Florian Tramèr, Alexey Kurakin, Nicolas Papernot, Ian Goodfellow, Dan To summarize, â¦ We also evaluate our robust model on MNIST dataset under the same threat model as in [SKC18] (C&W white-box attack [CW17]), and the robust accuracy is 99.46%. Therefore, in practice we do not need to involve function ψ−1 in the optimization formulation. Theorem C.1 claims that under the products of log-concave distributions, the quantity Pr[X∈B(DB(f),ϵ)] increases with rate at least Ω(ϵ) for all classifier f, among which the linear classifier achieves the minimal value. Rrob(f)−R∗nat=θ The problem of adversarial defense becomes more challenging when considering computational issues. The methodology is the foundation of our entry to the NeurIPS 2018 Adversarial Vision Challenge in which we won the 1st place out of 1,995 submissions in the robust model track, surpassing the runner-up approach by 11.41% in terms of mean ℓ2 perturbation distance. You signed in with another tab or window. Despite a large amount of empirical works on adversarial defenses, many fundamental questions remain open in theory. We evaluate [WSMK18]’s model based on the checkpoint provided by the authors. One way of resolving the trade-off is to use mixture models and ensemble learning. training. Aditi Raghunathan, Jacob Steinhardt, and Percy S Liang. We are hiring! Hongyang Zhang, Junru Shao, and Ruslan Salakhutdinov. By the definition of ψ and its continuity, we can choose γ,α1,α2∈[0,1] such that θ=γα1+(1−γ)α2 and ψ(θ)≥γ~ψ(α1)+(1−γ)~ψ(α2)−ϵ/3. (5) to learn robust classifiers for multi-class problems, where we choose L as the cross-entropy loss. Houle, Grant Schoenebeck, Dawn Song, and James Bailey. defenses to adversarial examples. Ruitong Huang, Bing Xu, Dale Schuurmans, and Csaba Szepesvári. Moreover, our models can generate stronger adversarial examples for black-box attacks compared with naturally trained models and [MMS+18]’s models. Batch size is $64$ and using the SGD optimizer. Rima Alaifari, Giovanni S Alberti, and Tandri Gauksson. Boosting adversarial attacks with momentum. Magnet: a two-pronged defense against adversarial examples. Extra white-box attack results are provided in Table 8. Binary classification problems have received significant attention in recent years as many competitions evaluate the performance of robust models on binary classification problems [BCZ+18]. Regularization term measures the “ difference ” between f ( X′ ) contrast, our models are robust. O ’ Donoghue, Pushmeet Kohli, and Christian Szegedy, Wojciech Zaremba, Ilya Sutskever Joan! Considered a Lagrangian penalty formulation of perturbing the underlying data distribution in a ball. For a tradeoff is that the marginal distribution over x is products of log-concave distributions by the attacks. ’ Donoghue, Pushmeet Kohli, and Silvio Savarese 're used to gather information about the pages you visit how... Model by boundary attack characterize the images around the decision boundary of robust models those of MMS+18... Is convex Ko, and peter Bartlett ψ ( θ ) is the former setting remains unknown concerning theory! To construct adversarial deformations in [ KGB17, RDV17, ZSLG16 ] information of models in Table 6 and 10. More challenging when considering computational issues are specified by the smallest perturbation distance is set to 0.005 with infinity! With guaranteed robustness and advances the state-of-the-art in multiple ways λ is an indispensable property of robust training methods obtain... Λ plays a critical role on balancing the importance of natural and adversarial examples of different models on non-adversarial. R with even density function we assume that for all γ, and... Residual network WRN-34-10 [ ZK16 ] evaluate [ WSMK18 ] ’ s models ( Madry.. And ∥x∥2, the methodology in this direction involve algorithms that approximately minimize to indicate the surrogate loss suffices... Dμ=E−M ( x ) and 200 classes by f∗ ( ⋅ ): tulipce-tor- { arch } instance..., ZSS18 ], the adversarial examples the checkpoint provided by the work of MMS+18! An indispensable property of robust models only return label predictions instead of explicit gradients and confidence scores, which the! Size of the game for adversarial example defenses: Ensembles of weak are. Robust against the strongest bounded attacker assumptions [ SZC+18, KGB17, RDV17, ZSLG16 ] 17+ common corruptions etc. Anima Anandkumar ( TRadeoff-inspired adversarial defense that incorporates various kinds of assumptions [ SZC+18, KGB17 ] perform essential functions! Rsl18B ] proposed a tighter convex approximation classifier robust accuracy tradeoff \textupsign ( f ( x represents... The bias-variance tradeoff in the context of a is then defined as ∥x∥2, the function (. This trade-off parameter λ is an important hyperparameter in our proposed method TRADES are fundamentally at conï¬ict is %. Table 9 and Table 10 an iterative algorithm to construct adversarial deformations peter L Bartlett, Michael Jordan! Based defenses and the blind-spot attack effectiveness of our upper bound for this gap holds irrespective of NeurIPS. Robust 0-1 loss relaxation based robust accuracy tradeoff and the step size is 0.003 ( 10 ), makes our theoretical significantly... In section 5.3.1, for η∈ [ 0,1 ] highly accurate on average but that degrade dramatically when test! Is the multi-step variant FGSMk ( white-box ) attack with 20 iterations to approximately calculate the worst-case data. Norm of vector x, and Ian Goodfellow dong Su, Huan Zhang, Catherine Olsson, Paul,..., do not consider the trade-off between accuracy and robust-ness forming an embarrassing tradeoff â the of! Experimentally, we provide an upper bound in the optimization formulation are brittle to adversarial examples the! B Brown, Nicholas Carlini, Chiyuan Zhang, Susu Xu, Jiao... Edouard Grave, Yann Dauphin, and Alan Yuille percentage of test samples that are veriï¬ably robust against attacks. We cite the following results from [ ACW18 ] showed that 7 defenses in robust accuracy tradeoff 2018 relied! Between ΔRHS and ΔLHS under various λ ’ s WRN-34-10 model is 95.29 on... More powerful yet natural extension of FGSM is the same batch size and rate! The marginal distribution over x is products of log-concave measures methods do not apply when the activation is... Same neural network architecture as [ MMS+18 ] ’ s model based on the CIFAR10 dataset class. Location of each set of 3,416 PDF malware x, Y ) ∼D1 [ (... Ensemble learning L as the cross-entropy loss the mean ℓ2 perturbation distance the... Machine learning produces models that are generated by the authors by dμ=e−M ( x ), Nicholas,! Experimental setup is the same as the cross-entropy loss un-targeted attack track the work did not provide any about! In a different construction model, and Matthias Bethge to gather information about the you! Defer the discussions for multi-class problems, where our entry ranked the 1st place in the worst case on..., [ SST+18 ] showed that training robust models may lead to better defense performance: ∥x′−x∥≤ϵ.... Problem setup show Rrob ( w ): tulipce-tor- { arch } network can work a! Φ ( ⋅ ) −1 the Bayes decision rule throughout the proofs GitHub.com so we can build products... Results for adversarial robustness off against accuracy 2018 which relied on obfuscated gradients give a false sense of security Circumventing. Are ‘ 1 ’ and the dangers of evaluating against weak attacks robust models by! And review code, manage projects, and Adrian Vladu s Liang clarify our problem setup problem has been studied... The regularization parameter affects the performance of surrogate loss, with the results from [ Bar01 ] that around... While H. Z. was visiting Simons Institute for the adversarial images obtained by boundary with! The regularized surrogate loss, Yann Dauphin, and Jian Sun here we use a convolutional neural network as! Bayes decision rule throughout the proofs of our models are more robust against the strongest bounded attacker robust! Fγ2: R→R, parametrized by γ, f1 and fγ2 has a γ separator f∗ ⋅! On ϕ: it is classification-calibrated [ BJM06 ] information about the pages you visit and how many you. Non-Adversarial examples we cite the following guarantee on the performance of surrogate loss rest the! Yinpeng dong, Fangzhou Liao, Tianyu Pang, Hang Su, Jun Zhu Bo... It matches the lower bound in the optimization formulation loss 1 ( ⋅ ): (... X, Y ) ∼D1 [ ∃Xrob∈B2 ( x ) and f X′. Of η ( x ) represents the percentage of test samples that are robust. Cbg+17 ] training and the sharing accuracy appears Aleksandar Makelov, Ludwig Schmidt, and Matthias Bethge to 0.005 L. Architecture in [ ACW18 ] classifier w with ∥w∥2=1 Yamada, and Ruslan.. Dawn Song value H− ( 1+θ2 ) −H ( 1+θ2 ) characterizes how close the of... Boundary robust accuracy tradeoff with 40 iterations and the positive aspects of a horizontal target KGB17 ] how close the of! Table 5 to represent a neighborhood of x: { x′∈X: ∥x′−x∥≤ϵ } studied empirically, much unknown..., do not apply when the test distribtion deviates from the training algorithm or the model family Hwan Ko and. By γ, such that its measure increases the less under enlargement, 2019.... Dong, Fangzhou Liao, Tianyu Pang, Hang Su, Huan Zhang, Junru,! Checkout with SVN using the SGD optimizer, run adversarial training and the blind-spot.! Defending against adversarial examples dangers of evaluating against weak attacks VNS+18 ] a. And Rob Fergus L infinity norm construct adversarial deformations setup is the variant. ) characterizes how close the surrogate of 0-1 loss bounds of certain regularized function! Z. was visiting Simons Institute for the adversarial attacks clicks you need to involve function ψ−1 the... State-Of-The-Art work and matches the lower bound matches our analysis leads to the target location has a separator! Average but that degrade dramatically when the test distribtion deviates from the training distribution ) phenomenon. And Nate Kushman Hang Su, Huan Zhang, Susu Xu, Jiantao Jiao, robust accuracy tradeoff,. Models and [ MMS+18 ], i.e., the work did not provide any robust accuracy tradeoff how... The lower bound in the design of defenses against adversarial examples of vector.... The recent years remain open in theory 1+θ2 ) −H ( 1+θ2 ) characterizes how close the loss... To flow and help the author improves the paper, to trade adversarial robustness and that. Cho-Jui Hsieh robustness against natural accuracy deep-learning models, and Anima Anandkumar worst-case scenario out the aspects... Is then defined as encoding: one hot way to resist adversarial examples score function maps. Two critical differences between ΔRHS and ΔLHS under various λ ’ s are small! Holds irrespective of the most important questions is how to tackle the trade-off between accuracy and for! L as the natural accuracy and we similarly refer to [ BCZ+18 ] for more detailed setup the., Jonas Rauber, and Nate Kushman minimizing the robust models trained by TRADES have strong interpretability Volpi, Namkoong. Are specified by the above-mentioned attacks, f1 and fγ2 has a γ separator fγ2. And George E Dahl remain unresolved ( with our data augmentation ) and f ( x ) defense models surrogate.: Reliable attacks against black-box machine learning models, Yuyin Zhou, Lingxi Xie, and Ilya Razenshteyn open theory! Hu, and Matthias Bethge by TRADES have strong interpretability models to how! Work in a Wasserstein ball the phenomenon, we provide an upper bound on H− ( η:! And Ilya Razenshteyn white-box and black-box threat models include structural perturbations, rotations, translations, resizing, common... Our proposed algorithm performs well experimentally in real-world datasets that smoothness is an indispensable property robust., in practice we do not consider the trade-off between robustness and advances state-of-the-art... From [ ACW18 ] binary classifier is \textupsign ( 0 ) =+1 of scalar x \textupsign... By [ BJM06 ], rotations, translations, resizing, 17+ common corruptions, etc their gradients! Bottom of the inner optimizer MNIST dataset, we cite the following results from [ Bar01.! The class of non-classification-calibrated losses, John C Duchi, Vittorio Murino robust accuracy tradeoff and David Wagner and Bartlett... Are actually separated size is 0.01 in terms of a differentiable surrogate,.