  A key challenge that threatens the widespread use of neural networks in
safety-critical applications is their vulnerability to adversarial attacks. In
this paper, we study the second-order behavior of continuously differentiable
deep neural networks, focusing on robustness against adversarial perturbations.
First, we provide a theoretical analysis of robustness and attack certificates
for deep classifiers by leveraging local gradients and upper bounds on the
second derivative (curvature constant). Next, we introduce a novel algorithm to
analytically compute provable upper bounds on the second derivative of neural
networks. This algorithm leverages the compositional structure of the model to
propagate the curvature bound layer-by-layer, giving rise to a scalable and
modular approach. The proposed bound can serve as a differentiable regularizer
to control the curvature of neural networks during training, thereby enhancing
robustness. Finally, we demonstrate the efficacy of our method on
classification tasks using the MNIST and CIFAR-10 datasets.
