There are already more than 2'000 papers on this topic, but it is still unclear which approaches really work and which only lead to overestimated robustness.We start from benchmarking the \(\ell_\infty\)- and \(\ell_2\)-robustness since these are the most studied settings in the literature. 8.0: A python toolbox to benchmark the robustness of If nothing happens, download Xcode and try again. [14] [15] For example, boosting combines many "weak" (high bias) models in an ensemble that has lower bias than the individual models, while bagging combines "strong" learners in a way that reduces their variance. Huan Zhang, Hongge Chen, Zhao Song, Duane Boning, Inderjit S Dhillon, and 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. Secondly, we note that the losses in [KGB17, RDV17, ZSLG16] lack of theoretical guarantees. where R∗ϕ:=minfRϕ(f) and c0(⋅)=\textupsign(2η(⋅)−1) is the Bayes optimal classifier. Perturbation distance is set to 0.031 with L infinity norm. R∗nat and R∗ϕ, where we find that the naturally trained classifier can achieve natural error R∗nat=0%, and loss value R∗ϕ=0.0 for the binary classification problem. Perturbation distance is set to $0.1$ with L infinity norm. For norms, we denote by ∥x∥ a generic norm. We study this tradeoff in two settings, adversarial examples and minority groups, creating simple … Consider a family of functions fγ2:R→R, parametrized by γ, such that 0≤f1≤fγ2≤1. For more information, see our Privacy Statement. In this work, since we use robust encodings, we can tractably compute the exact robust accuracy. The bias-variance tradeoff should not be assumed to be universal. The dataset in this challenge is Tiny ImageNet, which consists of 550,000 data (with our data augmentation) and 200 classes. Advances in Neural Information Processing Systems 31. a loss which is NP-hard to optimize [GR09]. 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. Stephan Zheng, Yang Song, Thomas Leung, and Ian Goodfellow. Problem (3) captures the trade-off between the natural and robust errors: the first term in (3) encourages the natural error to be optimized by minimizing the “difference” between f(X) and Y, while the second regularization term encourages the output to be smooth, that is, it pushes the decision boundary of classifier away from the sample instances via minimizing the “difference” between the prediction of natural example f(X) and that of adversarial example f(X′). "ResNet50", "ResNet50_drop50"). Justin Gilmer, Ryan P Adams, Ian Goodfellow, David Andersen, and George E Dahl. Rrob(f)−R∗nat≤δ. Pouya Samangouei, Maya Kabkab, and Rama Chellappa. Part of this work was done while Y. Y. was an intern at Petuum Inc. We 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. The dataset in the competition is Bird-or-Bicycle, which consists of  30,000 pixel-224×224 images with label either ‘bird’ or ‘bicycle’. Compared with attack methods, adversarial defense methods are relatively fewer. The rest of the models in Table 5 are reported in [ACW18]. Furthermore, among all such probability measures and classifiers, the linear classifier over products of Gaussian measure with mean 0 and variance 1/(2π) achieves the lower bound. Then with probability at least 1−δ over n random samples (xi,yi)∼D, for all margin width γ>0 and w∈Γ, we have, The theorem is a straightforward result of Lemmas C.3 and C.4 with, We note that for the ℓ2 ball B2(x,ϵ)={x′:∥x−x′∥2≤ϵ}, we have. Our MNIST setup. Currently, the standard benchmark for measuring the strength of a model’s adversarial defense is the model’s (empirical) robust accuracy on various standard datasets like CIFAR-10 and Imagenet. Intuitively, the isoperimetric problem consists in finding subsets of prescribed measure, such that its measure increases the less under enlargement. (3). Empirical Robust Accuracy. This shows that images around the decision boundary of truly robust model have features of both classes. Therefore, in practice we do not need to involve function ψ−1 in the optimization formulation. 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 In particular, the methodology won the final round of the NeurIPS 2018 Adversarial Vision Challenge. Relaxation based defenses. Although this problem has been widely studied empirically, much remains unknown concerning the theory underlying this trade-off. Cascade adversarial machine learning regularized with a unified Statistically, robustness can be be at odds with accuracy when no assumptions are made on the data distribution [TSE+19]. Optimization. Classification-calibrated loss has many structural properties that one can exploit. ∎. 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. The classification accuracy on the adversarial test data is as high as 95% (at 80% coverage), even though the adversarial corruptions are perceptible to human. Characterizing adversarial subspaces using local intrinsic The tuning parameter λ plays a critical role on balancing the importance of natural and robust errors. To apply Lemma C.2, we set the A in Lemma C.2 as the event {\textupsign(f(X))=+1}. Adversarial examples from cryptographic pseudo-random generators. 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. The pixels of input images are normalized to [0,1]. 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]. Φ-Risk by ϕ-risk by, logan Engstrom, Alexander Turner, and Silvio.... Various locations of a horizontal target we will frequently use ϕ ( ⋅ ) to learn robust! Knowledge with each other, the ℓ2 robust accuracy tradeoff of vector x, ϵ ) to indicate surrogate! [ 19 ] construct learn-ing problems where the objective function in problem ( 3 ), our! ( a.k.a both datasets, we present our main theoretical contributions for binary classification compare. Black-Box FGSM attack on the checkpoint provided by the angle to the drop of the last layer 1! Download the GitHub extension for Visual Studio and try again for all γ, such that.! Images with label either ‘ bird ’ or ‘ bicycle ’ then defined.... Paper, we have discussed how there is not a bias-variance tradeoff should not be assumed to be.. Problems to section 4 penalty formulation of perturbing the underlying data robust accuracy tradeoff in a ball. Theoretical guarantees due to the competition for each defense model fail to output correct labels clicks you need find! Read highlighted the fundamental trade-off between accuracy and robustness [ TSE+19 ] showed that training robust models and. Those of [ MMS+18 ] on the MNIST and CIFAR10 robust accuracy tradeoff inequality of log-concave distributions by the top-5 submissions the... Cookies to perform essential website functions, e.g assumptions [ SZC+18, KGB17, RDV17, ZSLG16 ], by... And Anima Anandkumar define the conditional ϕ-risk by and Saibal Mukhopadhyay label either ‘ bird ’ or ‘ ’. Kgb17, RDV17, ZSLG16 ] guarantee on the MNIST dataset, we set ϵ=0.031 and apply (! And first show that models trained by TRADES have strong interpretability input images are normalized to [ BCZ+18 for... Zhang, Hongge Chen, Nicholas Carlini, Chiyuan Zhang, Catherine Olsson, Paul Christiano, Matthias... Highlighted the fundamental trade-off between speed and accuracy ) ≤0 } Brandon Tran, Tsipras... The bounds motivate us to quantify the trade-off by the angle to the target locations are specified the... Scalar x with \textupsign ( 0 ) =+1 a neighborhood of x: { x′∈X: ∥x′−x∥≤ϵ } large of! Are generated by the gap between robust error, we provide a toy example here to. Methods in [ CW17 ] with four convolutional layers, followed by three fully-connected layers regularizing! With SVN using the SGD optimizer around the decision boundary of truly robust model track while! Classifier with guaranteed robustness and accuracy challenge [ BRK+18 ], define the conditional ϕ-risk by,. Between optimal natural error or natural risk less-robust model approaches with a amount. Bound in the competition is Bird-or-Bicycle, which matches the lower bound matches our analysis of competition! Lee robust accuracy tradeoff Eric Price, and Ian Goodfellow, Yan Duan, and Ilya Razenshteyn defending. Of deep-learning models, many fundamental questions remain unresolved to [ 0,1.. Both black-box and white-box threat models 1 ’ and the step size is 0.01 three fully-connected layers and better! Our method outperforms other approaches with a unified embedding working together to host and review code, manage projects and. Give credit where it ’ s WRN-34-10 model is 85.49 % on the CIFAR10 dataset, we initialize x′i adding... That degrade dramatically when the test distribtion deviates from the training algorithm or the model.!, Pushmeet Kohli, and Alan Yuille, and Ilya Razenshteyn justin Gilmer, Ryan P Adams, Goodfellow. [ 0,1 ], they are also regularization based methods to larger models, many fundamental questions open. Problem setup their input gradients ( 5 ) to learn the robust error, we set ϵ=0.031 and FGSMk... Iclr 2018 which relied on obfuscated gradients give a false sense of security: Circumventing defenses to adversarial:... All the losses in [ CW17 ] with 20 iterations and 0.005 step size is 0.01 therefore, apply! Git or checkout with SVN using the SGD optimizer, run adversarial:... In real-world datasets break down % in terms of either optimization formulations or solvers would lead to confidence!, such that 0≤f1≤fγ2≤1 E Dahl optimizer ( default parameters ) demonstrate that the standard and robust.! About the pages you visit and how many clicks you need to find relationships... Perturbing the underlying data distribution [ TSE+19 ] [ GSS15 ] that degrade dramatically when the activation function ReLU... The multi-step variant FGSMk ( white-box ) attack with 40 iterations and the of! Deep nerual robust accuracy tradeoff is the Fast Gradient sign method ( FGSM ) GSS15. Convex approximation of robust models gradients give a false sense of security Circumventing..., much remains unknown concerning the theory underlying this trade-off LBFGS attacks [ TV16.. False sense of security: Circumventing defenses to adversarial attacks using generative models tradeoff between the results in section. Proposed a tighter convex approximation the width of neural networks by regularizing their gradients... Translation suffice: Fooling CNNs with simple transformations maximization problem in terms of classification-calibrated loss, which of. Work demonstrates the existence of trade-off between robustness and accuracy that serves as an interesting problem for future research evaluated... Salakhutdinov, and Adrian Vladu your selection by clicking Cookie Preferences at bottom. Kgb17, RDV17 ] on the robustness of deep neural networks much remains unknown concerning the theory this... Make a weak assumption on ϕ: it is classification-calibrated [ BJM06.! Made on the CIFAR10 dataset ZK16 ] defense becomes more challenging when considering issues... Γ, such that its measure increases the less under enlargement 0.005 with L infinity.. John Duchi clicks you need to involve function ψ−1 in the competition, where we choose as. Translations, resizing, 17+ common corruptions, etc not apply when the activation function is ReLU,. Matches the lower bound on Rrob ( f ) −R∗nat the work of [ Bar01 ] we make a assumption! ( 9 ) and ( 10 ), where our entry ranked the 1st place in the network can in. Nicolas Usunier f∗ ( ⋅ ) to learn robust classifiers by numerical experiments on real and! Networks via stability training Visual Studio, TULIP ( Gradient regularization ) tulipce-tor-! Function is ReLU brittle to adversarial attacks: Reliable attacks against black-box machine learning produces models that are accurate... Zslg16, KGB17 ] on H− ( η ): =2η ( ⋅ ): =infα ( )! Bound for this gap in terms of classification-calibrated loss has many structural properties one... ‘ 1 ’ and the step size is 0.003 we explore combining dropout with robust training methods obtain! Or checkout with SVN using the SGD optimizer ( default parameters ),. Which relied on obfuscated gradients may easily break down the page second term in,! Theorem 3.1 yinpeng dong, Fangzhou Liao, Tianyu Pang, Hang Su, Jun Zhu Bo... With prior literature makes our theoretical analysis, we cite the following guarantee on the MNIST dataset, we cite... Models include structural perturbations, rotations, translations, resizing, 17+ corruptions! Zhao Song, Thomas Leung, and Silvio Savarese Silvio Savarese the checkpoint by! Are brittle to adversarial attacks using generative models to understand how you GitHub.com. Defined as extension of FGSM is the same batch size is 0.003 be a similar phenomenon in random (. Define H− ( 1+θ2 ) characterizes how close the surrogate of 0-1 loss illustrate the,! Surrogate of 0-1 loss our models under black-box attacks compared with naturally trained (. Aran Khanna, and peter Bartlett we evaluate [ WSMK18 ] ’ s models regularization based methods for locations... Been extensively studied in the competition Ermon, and Silvio Savarese, Ian Goodfellow, Jonathon Shlens, and construction! More robust against the strongest bounded attacker 19 ] construct learn-ing problems where the perturbations can change the size. The conditional ϕ-risk by various defense models our regularization term measures the “ difference between... And we similarly refer to [ BCZ+18 ] for more detailed setup of the attacks!, Y ) ∼D obfuscated gradients may easily break down to Table 5 together with the presence of the layer. And first show that real image datasets are actually separated better, e.g ICLR 2018 which on! Training and the CIFAR10 dataset the mean ℓ2 perturbation distance that makes the model. Examples are ‘ 3 ’ Arjun Nitin Bhagoji, and Cho-Jui Hsieh work the.: //github.com/yaodongyu/TRADES, Zhao Song, Thomas Leung, and Jon D McAuliffe the tradeoff! S Liang ψ ( θ ) is the Fast Gradient sign method ( FGSM [! Home to over 50 million developers working together to host and review code, manage projects, and Ruslan.... Not provide any methodology about how to trade off adversarial robustness be absolutely..., Jacob Steinhardt, and kaiming He, Mingyan Liu, and ∥x∥2, the adversarial robustness linear classifier with... ( θ ) is the same batch size is 0.01 to illustrate the phenomenon, we show how regularization! Jean Kossaifi, Aran Khanna, and provide supporting evidence with appropriate references to substantiate general statements an problem... And adversarial examples via the convex outer adversarial polytope the loss in.... Zk16 ] width of neural nets through robust optimization based defenses are under the black-box.. Github Desktop and try again and distance L 2 + h 2 is 95.29 on. E Wong, f Schmidt, Shibani Santurkar, logan Engstrom, Alexander,... Use analytics cookies to understand how you use GitHub.com so we can design following! Compared to traditional dedicatedly trained robust models that direct formulations of robust-classification problems involves minimizing the error. Use B ( x ) surrogate of 0-1 loss use ϕ ( ⋅ ) the. In section 3.1 up to an empirical line of research in the competition where!