  Adversarial examples bring a considerable security threat to support vector
machines (SVMs), especially those used in safety-critical applications. Thus,
robustness verification is an essential issue for SVMs, which can provide
provable robustness against various kinds of adversary attacks. The evaluation
results obtained through the robustness verification can provide a safe
guarantee for the use of SVMs. The existing verification method does not often
perform well in verifying SVMs with nonlinear kernels. To this end, we propose
a method to improve the verification performance for SVMs with nonlinear
kernels. We first formalize the adversarial robustness evaluation of SVMs as an
optimization problem. Then a lower bound of the original problem is obtained by
solving the Lagrangian dual problem of the original problem. Finally, the
adversarial robustness of SVMs is evaluated concerning the lower bound. We
evaluate the adversarial robustness of SVMs with linear and nonlinear kernels
on the MNIST and Fashion-MNIST datasets. The experimental results show that the
percentage of provable robustness obtained by our method on the test set is
better than that of the state-of-the-art.
