  Adversarial training can considerably robustify deep neural networks to
resist adversarial attacks. However, some works suggested that adversarial
training might comprise the privacy-preserving and generalization abilities.
This paper establishes and quantifies the privacy-robustness trade-off and
generalization-robustness trade-off in adversarial training from both
theoretical and empirical aspects. We first define a notion, {\it robustified
intensity} to measure the robustness of an adversarial training algorithm. This
measure can be approximate empirically by an asymptotically consistent
empirical estimator, {\it empirical robustified intensity}. Based on the
robustified intensity, we prove that (1) adversarial training is $(\varepsilon,
\delta)$-differentially private, where the magnitude of the differential
privacy has a positive correlation with the robustified intensity; and (2) the
generalization error of adversarial training can be upper bounded by an
$\mathcal O(\sqrt{\log N}/N)$ on-average bound and an $\mathcal O(1/\sqrt{N})$
high-probability bound, both of which have positive correlations with the
robustified intensity. Additionally, our generalization bounds do not
explicitly rely on the parameter size which would be prohibitively large in
deep learning. Systematic experiments on standard datasets, CIFAR-10 and
CIFAR-100, are in full agreement with our theories. The source code package is
available at \url{https://github.com/fshp971/RPG}.
