A robustness certificate is the minimum distance of a given input to the
decision boundary of the classifier (or its lower bound). For {\it any} input
perturbations with a magnitude smaller than the certificate value, the
classification output will provably remain unchanged. Exactly computing the
robustness certificates for neural networks is difficult since it requires
solving a non-convex optimization. In this paper, we provide
computationally-efficient robustness certificates for neural networks with
differentiable activation functions in two steps. First, we show that if the
eigenvalues of the Hessian of the network are bounded, we can compute a
robustness certificate in the $l_2$ norm efficiently using convex optimization.
Second, we derive a computationally-efficient differentiable upper bound on the
curvature of a deep network. We also use the curvature bound as a
regularization term during the training of the network to boost its certified
robustness. Putting these results together leads to our proposed {\bf
C}urvature-based {\bf R}obustness {\bf C}ertificate (CRC) and {\bf
C}urvature-based {\bf R}obust {\bf T}raining (CRT). Our numerical results show
that CRT leads to significantly higher certified robust accuracy compared to
interval-bound propagation (IBP) based training. We achieve certified robust
accuracy 69.79\%, 57.78\% and 53.19\% while IBP-based methods achieve 44.96\%,
44.74\% and 44.66\% on 2,3 and 4 layer networks respectively on the
MNIST-dataset.

通过计算神经网络的 Hessian 矩阵的特征值上界，提出了基于曲率的神经网络鲁棒性证明和训练方法，相较于基于区间传递的方法，在 MNIST 上取得了更高的鲁棒性准确率。