Lower bounds on the robustness to adversarial perturbations

The input-output mappings learned by state-of-the-art neural networks are significantly discontinuous. It is possible to cause a neural network used for image recognition to misclassify its input by applying very specific, hardly perceptible perturbations to the input, called adversarial perturbations. Many hypotheses have been proposed to explain the existence of these peculiar samples as well as several methods to mitigate them, but a proven explanation remains elusive. In this work, we take steps towards a formal characterization of adversarial perturbations by deriving lower bounds on the magnitudes of perturbations necessary to change the classification of neural networks. The proposed bounds can be computed efficiently, requiring time at most linear in the number of parameters and hyperparameters of the model for any given sample. This makes them suitable for use in model selection, when one wishes to find out which of several proposed classifiers is most robust to adversarial perturbations. They may also be used as a basis for developing techniques to increase the robustness of classifiers, since they enjoy the theoretical guarantee that no adversarial perturbation could possibly be any smaller than the quantities provided by the bounds. We experimentally verify the bounds on the MNIST and CIFAR-10 data sets and find no violations. Additionally, the experimental results suggest that very small adversarial perturbations may occur with non-zero probability on natural samples

Certifiable Robustness to Adversarial State Uncertainty in Deep Reinforcement Learning

Deep Neural Network-based systems are now the state-of-the-art in many robotics tasks, but their application in safety-critical domains remains dangerous without formal guarantees on network robustness. Small perturbations to sensor inputs (from noise or adversarial examples) are often enough to change network-based decisions, which was recently shown to cause an autonomous vehicle to swerve into another lane. In light of these dangers, numerous algorithms have been developed as defensive mechanisms from these adversarial inputs, some of which provide formal robustness guarantees or certificates. This work leverages research on certified adversarial robustness to develop an online certifiably robust for deep reinforcement learning algorithms. The proposed defense computes guaranteed lower bounds on state-action values during execution to identify and choose a robust action under a worst-case deviation in input space due to possible adversaries or noise. Moreover, the resulting policy comes with a certificate of solution quality, even though the true state and optimal action are unknown to the certifier due to the perturbations. The approach is demonstrated on a Deep Q-Network policy and is shown to increase robustness to noise and adversaries in pedestrian collision avoidance scenarios and a classic control task. This work extends one of our prior works with new performance guarantees, extensions to other RL algorithms, expanded results aggregated across more scenarios, an extension into scenarios with adversarial behavior, comparisons with a more computationally expensive method, and visualizations that provide intuition about the robustness algorithm.Comment: arXiv admin note: text overlap with arXiv:1910.1290

Analysis of classifiers' robustness to adversarial perturbations

The goal of this paper is to analyze an intriguing phenomenon recently discovered in deep networks, namely their instability to adversarial perturbations (Szegedy et. al., 2014). We provide a theoretical framework for analyzing the robustness of classifiers to adversarial perturbations, and show fundamental upper bounds on the robustness of classifiers. Specifically, we establish a general upper bound on the robustness of classifiers to adversarial perturbations, and then illustrate the obtained upper bound on the families of linear and quadratic classifiers. In both cases, our upper bound depends on a distinguishability measure that captures the notion of difficulty of the classification task. Our results for both classes imply that in tasks involving small distinguishability, no classifier in the considered set will be robust to adversarial perturbations, even if a good accuracy is achieved. Our theoretical framework moreover suggests that the phenomenon of adversarial instability is due to the low flexibility of classifiers, compared to the difficulty of the classification task (captured by the distinguishability). Moreover, we show the existence of a clear distinction between the robustness of a classifier to random noise and its robustness to adversarial perturbations. Specifically, the former is shown to be larger than the latter by a factor that is proportional to \sqrt{d} (with d being the signal dimension) for linear classifiers. This result gives a theoretical explanation for the discrepancy between the two robustness properties in high dimensional problems, which was empirically observed in the context of neural networks. To the best of our knowledge, our results provide the first theoretical work that addresses the phenomenon of adversarial instability recently observed for deep networks. Our analysis is complemented by experimental results on controlled and real-world data