Analyzing the robustness of neural networks against norm-bounded uncertainties and adversarial attacks has found many applications ranging from safety verification to robust training. In this paper, we propose a semidefinite programming (SDP) framework for safety verification and robustness analysis of neural networks with general activation functions. Our main idea is to abstract various properties of activation functions (e.g., monotonicity, bounded slope, bounded values, and repetition across layers) with the formalism of quadratic constraints. We then analyze the safety properties of the abstracted network via the S-procedure and semidefinite programming. Compared to other semidefinite relaxations proposed in the literature, our method is less conservative, especially for deep networks, with an order of magnitude reduction in computational complexity. Furthermore, our approach is applicable to any activation functions.
The performance of certification algorithms for neural networks can be measured along three axes. The first axis is the tightness of the certification bounds; the second axis is the computational complexity, and, the third axis is applicability across various models (e.g. different activation functions). These axes conflict. For instance, the conservatism of the algorithm is typically at odds with the computational complexity. On the other hand, generalizable algorithms tend to be more conservative. The relative advantage of any of these algorithms is application specific. For example, reachability analysis and safety verification applications call for less conservative algorithms, whereas in robust training, computationally fast algorithms are desirable @cite_3 .
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Neural networks have demonstrated considerable success on a wide variety of real-world problems. However, networks trained only to optimize for training accuracy can often be fooled by adversarial examples - slightly perturbed inputs that are misclassified with high confidence. Verification of networks enables us to gauge their vulnerability to such adversarial examples. We formulate verification of piecewise-linear neural networks as a mixed integer program. On a representative task of finding minimum adversarial distortions, our verifier is two to three orders of magnitude quicker than the state-of-the-art. We achieve this computational speedup via tight formulations for non-linearities, as well as a novel presolve algorithm that makes full use of all information available. The computational speedup allows us to verify properties on convolutional networks with an order of magnitude more ReLUs than networks previously verified by any complete verifier. In particular, we determine for the first time the exact adversarial accuracy of an MNIST classifier to perturbations with bounded @math norm @math : for this classifier, we find an adversarial example for 4.38 of samples, and a certificate of robustness (to perturbations with bounded norm) for the remainder. Across all robust training procedures and network architectures considered, we are able to certify more samples than the state-of-the-art and find more adversarial examples than a strong first-order attack. Neural networks have demonstrated considerable success on a wide variety of real-world problems. However, neural networks can be fooled by adversarial examples – slightly perturbed inputs that are misclassified with high confidence. Verification of networks enables us to gauge their vulnerability to such adversarial examples. We formulate verification of piecewise-linear neural networks as a mixed integer program. Our verifier finds minimum adversarial distortions two to three orders of magnitude more quickly than the state-of-the-art. We achieve this via tight formulations for non-linearities, as well as a novel presolve algorithm that makes full use of all information available. The computational speedup enables us to verify properties on convolutional networks with an order of magnitude more ReLUs than had been previously verified by any complete verifier, and we determine for the first time the exact adversarial accuracy of an MNIST classifier to perturbations with bounded l[infinity] norm e = 0:1. On this network, we find an adversarial example for 4.38 of samples, and a certificate of robustness for the remainder. Across a variety of robust training procedures, we are able to certify more samples than the state-of-the-art and find more adversarial examples than a strong first-order attack for every network. While neural networks have achieved high accuracy on standard image classification benchmarks, their accuracy drops to nearly zero in the presence of small adversarial perturbations to test inputs. Defenses based on regularization and adversarial training have been proposed, but often followed by new, stronger attacks that defeat these defenses. Can we somehow end this arms race? In this work, we study this problem for neural networks with one hidden layer. We first propose a method based on a semidefinite relaxation that outputs a certificate that for a given network and test input, no attack can force the error to exceed a certain value. Second, as this certificate is differentiable, we jointly optimize it with the network parameters, providing an adaptive regularizer that encourages robustness against all attacks. On MNIST, our approach produces a network and a certificate that no that perturbs each pixel by at most @math can cause more than @math test error. While neural networks have achieved high accuracy on standard image classification benchmarks, their accuracy drops to nearly zero in the presence of small adversarial perturbations to test inputs. Defenses based on regularization and adversarial training have been proposed, but often followed by new, stronger attacks that defeat these defenses. Can we somehow end this arms race? In this work, we study this problem for neural networks with one hidden layer. We first propose a method based on a semidefinite relaxation that outputs a certificate that for a given network and test input, no attack can force the error to exceed a certain value. Second, as this certificate is differentiable, we jointly optimize it with the network parameters, providing an adaptive regularizer that encourages robustness against all attacks. On MNIST, our approach produces a network and a certificate that no attack that perturbs each pixel by at most = 0.1 can cause more than 35 test error.