  While additional training data improves the robustness of deep neural
networks against adversarial examples, it presents the challenge of curating a
large number of specific real-world samples. We circumvent this challenge by
using additional data from proxy distributions learned by advanced generative
models. We first seek to formally understand the transfer of robustness from
classifiers trained on proxy distributions to the real data distribution. We
prove that the difference between the robustness of a classifier on the two
distributions is upper bounded by the conditional Wasserstein distance between
them. Next we use proxy distributions to significantly improve the performance
of adversarial training on five different datasets. For example, we improve
robust accuracy by up to 7.5% and 6.7% in $\ell_{\infty}$ and $\ell_2$ threat
model over baselines that are not using proxy distributions on the CIFAR-10
dataset. We also improve certified robust accuracy by 7.6% on the CIFAR-10
dataset. We further demonstrate that different generative models bring a
disparate improvement in the performance in robust training. We propose a
robust discrimination approach to characterize the impact of individual
generative models and further provide a deeper understanding of why current
state-of-the-art in diffusion-based generative models are a better choice for
proxy distribution than generative adversarial networks.
