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Register a new userAdversarial Robustness of Stabilized NeuralODEs Might be from Obfuscated Gradients
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In this paper we introduce a provably stable architecture for Neural Ordinary Differential Equations (ODEs) which achieves non-trivial adversarial robustness under white-box adversarial attacks even when the network is trained naturally. For most existing defense methods withstanding strong white-box attacks, to improve robustness of neural networks, they need to be trained adversarially, hence have to strike a trade-off between natural accuracy and adversarial robustness. Inspired by dynamical system theory, we design a stabilized neural ODE network named SONet whose ODE blocks are skew-symmetric and proved to be input-output stable. With natural training, SONet can achieve comparable robustness with the state-of-the-art adversarial defense methods, without sacrificing natural accuracy. Even replacing only the first layer of a ResNet by such a ODE block can exhibit further improvement in robustness, e.g., under PGD-20 ($ell_infty=0.031$) attack on CIFAR-10 dataset, it achieves 91.57% and natural accuracy and 62.35% robust accuracy, while a counterpart architecture of ResNet trained with TRADES achieves natural and robust accuracy 76.29% and 45.24%, respectively. To understand possible reasons behind this surprisingly good result, we further explore the possible mechanism underlying such an adversarial robustness. We show that the adaptive stepsize numerical ODE solver, DOPRI5, has a gradient masking effect that fails the PGD attacks which are sensitive to gradient information of training loss; on the other hand, it cannot fool the CW attack of robust gradients and the SPSA attack that is gradient-free. This provides a new explanation that the adversarial robustness of ODE-based networks mainly comes from the obfuscated gradients in numerical ODE solvers.
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We show a hardness result for random smoothing to achieve certified adversarial robustness against attacks in the $ell_p$ ball of radius $epsilon$ when $p>2$. Although random smoothing has been well understood for the $ell_2$ case using the Gaussian distribution, much remains unknown concerning the existence of a noise distribution that works for the case of $p>2$. This has been posed as an open problem by Cohen et al. (2019) and includes many significant paradigms such as the $ell_infty$ threat model. In this work, we show that any noise distribution $mathcal{D}$ over $mathbb{R}^d$ that provides $ell_p$ robustness for all base classifiers with $p>2$ must satisfy $mathbb{E}eta_i^2=Omega(d^{1-2/p}epsilon^2(1-delta)/delta^2)$ for 99% of the features (pixels) of vector $etasimmathcal{D}$, where $epsilon$ is the robust radius and $delta$ is the score gap between the highest-scored class and the runner-up. Therefore, for high-dimensional images with pixel values bounded in $[0,255]$, the required noise will eventually dominate the useful information in the images, leading to trivial smoothed classifiers.
Several recent results provide theoretical insights into the phenomena of adversarial examples. Existing results, however, are often limited due to a gap between the simplicity of the models studied and the complexity of those deployed in practice. In this work, we strike a better balance by considering a model that involves learning a representation while at the same time giving a precise generalization bound and a robustness certificate. We focus on the hypothesis class obtained by combining a sparsity-promoting encoder coupled with a linear classifier, and show an interesting interplay between the expressivity and stability of the (supervised) representation map and a notion of margin in the feature space. We bound the robust risk (to $ell_2$-bounded perturbations) of hypotheses parameterized by dictionaries that achieve a mild encoder gap on training data. Furthermore, we provide a robustness certificate for end-to-end classification. We demonstrate the applicability of our analysis by computing certified accuracy on real data, and compare with other alternatives for certified robustness.
Gaussian processes (GPs) enable principled computation of model uncertainty, making them attractive for safety-critical applications. Such scenarios demand that GP decisions are not only accurate, but also robust to perturbations. In this paper we present a framework to analyse adversarial robustness of GPs, defined as invariance of the models decision to bounded perturbations. Given a compact subset of the input space $Tsubseteq mathbb{R}^d$, a point $x^*$ and a GP, we provide provable guarantees of adversarial robustness of the GP by computing lower and upper bounds on its prediction range in $T$. We develop a branch-and-bound scheme to refine the bounds and show, for any $epsilon > 0$, that our algorithm is guaranteed to converge to values $epsilon$-close to the actual values in finitely many iterations. The algorithm is anytime and can handle both regression and classification tasks, with analytical formulation for most kernels used in practice. We evaluate our methods on a collection of synthetic and standard benchmark datasets, including SPAM, MNIST and FashionMNIST. We study the effect of approximate inference techniques on robustness and demonstrate how our method can be used for interpretability. Our empirical results suggest that the adversarial robustness of GPs increases with accurate posterior estimation.
Though deep neural networks have achieved significant progress on various tasks, often enhanced by model ensemble, existing high-performance models can be vulnerable to adversarial attacks. Many efforts have been devoted to enhancing the robustness of individual networks and then constructing a straightforward ensemble, e.g., by directly averaging the outputs, which ignores the interaction among networks. This paper presents a new method that explores the interaction among individual networks to improve robustness for ensemble models. Technically, we define a new notion of ensemble diversity in the adversarial setting as the diversity among non-maximal predictions of individual members, and present an adaptive diversity promoting (ADP) regularizer to encourage the diversity, which leads to globally better robustness for the ensemble by making adversarial examples difficult to transfer among individual members. Our method is computationally efficient and compatible with the defense methods acting on individual networks. Empirical results on various datasets verify that our method can improve adversarial robustness while maintaining state-of-the-art accuracy on normal examples.
We present a more general analysis of $H$-calibration for adversarially robust classification. By adopting a finer definition of calibration, we can cover settings beyond the restricted hypothesis sets studied in previous work. In particular, our results hold for most common hypothesis sets used in machine learning. We both fix some previous calibration results (Bao et al., 2020) and generalize others (Awasthi et al., 2021). Moreover, our calibration results, combined with the previous study of consistency by Awasthi et al. (2021), also lead to more general $H$-consistency results covering common hypothesis sets.
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