  Adversarial examples pose a security threat to many critical systems built on
neural networks (such as face recognition systems, and self-driving cars).
While many methods have been proposed to build robust models, how to build
certifiably robust yet accurate neural network models remains an open problem.
For example, adversarial training improves empirical robustness, but they do
not provide certification of the model's robustness. On the other hand,
certified training provides certified robustness but at the cost of a
significant accuracy drop. In this work, we propose a novel approach that aims
to achieve both high accuracy and certified probabilistic robustness. Our
method has two parts, i.e., a probabilistic robust training method with an
additional goal of minimizing variance in terms of divergence and a runtime
inference method for certified probabilistic robustness of the prediction. The
latter enables efficient certification of the model's probabilistic robustness
at runtime with statistical guarantees. This is supported by our training
objective, which minimizes the variance of the model's predictions in a given
vicinity, derived from a general definition of model robustness. Our approach
works for a variety of perturbations and is reasonably efficient. Our
experiments on multiple models trained on different datasets demonstrate that
our approach significantly outperforms existing approaches in terms of both
certification rate and accuracy.

Machine Learning