  This paper proposes a novel, abstraction-based, certified training method for
robust image classifiers. Via abstraction, all perturbed images are mapped into
intervals before feeding into neural networks for training. By training on
intervals, all the perturbed images that are mapped to the same interval are
classified as the same label, rendering the variance of training sets to be
small and the loss landscape of the models to be smooth. Consequently, our
approach significantly improves the robustness of trained models. For the
abstraction, our training method also enables a sound and complete black-box
verification approach, which is orthogonal and scalable to arbitrary types of
neural networks regardless of their sizes and architectures. We evaluate our
method on a wide range of benchmarks in different scales. The experimental
results show that our method outperforms state of the art by (i) reducing the
verified errors of trained models up to 95.64%; (ii) totally achieving up to
602.50x speedup; and (iii) scaling up to larger models with up to 138 million
trainable parameters. The demo is available at
https://github.com/zhangzhaodi233/ABSCERT.git.
