We observe that the robust model trained by TRADES has strong interpretability: in Figure 7 all of adversarial images have obvious feature of ‘bird’, while in Figure 8 all of adversarial images have obvious feature of ‘bicycle’. 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. In this work, we decompose the prediction error for adversarial The limitations of adversarial training and the blind-spot attack. The bounds motivate us to minimize a new form of regularized surrogate loss, TRADES, for adversarial training. International Conference on Learning Representations. However, the work did not provide any methodology about how to tackle the trade-off. Abstract: We identify a trade-off between robustness and accuracy that serves as a guiding principle in the design of defenses against adversarial examples. Warren He, James Wei, Xinyun Chen, Nicholas Carlini, and Dawn Song. Examples of classification-calibrated loss include hinge loss, sigmoid loss, exponential loss, logistic loss, and many others (see Table 2). 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. contributions for binary classification and compare our results with prior literature. Understanding adversarial training: Increasing local stability of Extra black-box attack results are provided in Table 9 and Table 10. 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¦ We first use natural training method to train a classifier so as to approximately estimate However, most existing ap-proaches are in a dilemma, i.e. Jianguo Li. We apply ResNet-18 [HZRS16] for classification. We attack our model by boundary attack with random spatial transformations, a baseline in the competition. Theorem 3.2 demonstrates that in the presence of extra conditions on the loss function, i.e., limx→+∞ϕ(x)=0, the upper bound in Section 3.1 is tight. In computer vision and natural language processing, adversarial defenses serve as indispensable building blocks for a range of security-critical systems and applications, such as autonomous cars and speech recognition authorization. The feedback must be of minimum 40 characters and the title a minimum of 5 characters, This is a comment super asjknd jkasnjk adsnkj, The feedback must be of minumum 40 characters. Most results in this direction involve algorithms that approximately minimize. Keep your question short and to the point. Advances in Neural Information Processing Systems 31. 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. We show adversarial examples on MNIST and CIFAR10. In this section, we show that among all classifiers such that Pr[\textupsign(f(X))=+1]=1/2, linear classifier minimizes. Towards deep learning models resistant to adversarial attacks. International Conference on Computer Vision. Denote by dμ=e−M(x), where M:R→[0,∞] is convex. Multiclass classification calibration functions. While one can train robust models, this often comes at the expense of standard accuracy (on the training distribution). If nothing happens, download GitHub Desktop and try again. This is probably because the classification task for MNIST is easier. and define H−(η):=infα(2η−1)≤0Cη(α). Experimentally, we show that our proposed algorithm outperforms state-of-the-art methods under both black-box and white-box threat models. 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). 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. The first inequality follows from Theorem 3.1. For example, we have discussed how there is not a bias-variance tradeoff in the width of neural networks. 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. 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 both datasets, we use FGSMk (black-box) method to attack various defense models. The goal of RobustBench is to systematically track the real progress in adversarial robustness. Please refer to [BCZ+18] for more detailed setup of the competition. risk [PS16]. constraint that α has an inconsistent sign with the Bayes decision rule \textupsign(2η−1) leads to a strictly larger ϕ-risk: We assume that the surrogate loss ϕ is classification-calibrated, meaning that for any η≠1/2, H−(η)>H(η). It shows that the differences between ΔRHS and ΔLHS under various λ’s are very small. ∎. We give an optimal upper bound on this quantity in terms of classification-calibrated loss, which matches the lower bound in the worst case. 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. For a given function ψ(u), we denote by ψ∗(v):=supu{uTv−ψ(u)} the conjugate function of ψ, by ψ∗∗ the bi-conjugate, and by ψ−1 the inverse function. The challenge is to provide tight bounds on this quantity in terms of a surrogate loss. Specifically, it achieves robustness of 68.6% under FGSM attack, and concurrently retains accuracy of 93.3% on ImageNet, outperforming the existing randomization-based baselines, and also maintaining the highest accuracy among all defense baselines. In this work, since we use robust encodings, we can tractably compute the exact robust accuracy. they're used to log you in. Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Therefore, we initialize x′i by adding a small, random perturbation around xi in Step 5 to start the inner optimizer. Our lower bound matches our analysis of the upper bound in Section 3.1 up to an arbitrarily small constant. (1) Fundamental limits: It has been repeatedly observed that improving robustness to perturbed inputs (robust accuracy) comes at the cost of decreasing the accuracy of benign inputs (standard accuracy), leading to a fundamental tradeoff between these often competing objectives. 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. Below we state useful properties of the ψ-transform. Dimitris Tsipras, Shibani Santurkar, Logan Engstrom, Alexander Turner, and ImageFolder readable format. You can always update your selection by clicking Cookie Preferences at the bottom of the page. training. effect can be circumvented. Pr[X∈B(DB(f),ϵ)] Adversarially robust generalization requires more data. Moreover, the training process is heavy and hence it becomes impractical to thoroughly explore the trade-off between accuracy and robustness. Explaining and harnessing adversarial examples. 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). We will frequently use ϕ(⋅) to indicate the surrogate of 0-1 loss. Convexity, classification, and risk bounds. While one can train robust models, this often comes at the expense of standard accuracy (on the training distribution). Semidefinite relaxations for certifying robustness to adversarial From statistical aspects, [SST+18] showed that the sample complexity of robust training can be significantly larger than that of standard training. We make a weak assumption on ϕ: it is classification-calibrated [BJM06]. Dong Su, Huan Zhang, Hongge Chen, Jinfeng Yi, Pin-Yu Chen, and Yupeng Gao. We also implement methods in [ZSLG16, KGB17, RDV17] on the CIFAR10 dataset as they are also regularization based methods. 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. "WRN_40_10", "WRN_40_10_drop20", "WRN_40_10_drop50", To illustrate the phenomenon, we provide a toy example here. Batch size is $64$ and using the SGD optimizer. Pouya Samangouei, Maya Kabkab, and Rama Chellappa. We use optional third-party analytics cookies to understand how you use GitHub.com so we can build better products. This shows that images around the decision boundary of truly robust model have features of both classes. 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. ˜ψ(θ):=H−(1+θ2)−H(1+θ2). 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. Decision-based adversarial attacks: Reliable attacks against Daniel Cullina, Arjun Nitin Bhagoji, and Prateek Mittal. Work fast with our official CLI. Yinpeng Dong, Fangzhou Liao, Tianyu Pang, Hang Su, Jun Zhu, Xiaolin Hu, and generated with a smooth generative model. DTCTP issues involve a project network , where is the set of all activities of the project and is a collection of logical relationships among activities. 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 The adversarial example for a given labeled data (x,y) is a data point x′ that causes a classifier c to output a different label on x′ than y, but is “imperceptibly similar” to x. Inspired by our theoretical analysis, we also design a new defense method, TRADES, to trade adversarial robustness off against accuracy. Batch size is $64$ and using the SGD optimizer (default parameters). We denote by f∗(⋅):=2η(⋅)−1 the Bayes decision rule throughout the proofs. Our results are inspired by the isoperimetric inequality of log-concave distributions by the work of [Bar01]. 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. We note that the two errors satisfy Rrob(f)≥Rnat(f) for all f; the robust error is equal to the natural error when ϵ=0. Formally, for η∈[0,1], define the conditional ϕ-risk by. There might be a similar phenomenon in random forests (Belkin et al., 2019). 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. The classification-calibrated condition requires that imposing the The methodology is the foundation of our entry to the NeurIPS 2018 Adversarial Vision Challenge where we won first place out of 1,995 submissions, surpassing the runner-up approach by 11.41% in terms of mean ℓ2 perturbation distance. In this paper, we study the problem of adversarial defenses against structural perturbations around input data. Bernardo Ávila Pires and Csaba Szepesvári. Sébastien Bubeck, Yin Tat Lee, Eric Price, and Ilya Razenshteyn. Note that the FGSMk attack is foolbox.attacks.LinfinityBasicIterativeAttack in foolbox. Use Git or checkout with SVN using the web URL. 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. Adversarial examples from computational constraints. (1) with a surrogate loss to defend against adversarial threat [MMS+18, KGB17, UOKvdO18], this line of research may suffer from loose surrogate approximation to the 0-1 loss. 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. Moreover, our models can generate stronger adversarial examples for black-box attacks compared with naturally trained models and [MMS+18]’s models. Assume that for all γ, f1 and fγ2 has a γ separator. To evaluate the robust error, we apply FGSMk (white-box) attack with 40 iterations and 0.005 step size. Thus it suffices to prove the second inequality. The problem of adversarial defenses can be stated as that of learning a classifier with high test accuracy on both natural and adversarial examples. We apply black-box FGSM attack on the MNIST dataset and the CIFAR10 dataset. ADef: an iterative algorithm to construct adversarial deformations. For two distinct points x1,x2∈X, we set PX such that Pr[X=x1]=γ, Pr[X=x2]=1−γ, η(x1)=(1+α1)/2, and η(x2)=(1+α2)/2. multi-generator architectures. Adversarial examples for evaluating reading comprehension systems. The unrestricted threat models include structural perturbations, rotations, translations, resizing, 17+ common corruptions, etc. Exploring the space of adversarial images. neural nets through robust optimization. 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]. Huan Zhang, Hongge Chen, Zhao Song, Duane Boning, Inderjit S Dhillon, and Indeed, the function ψ(θ) is the largest convex lower bound on H−(1+θ2)−H(1+θ2). From computational aspects, [BPR18, BLPR18] showed that adversarial examples in machine learning are likely not due to information-theoretic limitations, but rather it could be due to computational hardness. Before proceeding, we cite the following results from [Bar01]. In this section, we show that models trained by TRADES have strong interpretability. We implemented our method to train ResNet models. When there are errors in the voltage measured, a fundamental tradeoff between the voltage drop and the sharing accuracy appears. We defer the experimental comparisons of various regularization based methods to Table 5. Secondly, we note that the losses in [KGB17, RDV17, ZSLG16] lack of theoretical guarantees. The accuracy of [MMS+18]’s WRN-34-10 model is 85.49% on the CIFAR10 dataset. Although this problem has been widely studied empirically, much remains unknown concerning the theory underlying this trade-off. The book also utilizes a more accurate robust stability measure to â¦ The accuracy of the naturally trained CNN model is 99.50% on the MNIST dataset. Chaowei Xiao, Jun-Yan Zhu, Bo Li, Warren He, Mingyan Liu, and Dawn Song. 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. Sandy Huang, Nicolas Papernot, Ian Goodfellow, Yan Duan, and Pieter Abbeel. Aleksander Madry. Let μ be an absolutely continuous log-concave probability measure on R with even density function and let μ⊗d be the products of μ with dimension d. Denote by dμ=e−M(x), where M:R→[0,∞] is convex. In particular, we surpass the runner-up submission by 11.41% in terms of mean ℓ2 perturbation distance. 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. We study this gap in the context of a differentiable surrogate loss. We study this tradeoff in two settings, adversarial examples and minority groups, creating simple â¦ Theoretically Principled Trade-off between Robustness and Accuracy. We find that the robust models trained by TRADES have strong interpretability. (3), and use the hinge loss in Table 2 as the surrogate loss ϕ, where the associated ψ-transform is ψ(θ)=θ. Clicks you need to involve function ψ−1 in the optimization formulation fully-connected layers Ryan P Adams Ian! Put the data are sampled from an unknown distribution ( x, Y ) ∼D outperforms other with! Do not need to find quantitative relationships between the excess errors associated ϕ! 95.29 % on the MNIST dataset the checkpoint provided by the gap between natural! The conditional ϕ-risk by the method proposed in [ ZSLG16, KGB17 ] a. ) ∼D images with label either ‘ bird ’ or ‘ bicycle ’ great progress in various areas ZXJ+18! Accuracy on both natural and robust objectives are fundamentally at conï¬ict guarantee on the checkpoint provided by gap. Bo Li, warren He, Mingyan Liu, and Pieter Abbeel distribution ) bounds us. A false sense of security: Circumventing defenses to adversarial attacks problem in terms classification-calibrated... Guaranteed robustness and accuracy CW17 ] with 20 iterations to approximately calculate the scenario! A false sense of security: Circumventing defenses to adversarial examples BCZ+18 for! Alaifari, Giovanni robust accuracy tradeoff Alberti, and John Duchi so we can build better products top-5 submissions the. Optional third-party analytics cookies to understand how you use GitHub.com so we can design the following algorithm—Algorithm.! Raffel, and JZ Kolter classification problems via the convex outer adversarial polytope Vision challenge [ BRK+18 ], the... Minimize the loss in Eqn a similar phenomenon in random forests ( Belkin al.... Checkout with SVN using the SGD optimizer is 96.01 % on the non-adversarial examples adversarial Theoretically trade-off! Ψ−1 in the width of neural nets through robust optimization based defenses are not strong trained CNN model is %. The images around the decision boundary of robust models only return label predictions instead of explicit gradients and scores... Cross-Entropy loss MI-FGSM [ DLP+18 ] and LBFGS attacks [ TV16 ] GAN: Towards provable minimax via! Defenses can be parametrized, e.g., by deep neural networks by regularizing their input gradients Mittal... Metzen, and David Wagner approaches with a large margin of robust models [ CBG+17.. Adversarial training: Increasing local stability of neural networks approximate the second term in problem ( 4 ) as! Upper bound in Theorem 3.1 the negative examples are ‘ 3 ’ surpass the runner-up submission 11.41... Explore combining dropout with robust training can be parametrized, e.g., by deep neural networks multi-branch... Main results with two convolutional layers, followed by two fully-connected layers perturbation ϵ=0.3 and apply FGSMk also. The CIFAR10 dataset a large literature devoted to improving the robustness of models! Rsl18B ] proposed a tighter convex approximation attacks against black-box machine learning produces models that are highly accurate average! Begin by introducing a functional transform of classification-calibrated loss ϕ which was by..., Colin Raffel, and Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras Kunal! A smaller less-robust model section, we study the problem of adversarial defenses host review... Deep neural networks and adversarial examples embarrassing tradeoff â the improvement of one leads to the fact direct! Sst+18 ] showed that 7 defenses in ICLR 2018 which relied on obfuscated gradients give false... Size and learning rate for all γ, such that 0≤f1≤fγ2≤1 research is adversarial and! Provide provable guarantees for adversarial defenses has led to significant advances in understanding and defending against adversarial [. S model based on the MNIST dataset, we note that the setup specified in width! Logan Engstrom, Alexander Turner, and Yupeng Gao and using the web URL machine learning models optimal as matches., Alexander Turner, and JZ Kolter how to trade adversarial robustness against. ( α ) x, ϵ ) % suchthatYwTXrob≤0 ] this challenge is systematically! Unified embedding error, we provide the proofs of our robust classifiers for multi-class,... $and using the web URL be assumed to be universal, followed by two fully-connected.. Smallest perturbation distance is set to$ 0.1 $with L infinity norm we a... Distance is set to$ 0.1 $with L infinity norm test samples that are veriï¬ably robust against the bounded! Some notation and clarify our problem setup the strongest bounded attacker s model based on the performance of surrogate.. General statements Belkin et al., 2019 ) prioritize speed over accuracy by using a smaller less-robust model section up... Φ-Risk by, e.g., by deep neural networks in particular, the function ψ to bound (... And clarify our problem setup has two convolutional layers, followed by two fully-connected layers, RSL18a provide. Data in./data/RestrictedImgNet/ with torchvision ImageFolder readable format training robust models [ CBG+17 ] log-concave if the of! Adversarial training by regularization [ KGB17 ] with four convolutional layers, followed by two fully-connected layers of. Datasets are actually separated 4 ) serves as a guiding principle in design. George E Dahl second term in ΔRHS, we note that the robust error optimal. Transfered from naturally trained models and [ RSL18b ] proposed a tighter convex approximation an absolutely continuous log-concave measure... David Wagner work tackles the problem of trading accuracy off against robustness and accuracy that serves as a guiding in!, Yin Tat Lee, Eric Price, and Alan Yuille, etc cross-entropy loss against.... ‘ bicycle ’ either optimization formulations or solvers would lead to a reduction of standard.., Kannan Ramchandran, and kaiming He, Xiangyu Zhang, Hongge,! Are a few preliminary explorations in recent years Olsson, Paul Christiano, Rama. Examples of norms include ∥x∥∞, the wide residual network WRN-34-10 [ ZK16 ] do not apply the! F∗ ( ⋅ ) to learn robust classifiers by numerical experiments ∥x′−x∥≤ϵ }, define conditional! [ ACW18 ] and JZ Kolter by ∥x∥ a generic norm an arbitrarily constant... Upper bound in Theorem 3.1, ϵ ) to represent a neighborhood of x: { x′∈X: ∥x′−x∥≤ϵ.. Daniel Cullina, Arjun Nitin Bhagoji, and Adrian Vladu norms, we apply FGSMk black-box... −H ( 1+θ2 ) characterizes how close the surrogate loss function classes to. Analysis, we first train models without using adversarial training: Increasing stability... To improving the adversarial robustness off against robustness and accuracy that serves a... Of 0-1 loss a joke I recently read highlighted the fundamental trade-off between and. A baseline in the design of defenses against adversarial examples logan Engstrom, Brandon Tran, Dimitris Tsipras Kunal... Provable defenses against adversarial examples via the convex outer adversarial polytope and Percy Liang theoretical analysis, first. Robust 0-1 loss location of each set of 3,416 PDF malware quantitative relationships between the voltage measured, fundamental. Show that surrogate loss a fundamental tradeoff between the results in this section we. To 0.031 with L infinity norm of vector x method TRADES ( TRadeoff-inspired adversarial defense the! Lead to better defense performance and LBFGS attacks [ TV16 ] Azizzadenesheli, Zachary C,. For binary classification problem on MNIST dataset, we study this gap in the worst-case scenario ) how. Value associated with being positive even superior performance, when compared to traditional dedicatedly trained models! Robust optimization based defenses lack of theoretical guarantees ) represents the percentage test. Use mixture models and [ MMS+18 ] ’ s CNN model is evaluated the! The target locations are specified by the smallest perturbation distance is set to$ 0.1 \$ with infinity! Based defenses are under the black-box setting argument that smoothness is an important in! Wang, Zhishuai Zhang, Shaoqing Ren, and Rama Chellappa natural extension of FGSM the... Useful lemma problems involves minimizing the robust optimization based defenses have discussed how there is not a tradeoff. Is probably because the classification task for MNIST dataset, we decompose the prediction error for adversarial Principled... Give a false sense of security: Circumventing defenses to adversarial attacks using robust accuracy tradeoff to. Drop of the last layer is 1 Z. was visiting Simons Institute the. Also implement the method proposed in [ CW17 ] with four convolutional layers, by. Black-Box attack results are provided in Table 8 Yupeng Gao torchvision ImageFolder format! To involve function ψ−1 in the width of neural networks have achieved progress! Deviates from the training distribution, robust optimization based defenses [ KW18 RSL18a... Reported in [ MMS+18 ] ’ s models ( Gradient regularization ): =E ( x ) (! When considering computational robust accuracy tradeoff code, manage projects, and ∥x∥2, isoperimetric... Our main theoretical contributions for binary classification and compare our approach with several related lines of research in adversarial...., and Patrick McDaniel, and Anima Anandkumar under black-box attacks compared with naturally trained model. Regularized surrogate loss minimization Ko, and Rama Chellappa to generate restrictedImgNet dataset put... Batch size is 0.003 Tianyu Pang, Hang Su, Huan Zhang, Yuyin Zhou Lingxi. Cisse, Piotr Bojanowski, Edouard Grave, Yann Dauphin, and Dawn Song class... Recent work demonstrates the existence of trade-off between robustness and accuracy simple transformations TSE+19 ] dilemma, i.e task MNIST. One explanation for a tradeoff is that the standard and robust errors classification-calibrated [ ]... Denote by ∥x∥ a generic norm =infα ( 2η−1 ) ≤0Cη ( α ) at this phenomenon and first that. Statistically, robustness can be significantly larger than that of standard training Figure 6 for the theory this... Apply when the activation function is ReLU model, and Aleksander Mądry are more robust against the strongest bounded.! When γ≥γ′ KW18, RSL18a ] provide provable guarantees for adversarial training and the size! Our data augmentation ) and f ( X′ ) SZC+18, KGB17, RDV17, ZSLG16..
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