The fragility of modern machine learning models has drawn a considerable amount of attention from both academia and the public. While immense interests were in either crafting adversarial attacks as a way to measure the robustness of neural networks or devising worst-case analytical robustness verification with guarantees, few methods could enjoy both scalability and robustness guarantees at the same time. As an alternative to these attempts, randomized smoothing adopts a different prediction rule that enables statistical robustness arguments which easily scale to large networks. However, in this paper, we point out the side effects of current randomized smoothing workflows. Specifically, we articulate and prove two major points: 1) the decision boundaries of smoothed classifiers will shrink, resulting in disparity in class-wise accuracy; 2) applying noise augmentation in the training process does not necessarily resolve the shrinking issue due to the inconsistent learning objectives.
Carnegie Mellon University（卡内基梅隆大学）；Bosch Center for AI（博世人工智能中心）
We show how to turn any classifier that classifies well under Gaussian noise into a new classifier that is certifiably robust to adversarial perturbations under the $\ell_2$ norm. This "randomized smoothing" technique has been proposed recently in the literature, but existing guarantees are loose. We prove a tight robustness guarantee in $\ell_2$ norm for smoothing with Gaussian noise. We use randomized smoothing to obtain an ImageNet classifier with e.g. a certified top-1 accuracy of 49% under adversarial perturbations with $\ell_2$ norm less than 0.5 (=127/255). No certified defense has been…
Recent advances in machine learning (ML) algorithms, especially deep neural networks (DNNs), have demonstrated remarkable success (sometimes exceeding human-level performance) on several tasks, including face and speech recognition. However, ML algorithms are vulnerable to \emph{adversarial attacks}, such test-time, training-time, and backdoor attacks. In test-time attacks an adversary crafts adversarial examples, which are specially crafted perturbations imperceptible to humans which, when added to an input example, force a machine learning model to misclassify the given input example. Advers…
Randomized smoothing is a popular way of providing robustness guarantees against adversarial attacks: randomly-smoothed functions have a universal Lipschitz-like bound, allowing for robustness certificates to be easily computed. In this work, we show that there also exists a universal curvature-like bound for Gaussian random smoothing: given the exact value and gradient of a smoothed function, we compute a lower bound on the distance of a point to its closest adversarial example, called the Second-order Smoothing (SoS) robustness certificate. In addition to proving the correctness of this nove…
Randomized smoothing is the current state-of-the-art defense with provable robustness against $\ell_2$ adversarial attacks. Many works have devised new randomized smoothing schemes for other metrics, such as $\ell_1$ or $\ell_\infty$; however, substantial effort was needed to derive such new guarantees. This begs the question: can we find a general theory for randomized smoothing? We propose a novel framework for devising and analyzing randomized smoothing schemes, and validate its effectiveness in practice. Our theoretical contributions are: (1) we show that for an appropriate notion of "opti…
MIT（麻省理工学院）；MIT-IBM Watson AI Lab, IBM Research（麻省理工学院-IBM Watson人工智能实验室，IBM研究）
Randomized smoothing is a recently proposed defense against adversarial attacks that has achieved SOTA provable robustness against $\ell_2$ perturbations. A number of publications have extended the guarantees to other metrics, such as $\ell_1$ or $\ell_\infty$, by using different smoothing measures. Although the current framework has been shown to yield near-optimal $\ell_p$ radii, the total safety region certified by the current framework can be arbitrarily small compared to the optimal. In this work, we propose a framework to improve the certified safety region for these smoothed classifiers…
Mila（米拉）；University of Texas at Austin（得克萨斯大学奥斯汀分校）；Peking University（北京大学）
Randomized classifiers have been shown to provide a promising approach for achieving certified robustness against adversarial attacks in deep learning. However, most existing methods only leverage Gaussian smoothing noise and only work for $\ell_2$ perturbation. We propose a general framework of adversarial certification with non-Gaussian noise and for more general types of attacks, from a unified functional optimization perspective. Our new framework allows us to identify a key trade-off between accuracy and robustness via designing smoothing distributions, helping to design new families of n…
Randomized smoothing is a recent and celebrated solution to certify the robustness of any classifier. While it indeed provides a theoretical robustness against adversarial attacks, the dimensionality of current classifiers necessarily imposes Monte Carlo approaches for its application in practice. This paper questions the effectiveness of randomized smoothing as a defense, against state of the art black-box attacks. This is a novel perspective, as previous research works considered the certification as an unquestionable guarantee. We first formally highlight the mismatch between a theoretical …
University of Toronto（多伦多大学）；State University of New York at Buffalo（纽约州立大学布法罗分校）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
As a certified defensive technique, randomized smoothing has received considerable attention due to its scalability to large datasets and neural networks. However, several important questions remain unanswered, such as (i) whether the Gaussian mechanism is an appropriate option for certifying $\ell_2$-norm robustness, and (ii) whether there is an appropriate randomized (smoothing) mechanism to certify $\ell_\infty$-norm robustness. To shed light on these questions, we argue that the main difficulty is how to assess the appropriateness of each randomized mechanism. In this paper, we propose a g…
Randomized smoothing is a general technique for computing sample-dependent robustness guarantees against adversarial attacks for deep classifiers. Prior works on randomized smoothing against L_1 adversarial attacks use additive smoothing noise and provide probabilistic robustness guarantees. In this work, we propose a non-additive and deterministic smoothing method, Deterministic Smoothing with Splitting Noise (DSSN). To develop DSSN, we first develop SSN, a randomized method which involves generating each noisy smoothing sample by first randomly splitting the input space and then returning a …
School of Computation, Information & Technology, Technical Univ. of Munich（慕尼黑工业大学计算、信息与技术学院）；Fraunhofer Institute for Cognitive Systems IKS（弗劳恩霍夫认知系统研究所IKS）
Patch adversarial attacks on images, in which the attacker can distort pixels within a region of bounded size, are an important threat model since they provide a quantitative model for physical adversarial attacks. In this paper, we introduce a certifiable defense against patch attacks that guarantees for a given image and patch attack size, no patch adversarial examples exist. Our method is related to the broad class of randomized smoothing robustness schemes which provide high-confidence probabilistic robustness certificates. By exploiting the fact that patch attacks are more constrained tha…
In the last couple of years, several adversarial attack methods based on different threat models have been proposed for the image classification problem. Most existing defenses consider additive threat models in which sample perturbations have bounded L_p norms. These defenses, however, can be vulnerable against adversarial attacks under non-additive threat models. An example of an attack method based on a non-additive threat model is the Wasserstein adversarial attack proposed by Wong et al. (2019), where the distance between an image and its adversarial example is determined by the Wasserste…
National Technical University of Athens（希腊雅典国家技术大学）；Université Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CSX-AI（CSX-AI公司）；Carl von Ossietzky University of Oldenburg（奥尔登堡卡尔·冯·奥西特齐大学）；Chalmers University of Technology（查尔姆斯理工大学）
Recent works have shown the effectiveness of randomized smoothing as a scalable technique for building neural network-based classifiers that are provably robust to $\ell_2$-norm adversarial perturbations. In this paper, we employ adversarial training to improve the performance of randomized smoothing. We design an adapted attack for smoothed classifiers, and we show how this attack can be used in an adversarial training setting to boost the provable robustness of smoothed classifiers. We demonstrate through extensive experimentation that our method consistently outperforms all existing provabl…
Center for Data Science, Peking University（北京大学数据科学中心）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）；National School of Development, Peking University（北京大学国家发展研究院）；Beijing International Center for Mathematical Research（北京国际数学研究中心）；Institute for Artificial Inteligence, Peking University（北京大学人工智能研究所）
Randomized smoothing has achieved state-of-the-art certified robustness against $l_2$-norm adversarial attacks. However, it is not wholly resolved on how to find the optimal base classifier for randomized smoothing. In this work, we employ a Smoothed WEighted ENsembling (SWEEN) scheme to improve the performance of randomized smoothed classifiers. We show the ensembling generality that SWEEN can help achieve optimal certified robustness. Furthermore, theoretical analysis proves that the optimal SWEEN model can be obtained from training under mild assumptions. We also develop an adaptive predict…
Randomized smoothing, a method to certify a classifier's decision on an input is invariant under adversarial noise, offers attractive advantages over other certification methods. It operates in a black-box and so certification is not constrained by the size of the classifier's architecture. Here, we extend the work of Li et al. \cite{li2018second}, studying how the choice of divergence between smoothing measures affects the final robustness guarantee, and how the choice of smoothing measure itself can lead to guarantees in differing threat models. To this end, we develop a method to certify ro…
We present a method for provably defending any pretrained image classifier against $\ell_p$ adversarial attacks. This method, for instance, allows public vision API providers and users to seamlessly convert pretrained non-robust classification services into provably robust ones. By prepending a custom-trained denoiser to any off-the-shelf image classifier and using randomized smoothing, we effectively create a new classifier that is guaranteed to be $\ell_p$-robust to adversarial examples, without modifying the pretrained classifier. Our approach applies to both the white-box and the black-box…
Pennsylvania State University（宾夕法尼亚州立大学）；Microsoft Corporation（微软公司）；University of California, Los Angeles（加州大学洛杉矶分校）；Tsinghua University（清华大学）
Adversarial training is so far the most effective strategy in defending against adversarial examples. However, it suffers from high computational costs due to the iterative adversarial attacks in each training step. Recent studies show that it is possible to achieve fast Adversarial Training by performing a single-step attack with random initialization. However, such an approach still lags behind state-of-the-art adversarial training algorithms on both stability and model robustness. In this work, we develop a new understanding towards Fast Adversarial Training, by viewing random initializatio…
Despite the vulnerability of object detectors to adversarial attacks, very few defenses are known to date. While adversarial training can improve the empirical robustness of image classifiers, a direct extension to object detection is very expensive. This work is motivated by recent progress on certified classification by randomized smoothing. We start by presenting a reduction from object detection to a regression problem. Then, to enable certified regression, where standard mean smoothing fails, we propose median smoothing, which is of independent interest. We obtain the first model-agnostic…
The study of provable adversarial robustness has mostly been limited to classification tasks and models with one-dimensional real-valued outputs. We extend the scope of certifiable robustness to problems with more general and structured outputs like sets, images, language, etc. We model the output space as a metric space under a distance/similarity function, such as intersection-over-union, perceptual similarity, total variation distance, etc. Such models are used in many machine learning problems like image segmentation, object detection, generative models, image/audio-to-text systems, etc. B…
Technion – Israel Institute of Technology（以色列理工学院）；Intel – Artificial Intelligence Products Group (AIPG)（英特尔-人工智能产品集团(AIPG)）
Deep neural networks are known to be vulnerable to adversarial attacks. Current methods of defense from such attacks are based on either implicit or explicit regularization, e.g., adversarial training. Randomized smoothing, the averaging of the classifier outputs over a random distribution centered in the sample, has been shown to guarantee the performance of a classifier subject to bounded perturbations of the input. In this work, we study the application of randomized smoothing as a way to improve performance on unperturbed data as well as to increase robustness to adversarial attacks. The p…
Randomized smoothing has been shown to provide good certified-robustness guarantees for high-dimensional classification problems. It uses the probabilities of predicting the top two most-likely classes around an input point under a smoothing distribution to generate a certified radius for a classifier's prediction. However, most smoothing methods do not give us any information about the confidence with which the underlying classifier (e.g., deep neural network) makes a prediction. In this work, we propose a method to generate certified radii for the prediction confidence of the smoothed classi…
National University of Singapore（新加坡国立大学）；Technical University of Munich（慕尼黑工业大学）
The current state-of-the-art defense methods against adversarial examples typically focus on improving either empirical or certified robustness. Among them, adversarially trained (AT) models produce empirical state-of-the-art defense against adversarial examples without providing any robustness guarantees for large classifiers or higher-dimensional inputs. In contrast, existing randomized smoothing based models achieve state-of-the-art certified robustness while significantly degrading the empirical robustness against adversarial examples. In this paper, we propose a novel method, called \emph…
Randomized smoothing, using just a simple isotropic Gaussian distribution, has been shown to produce good robustness guarantees against $\ell_2$-norm bounded adversaries. In this work, we show that extending the smoothing technique to defend against other attack models can be challenging, especially in the high-dimensional regime. In particular, for a vast class of i.i.d.~smoothing distributions, we prove that the largest $\ell_p$-radius that can be certified decreases as $O(1/d^{\frac{1}{2} - \frac{1}{p}})$ with dimension $d$ for $p > 2$. Notably, for $p \geq 2$, this dependence on $d$ is no …
Huawei Moscow Research Center（华为莫斯科研究中心）；Lomonosov Moscow State University（罗蒙诺索夫莫斯科国立大学）；Nuro, Inc.（Nuro公司）
Currently the most popular method of providing robustness certificates is randomized smoothing where an input is smoothed via some probability distribution. We propose a novel approach to randomized smoothing over multiplicative parameters. Using this method we construct certifiably robust classifiers with respect to a gamma correction perturbation and compare the result with classifiers obtained via other smoothing distributions (Gaussian, Laplace, uniform). The experiments show that asymmetrical Rayleigh distribution allows to obtain better certificates for some values of perturbation parame…
King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）；University of Oxford（牛津大学）
Deep neural networks are vulnerable to input deformations in the form of vector fields of pixel displacements and to other parameterized geometric deformations e.g. translations, rotations, etc. Current input deformation certification methods either 1. do not scale to deep networks on large input datasets, or 2. can only certify a specific class of deformations, e.g. only rotations. We reformulate certification in randomized smoothing setting for both general vector field and parameterized deformations and propose DeformRS-VF and DeformRS-Par, respectively. Our new formulation scales to large …
We extend randomized smoothing to cover parameterized transformations (e.g., rotations, translations) and certify robustness in the parameter space (e.g., rotation angle). This is particularly challenging as interpolation and rounding effects mean that image transformations do not compose, in turn preventing direct certification of the perturbed image (unlike certification with $\ell^p$ norms). We address this challenge by introducing three different kinds of defenses, each with a different guarantee (heuristic, distributional and individual) stemming from the method used to bound the interpol…
Institute of Information Science, Academia Sinica（中央研究院资讯科学研究所）
Randomized smoothing has established state-of-the-art provable robustness against $\ell_2$ norm adversarial attacks with high probability. However, the introduced Gaussian data augmentation causes a severe decrease in natural accuracy. We come up with a question, "Is it possible to construct a smoothed classifier without randomization while maintaining natural accuracy?". We find the answer is definitely yes. We study how to transform any classifier into a certified robust classifier based on a popular and elegant mathematical tool, Bernstein polynomial. Our method provides a deterministic alg…
Backdoor attack is a severe security threat to deep neural networks (DNNs). We envision that, like adversarial examples, there will be a cat-and-mouse game for backdoor attacks, i.e., new empirical defenses are developed to defend against backdoor attacks but they are soon broken by strong adaptive backdoor attacks. To prevent such cat-and-mouse game, we take the first step towards certified defenses against backdoor attacks. Specifically, in this work, we study the feasibility and effectiveness of certifying robustness against backdoor attacks using a recent technique called randomized smooth…
It is well-known that classifiers are vulnerable to adversarial perturbations. To defend against adversarial perturbations, various certified robustness results have been derived. However, existing certified robustnesses are limited to top-1 predictions. In many real-world applications, top-$k$ predictions are more relevant. In this work, we aim to derive certified robustness for top-$k$ predictions. In particular, our certified robustness is based on randomized smoothing, which turns any classifier to a new classifier via adding noise to an input example. We adopt randomized smoothing because…
King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）；University of Oxford（牛津大学）
Randomized smoothing is a recent technique that achieves state-of-art performance in training certifiably robust deep neural networks. While the smoothing family of distributions is often connected to the choice of the norm used for certification, the parameters of these distributions are always set as global hyper parameters independent from the input data on which a network is certified. In this work, we revisit Gaussian randomized smoothing and show that the variance of the Gaussian distribution can be optimized at each input so as to maximize the certification radius for the construction o…
Institut de Recherche Technologique SystemX（SystemX技术研究院）；Univ. Bordeaux（波尔多大学）；CNRS（法国国家科学研究中心）；Bordeaux INP（波尔多国立理工学院）；IMB（波尔多数学研究所）
Certification of neural networks is an important and challenging problem that has been attracting the attention of the machine learning community since few years. In this paper, we focus on randomized smoothing (RS) which is considered as the state-of-the-art method to obtain certifiably robust neural networks. In particular, a new data-dependent RS technique called ANCER introduced recently can be used to certify ellipses with orthogonal axis near each input data of the neural network. In this work, we remark that ANCER is not invariant under rotation of input data and propose a new rotationa…
Technical University of Munich（慕尼黑工业大学）；University of Maryland, College Park（马里兰大学帕克分校）
Randomized Smoothing (RS), being one of few provable defenses, has been showing great effectiveness and scalability in terms of defending against $\ell_2$-norm adversarial perturbations. However, the cost of MC sampling needed in RS for evaluation is high and computationally expensive. To address this issue, we investigate the possibility of performing randomized smoothing and establishing the robust certification in the latent space of a network, so that the overall dimensionality of tensors involved in computation could be drastically reduced. To this end, we propose Latent Space Randomized …
We present a new certification method for image and point cloud segmentation based on randomized smoothing. The method leverages a novel scalable algorithm for prediction and certification that correctly accounts for multiple testing, necessary for ensuring statistical guarantees. The key to our approach is reliance on established multiple-testing correction mechanisms as well as the ability to abstain from classifying single pixels or points while still robustly segmenting the overall input. Our experimental evaluation on synthetic data and challenging datasets, such as Pascal Context, Citysc…
Certified Robustness under Heterogeneous Perturbations via Hybrid Randomized Smoothing
Department of Computer Science, School of Computing, Institute of Science Tokyo, Tokyo, Japan（东京科学研究所计算机科学系）；College of Computing and Data Science, Nanyang Technological University, Singapore（南洋理工大学计算机与数据科学学院）；PSL University, Paris, France（巴黎高等师范学院）
