This study uses CAPM (Convex Adversarial Polytope for Maxpool-based CNN) to improve the verified bound for general purpose maxpool-based convolutional neural networks (CNNs) under bounded norm adversarial perturbations. The maxpool function is decomposed as a series of ReLU functions to extend the convex relaxation technique to maxpool functions, by which the verified bound can be efficiently computed through a dual network. The experimental results demonstrate that this technique allows the state-of-the-art verification precision for maxpool-based CNNs and involves a much lower computational cost than current verification methods, such as DeepZ, DeepPoly and PRIMA. This method is also applicable to large-scale CNNs, which previous studies show to be often computationally prohibitively expensive. Under certain circumstances, CAPM is 40-times, 20-times or twice as fast and give a significantly higher verification bound (CAPM 98% vs. PRIMA 76%/DeepPoly 73%/DeepZ 8%) as compared to PRIMA/DeepPoly/DeepZ. Furthermore, we additionally present the time complexity of our algorithm as $O(W^2NK)$, where $W$ is the maximum width of the neural network, $N$ is the number of neurons, and $K$ is the size of the maxpool layer's kernel.

Deep neural networks (DNNs) could be deceived by generating human-imperceptible perturbations of clean samples. Therefore, enhancing the robustness of DNNs against adversarial attacks is a crucial task. In this paper, we aim to train robust DNNs by limiting the set of outputs reachable via a norm-bounded perturbation added to a clean sample. We refer to this set as adversarial polytope, and each clean sample has a respective adversarial polytope. Indeed, if the respective polytopes for all the samples are compact such that they do not intersect the decision boundaries of the DNN, then the DNN is robust against adversarial samples. Hence, the inner-working of our algorithm is based on learning \textbf{c}onfined \textbf{a}dversarial \textbf{p}olytopes (CAP). By conducting a thorough set of experiments, we demonstrate the effectiveness of CAP over existing adversarial robustness methods in improving the robustness of models against state-of-the-art attacks including AutoAttack.

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 certified defense 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. 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 our previous paper with new performance guarantees, 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.

We present an efficient technique, which allows to train classification networks which are verifiably robust against norm-bounded adversarial attacks. This framework is built upon the work of Gowal et al., who applies the interval arithmetic to bound the activations at each layer and keeps the prediction invariant to the input perturbation. While that method is faster than competitive approaches, it requires careful tuning of hyper-parameters and a large number of epochs to converge. To speed up and stabilize training, we supply the cost function with an additional term, which encourages the model to keep the interval bounds at hidden layers small. Experimental results demonstrate that we can achieve comparable (or even better) results using a smaller number of training iterations, in a more stable fashion. Moreover, the proposed model is not so sensitive to the exact specification of the training process, which makes it easier to use by practitioners.

We propose a method to learn deep ReLUbased classifiers that are provably robust against normbounded adversarial perturbations on the training data. For previously unseen examples, the approach is guaranteed to detect all adversarial examples, though it may flag some non-adversarial examples as well. The basic idea is to consider a convex outer approximation of the set of activations reachable through a norm-bounded perturbation, and we develop a robust optimization procedure that minimizes the worst case loss over this outer region (via a linear program). Crucially, we show that the dual problem to this linear program can be represented itself as a deep network similar to the backpropagation network, leading to very efficient optimization approaches that produce guaranteed bounds on the robust loss. The end result is that by executing a few more forward and backward passes through a slightly modified version of the original network (though possibly with much larger batch sizes), we can learn a classifier that is provably robust to any norm-bounded adversarial attack. We illustrate the approach on a number of tasks to train classifiers with robust adversarial guarantees (e.g. for MNIST, we produce a convolutional classifier that provably has less than 5.8% test error for any adversarial attack with bounded l