  We propose a scalable robust learning algorithm combining kernel smoothing
and robust optimization. Our method is motivated by the convex analysis
perspective of distributionally robust optimization based on probability
metrics, such as the Wasserstein distance and the maximum mean discrepancy. We
adapt the integral operator using supremal convolution in convex analysis to
form a novel function majorant used for enforcing robustness. Our method is
simple in form and applies to general loss functions and machine learning
models. Exploiting a connection with optimal transport, we prove theoretical
guarantees for certified robustness under distribution shift. Furthermore, we
report experiments with general machine learning models, such as deep neural
networks, to demonstrate competitive performance with the state-of-the-art
certifiable robust learning algorithms based on the Wasserstein distance.
