Alexander Levine

Alexander Levine, Soheil Feizi

Many applications of reinforcement learning can be formalized as goal-conditioned environments, where, in each episode, there is a "goal" that affects the rewards obtained during that episode but does not affect the dynamics. Various techniques have been proposed to improve performance in goal-conditioned environments, such as automatic curriculum generation and goal relabeling. In this work, we explore a connection between off-policy reinforcement learning in goal-conditioned settings and knowledge distillation. In particular: the current Q-value function and the target Q-value estimate are both functions of the goal, and we would like to train the Q-value function to match its target for all goals. We therefore apply Gradient-Based Attention Transfer (Zagoruyko and Komodakis 2017), a knowledge distillation technique, to the Q-function update. We empirically show that this can improve the performance of goal-conditioned off-policy reinforcement learning when the space of goals is high-dimensional. We also show that this technique can be adapted to allow for efficient learning in the case of multiple simultaneous sparse goals, where the agent can attain a reward by achieving any one of a large set of objectives, all specified at test time. Finally, to provide theoretical support, we give examples of classes of environments where (under some assumptions) standard off-policy algorithms such as DDPG require at least O(d^2) replay buffer transitions to learn an optimal policy, while our proposed technique requires only O(d) transitions, where d is the dimensionality of the goal and state space. Code is available at https://github.com/alevine0/ReenGAGE.

Alexander Levine, Soheil Feizi

In recent years, researchers have extensively studied adversarial robustness in a variety of threat models, including L_0, L_1, L_2, and L_infinity-norm bounded adversarial attacks. However, attacks bounded by fractional L_p "norms" (quasi-norms defined by the L_p distance with 0<p<1) have yet to be thoroughly considered. We proactively propose a defense with several desirable properties: it provides provable (certified) robustness, scales to ImageNet, and yields deterministic (rather than high-probability) certified guarantees when applied to quantized data (e.g., images). Our technique for fractional L_p robustness constructs expressive, deep classifiers that are globally Lipschitz with respect to the L_p^p metric, for any 0<p<1. However, our method is even more general: we can construct classifiers which are globally Lipschitz with respect to any metric defined as the sum of concave functions of components. Our approach builds on a recent work, Levine and Feizi (2021), which provides a provable defense against L_1 attacks. However, we demonstrate that our proposed guarantees are highly non-vacuous, compared to the trivial solution of using (Levine and Feizi, 2021) directly and applying norm inequalities. Code is available at https://github.com/alevine0/fractionalLpRobustness.

Wenxiao Wang, Alexander Levine, Soheil Feizi

Data poisoning considers an adversary that distorts the training set of machine learning algorithms for malicious purposes. In this work, we bring to light one conjecture regarding the fundamentals of data poisoning, which we call the Lethal Dose Conjecture. The conjecture states: If $n$ clean training samples are needed for accurate predictions, then in a size-$N$ training set, only $Θ(N/n)$ poisoned samples can be tolerated while ensuring accuracy. Theoretically, we verify this conjecture in multiple cases. We also offer a more general perspective of this conjecture through distribution discrimination. Deep Partition Aggregation (DPA) and its extension, Finite Aggregation (FA) are recent approaches for provable defenses against data poisoning, where they predict through the majority vote of many base models trained from different subsets of training set using a given learner. The conjecture implies that both DPA and FA are (asymptotically) optimal -- if we have the most data-efficient learner, they can turn it into one of the most robust defenses against data poisoning. This outlines a practical approach to developing stronger defenses against poisoning via finding data-efficient learners. Empirically, as a proof of concept, we show that by simply using different data augmentations for base learners, we can respectively double and triple the certified robustness of DPA on CIFAR-10 and GTSRB without sacrificing accuracy.

Priyatham Kattakinda, Alexander Levine, Soheil Feizi

Though the background is an important signal for image classification, over reliance on it can lead to incorrect predictions when spurious correlations between foreground and background are broken at test time. Training on a dataset where these correlations are unbiased would lead to more robust models. In this paper, we propose such a dataset called Diffusion Dreamed Distribution Shifts (D3S). D3S consists of synthetic images generated through StableDiffusion using text prompts and image guides obtained by pasting a sample foreground image onto a background template image. Using this scalable approach we generate 120K images of objects from all 1000 ImageNet classes in 10 diverse backgrounds. Due to the incredible photorealism of the diffusion model, our images are much closer to natural images than previous synthetic datasets. D3S contains a validation set of more than 17K images whose labels are human-verified in an MTurk study. Using the validation set, we evaluate several popular DNN image classifiers and find that the classification performance of models generally suffers on our background diverse images. Next, we leverage the foreground & background labels in D3S to learn a foreground (background) representation that is invariant to changes in background (foreground) by penalizing the mutual information between the foreground (background) features and the background (foreground) labels. Linear classifiers trained on these features to predict foreground (background) from foreground (background) have high accuracies at 82.9% (93.8%), while classifiers that predict these labels from background and foreground have a much lower accuracy of 2.4% and 45.6% respectively. This suggests that our foreground and background features are well disentangled. We further test the efficacy of these representations by training classifiers on a task with strong spurious correlations.

Sarthak Dayal, Abhinav Peri, Carl Qi et al.

Hierarchical Reinforcement Learning (HRL) promises to solve long-horizon Reinforcement Learning (RL) tasks more efficiently than non-hierarchical counterparts by discovering and reusing temporally-extended skills. However, obtaining skills that are actually reusable remains an open challenge. Towards this end, we focus on abstractions that exploit the intuition of local dynamics: local transitions in different global contexts require similar kinds of action sequences. By aligning these contexts with the action sequences they require, we are able to learn which skills to reuse and where to reuse them. In principle, this information should benefit many HRL algorithms, where high-level policies have to reason about the low-level skills they use. The resulting algorithm CARL (Contrastive Action-based Representations for Reusable Local Control) shows both qualitative clustering of meaningful skills in complex humanoid environments and improved downstream performance on the OGBench benchmark when integrated with HIQL.

Alexander Levine, Peter Stone, Amy Zhang

In real-world control settings, the observation space is often unnecessarily high-dimensional and subject to time-correlated noise. However, the controllable dynamics of the system are often far simpler than the dynamics of the raw observations. It is therefore desirable to learn an encoder to map the observation space to a simpler space of control-relevant variables. In this work, we consider the Ex-BMDP model, first proposed by Efroni et al. (2022), which formalizes control problems where observations can be factorized into an action-dependent latent state which evolves deterministically, and action-independent time-correlated noise. Lamb et al. (2022) proposes the "AC-State" method for learning an encoder to extract a complete action-dependent latent state representation from the observations in such problems. AC-State is a multistep-inverse method, in that it uses the encoding of the the first and last state in a path to predict the first action in the path. However, we identify cases where AC-State will fail to learn a correct latent representation of the agent-controllable factor of the state. We therefore propose a new algorithm, ACDF, which combines multistep-inverse prediction with a latent forward model. ACDF is guaranteed to correctly infer an action-dependent latent state encoder for a large class of Ex-BMDP models. We demonstrate the effectiveness of ACDF on tabular Ex-BMDPs through numerical simulations; as well as high-dimensional environments using neural-network-based encoders. Code is available at https://github.com/midi-lab/acdf.

Jiang Liu, Alexander Levine, Chun Pong Lau et al.

Object detection plays a key role in many security-critical systems. Adversarial patch attacks, which are easy to implement in the physical world, pose a serious threat to state-of-the-art object detectors. Developing reliable defenses for object detectors against patch attacks is critical but severely understudied. In this paper, we propose Segment and Complete defense (SAC), a general framework for defending object detectors against patch attacks through detection and removal of adversarial patches. We first train a patch segmenter that outputs patch masks which provide pixel-level localization of adversarial patches. We then propose a self adversarial training algorithm to robustify the patch segmenter. In addition, we design a robust shape completion algorithm, which is guaranteed to remove the entire patch from the images if the outputs of the patch segmenter are within a certain Hamming distance of the ground-truth patch masks. Our experiments on COCO and xView datasets demonstrate that SAC achieves superior robustness even under strong adaptive attacks with no reduction in performance on clean images, and generalizes well to unseen patch shapes, attack budgets, and unseen attack methods. Furthermore, we present the APRICOT-Mask dataset, which augments the APRICOT dataset with pixel-level annotations of adversarial patches. We show SAC can significantly reduce the targeted attack success rate of physical patch attacks. Our code is available at https://github.com/joellliu/SegmentAndComplete.

Alexander Levine, Soheil Feizi

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 representation of the center of the subdivision occupied by the input sample. In contrast to uniform additive smoothing, the SSN certification does not require the random noise components used to be independent. Thus, smoothing can be done effectively in just one dimension and can therefore be efficiently derandomized for quantized data (e.g., images). To the best of our knowledge, this is the first work to provide deterministic "randomized smoothing" for a norm-based adversarial threat model while allowing for an arbitrary classifier (i.e., a deep model) to be used as a base classifier and without requiring an exponential number of smoothing samples. On CIFAR-10 and ImageNet datasets, we provide substantially larger L_1 robustness certificates compared to prior works, establishing a new state-of-the-art. The determinism of our method also leads to significantly faster certificate computation. Code is available at: https://github.com/alevine0/smoothingSplittingNoise

Alexander Levine, Aounon Kumar, Thomas Goldstein et al.

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 novel certificate, we show that SoS certificates are realizable and therefore tight. Interestingly, we show that the maximum achievable benefits, in terms of certified robustness, from using the additional information of the gradient norm are relatively small: because our bounds are tight, this is a fundamental negative result. The gain of SoS certificates further diminishes if we consider the estimation error of the gradient norms, for which we have developed an estimator. We therefore additionally develop a variant of Gaussian smoothing, called Gaussian dipole smoothing, which provides similar bounds to randomized smoothing with gradient information, but with much-improved sample efficiency. This allows us to achieve (marginally) improved robustness certificates on high-dimensional datasets such as CIFAR-10 and ImageNet. Code is available at https://github.com/alevine0/smoothing_second_order.

Aounon Kumar, Alexander Levine, Soheil Feizi et al.

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 classifier. We consider two notions for quantifying confidence: average prediction score of a class and the margin by which the average prediction score of one class exceeds that of another. We modify the Neyman-Pearson lemma (a key theorem in randomized smoothing) to design a procedure for computing the certified radius where the confidence is guaranteed to stay above a certain threshold. Our experimental results on CIFAR-10 and ImageNet datasets show that using information about the distribution of the confidence scores allows us to achieve a significantly better certified radius than ignoring it. Thus, we demonstrate that extra information about the base classifier at the input point can help improve certified guarantees for the smoothed classifier. Code for the experiments is available at https://github.com/aounon/cdf-smoothing.

Alexander Levine, Soheil Feizi

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 than general sparse attacks, we derive meaningfully large robustness certificates against them. Additionally, in contrast to smoothing-based defenses against L_p and sparse attacks, our defense method against patch attacks is de-randomized, yielding improved, deterministic certificates. Compared to the existing patch certification method proposed by Chiang et al. (2020), which relies on interval bound propagation, our method can be trained significantly faster, achieves high clean and certified robust accuracy on CIFAR-10, and provides certificates at ImageNet scale. For example, for a 5-by-5 patch attack on CIFAR-10, our method achieves up to around 57.6% certified accuracy (with a classifier with around 83.8% clean accuracy), compared to at most 30.3% certified accuracy for the existing method (with a classifier with around 47.8% clean accuracy). Our results effectively establish a new state-of-the-art of certifiable defense against patch attacks on CIFAR-10 and ImageNet. Code is available at https://github.com/alevine0/patchSmoothing.

Alexander Levine, Soheil Feizi

Recently, techniques have been developed to provably guarantee the robustness of a classifier to adversarial perturbations of bounded L_1 and L_2 magnitudes by using randomized smoothing: the robust classification is a consensus of base classifications on randomly noised samples where the noise is additive. In this paper, we extend this technique to the L_0 threat model. We propose an efficient and certifiably robust defense against sparse adversarial attacks by randomly ablating input features, rather than using additive noise. Experimentally, on MNIST, we can certify the classifications of over 50% of images to be robust to any distortion of at most 8 pixels. This is comparable to the observed empirical robustness of unprotected classifiers on MNIST to modern L_0 attacks, demonstrating the tightness of the proposed robustness certificate. We also evaluate our certificate on ImageNet and CIFAR-10. Our certificates represent an improvement on those provided in a concurrent work (Lee et al. 2019) which uses random noise rather than ablation (median certificates of 8 pixels versus 4 pixels on MNIST; 16 pixels versus 1 pixel on ImageNet.) Additionally, we empirically demonstrate that our classifier is highly robust to modern sparse adversarial attacks on MNIST. Our classifications are robust, in median, to adversarial perturbations of up to 31 pixels, compared to 22 pixels reported as the state-of-the-art defense, at the cost of a slight decrease (around 2.3%) in the classification accuracy. Code is available at https://github.com/alevine0/randomizedAblation/.

Alexander Levine, Peter Stone, Amy Zhang

While sequential decision-making environments often involve high-dimensional observations, not all features of these observations are relevant for control. In particular, the observation space may capture factors of the environment which are not controllable by the agent, but which add complexity to the observation space. The need to ignore these "noise" features in order to operate in a tractably-small state space poses a challenge for efficient policy learning. Due to the abundance of video data available in many such environments, task-independent representation learning from action-free offline data offers an attractive solution. However, recent work has highlighted theoretical limitations in action-free learning under the Exogenous Block MDP (Ex-BMDP) model, where temporally-correlated noise features are present in the observations. To address these limitations, we identify a realistic setting where representation learning in Ex-BMDPs becomes tractable: when action-free video data from multiple agents with differing policies are available. Concretely, this paper introduces CRAFT (Comparison-based Representations from Action-Free Trajectories), a sample-efficient algorithm leveraging differences in controllable feature dynamics across agents to learn representations. We provide theoretical guarantees for CRAFT's performance and demonstrate its feasibility on a toy example, offering a foundation for practical methods in similar settings.

Wenxiao Wang, Alexander Levine, Soheil Feizi

Data poisoning attacks aim at manipulating model behaviors through distorting training data. Previously, an aggregation-based certified defense, Deep Partition Aggregation (DPA), was proposed to mitigate this threat. DPA predicts through an aggregation of base classifiers trained on disjoint subsets of data, thus restricting its sensitivity to dataset distortions. In this work, we propose an improved certified defense against general poisoning attacks, namely Finite Aggregation. In contrast to DPA, which directly splits the training set into disjoint subsets, our method first splits the training set into smaller disjoint subsets and then combines duplicates of them to build larger (but not disjoint) subsets for training base classifiers. This reduces the worst-case impacts of poison samples and thus improves certified robustness bounds. In addition, we offer an alternative view of our method, bridging the designs of deterministic and stochastic aggregation-based certified defenses. Empirically, our proposed Finite Aggregation consistently improves certificates on MNIST, CIFAR-10, and GTSRB, boosting certified fractions by up to 3.05%, 3.87% and 4.77%, respectively, while keeping the same clean accuracies as DPA's, effectively establishing a new state of the art in (pointwise) certified robustness against data poisoning.

Aounon Kumar, Alexander Levine, Tom Goldstein et al.

Certified robustness in machine learning has primarily focused on adversarial perturbations of the input with a fixed attack budget for each point in the data distribution. In this work, we present provable robustness guarantees on the accuracy of a model under bounded Wasserstein shifts of the data distribution. We show that a simple procedure that randomizes the input of the model within a transformation space is provably robust to distributional shifts under the transformation. Our framework allows the datum-specific perturbation size to vary across different points in the input distribution and is general enough to include fixed-sized perturbations as well. Our certificates produce guaranteed lower bounds on the performance of the model for any (natural or adversarial) shift of the input distribution within a Wasserstein ball around the original distribution. We apply our technique to: (i) certify robustness against natural (non-adversarial) transformations of images such as color shifts, hue shifts and changes in brightness and saturation, (ii) certify robustness against adversarial shifts of the input distribution, and (iii) show provable lower bounds (hardness results) on the performance of models trained on so-called "unlearnable" datasets that have been poisoned to interfere with model training.

Aounon Kumar, Alexander Levine, Soheil Feizi

The study of provable adversarial robustness for deep neural networks (DNNs) has mainly focused on static supervised learning tasks such as image classification. However, DNNs have been used extensively in real-world adaptive tasks such as reinforcement learning (RL), making such systems vulnerable to adversarial attacks as well. Prior works in provable robustness in RL seek to certify the behaviour of the victim policy at every time-step against a non-adaptive adversary using methods developed for the static setting. But in the real world, an RL adversary can infer the defense strategy used by the victim agent by observing the states, actions, etc., from previous time-steps and adapt itself to produce stronger attacks in future steps. We present an efficient procedure, designed specifically to defend against an adaptive RL adversary, that can directly certify the total reward without requiring the policy to be robust at each time-step. Our main theoretical contribution is to prove an adaptive version of the Neyman-Pearson Lemma -- a key lemma for smoothing-based certificates -- where the adversarial perturbation at a particular time can be a stochastic function of current and previous observations and states as well as previous actions. Building on this result, we propose policy smoothing where the agent adds a Gaussian noise to its observation at each time-step before passing it through the policy function. Our robustness certificates guarantee that the final total reward obtained by policy smoothing remains above a certain threshold, even though the actions at intermediate time-steps may change under the attack. Our experiments on various environments like Cartpole, Pong, Freeway and Mountain Car show that our method can yield meaningful robustness guarantees in practice.

Wei-An Lin, Chun Pong Lau, Alexander Levine et al.

Adversarial training is a popular defense strategy against attack threat models with bounded Lp norms. However, it often degrades the model performance on normal images and the defense does not generalize well to novel attacks. Given the success of deep generative models such as GANs and VAEs in characterizing the underlying manifold of images, we investigate whether or not the aforementioned problems can be remedied by exploiting the underlying manifold information. To this end, we construct an "On-Manifold ImageNet" (OM-ImageNet) dataset by projecting the ImageNet samples onto the manifold learned by StyleGSN. For this dataset, the underlying manifold information is exact. Using OM-ImageNet, we first show that adversarial training in the latent space of images improves both standard accuracy and robustness to on-manifold attacks. However, since no out-of-manifold perturbations are realized, the defense can be broken by Lp adversarial attacks. We further propose Dual Manifold Adversarial Training (DMAT) where adversarial perturbations in both latent and image spaces are used in robustifying the model. Our DMAT improves performance on normal images, and achieves comparable robustness to the standard adversarial training against Lp attacks. In addition, we observe that models defended by DMAT achieve improved robustness against novel attacks which manipulate images by global color shifts or various types of image filtering. Interestingly, similar improvements are also achieved when the defended models are tested on out-of-manifold natural images. These results demonstrate the potential benefits of using manifold information in enhancing robustness of deep learning models against various types of novel adversarial attacks.

Alexander Levine, Soheil Feizi

Adversarial poisoning attacks distort training data in order to corrupt the test-time behavior of a classifier. A provable defense provides a certificate for each test sample, which is a lower bound on the magnitude of any adversarial distortion of the training set that can corrupt the test sample's classification. We propose two novel provable defenses against poisoning attacks: (i) Deep Partition Aggregation (DPA), a certified defense against a general poisoning threat model, defined as the insertion or deletion of a bounded number of samples to the training set -- by implication, this threat model also includes arbitrary distortions to a bounded number of images and/or labels; and (ii) Semi-Supervised DPA (SS-DPA), a certified defense against label-flipping poisoning attacks. DPA is an ensemble method where base models are trained on partitions of the training set determined by a hash function. DPA is related to both subset aggregation, a well-studied ensemble method in classical machine learning, as well as to randomized smoothing, a popular provable defense against evasion attacks. Our defense against label-flipping attacks, SS-DPA, uses a semi-supervised learning algorithm as its base classifier model: each base classifier is trained using the entire unlabeled training set in addition to the labels for a partition. SS-DPA significantly outperforms the existing certified defense for label-flipping attacks on both MNIST and CIFAR-10: provably tolerating, for at least half of test images, over 600 label flips (vs. < 200 label flips) on MNIST and over 300 label flips (vs. 175 label flips) on CIFAR-10. Against general poisoning attacks, where no prior certified defenses exists, DPA can certify >= 50% of test images against over 500 poison image insertions on MNIST, and nine insertions on CIFAR-10. These results establish new state-of-the-art provable defenses against poisoning attacks.

Aounon Kumar, Alexander Levine, Tom Goldstein et al.

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 better than that of the $\ell_p$-radius that can be certified using isotropic Gaussian smoothing, essentially putting a matching lower bound on the robustness radius. When restricted to {\it generalized} Gaussian smoothing, these two bounds can be shown to be within a constant factor of each other in an asymptotic sense, establishing that Gaussian smoothing provides the best possible results, up to a constant factor, when $p \geq 2$. We present experimental results on CIFAR to validate our theory. For other smoothing distributions, such as, a uniform distribution within an $\ell_1$ or an $\ell_\infty$-norm ball, we show upper bounds of the form $O(1 / d)$ and $O(1 / d^{1 - \frac{1}{p}})$ respectively, which have an even worse dependence on $d$.

Alexander Levine, Soheil Feizi

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 Wasserstein metric ("earth-mover distance") between their normalized pixel intensities. Until now, there has been no certifiable defense against this type of attack. In this work, we propose the first defense with certified robustness against Wasserstein Adversarial attacks using randomized smoothing. We develop this certificate by considering the space of possible flows between images, and representing this space such that Wasserstein distance between images is upper-bounded by L_1 distance in this flow-space. We can then apply existing randomized smoothing certificates for the L_1 metric. In MNIST and CIFAR-10 datasets, we find that our proposed defense is also practically effective, demonstrating significantly improved accuracy under Wasserstein adversarial attack compared to unprotected models.

Alexander Levine, Sahil Singla, Soheil Feizi

Deep learning interpretation is essential to explain the reasoning behind model predictions. Understanding the robustness of interpretation methods is important especially in sensitive domains such as medical applications since interpretation results are often used in downstream tasks. Although gradient-based saliency maps are popular methods for deep learning interpretation, recent works show that they can be vulnerable to adversarial attacks. In this paper, we address this problem and provide a certifiable defense method for deep learning interpretation. We show that a sparsified version of the popular SmoothGrad method, which computes the average saliency maps over random perturbations of the input, is certifiably robust against adversarial perturbations. We obtain this result by extending recent bounds for certifiably robust smooth classifiers to the interpretation setting. Experiments on ImageNet samples validate our theory.