  Training Deep Neural Networks (DNNs) with adversarial examples often results
in poor generalization to test-time adversarial data. This paper investigates
this issue, known as adversarially robust generalization, through the lens of
Rademacher complexity. Building upon the studies by Khim and Loh (2018); Yin et
al. (2019), numerous works have been dedicated to this problem, yet achieving a
satisfactory bound remains an elusive goal. Existing works on DNNs either apply
to a surrogate loss instead of the robust loss or yield bounds that are notably
looser compared to their standard counterparts. In the latter case, the bounds
have a higher dependency on the width $m$ of the DNNs or the dimension $d$ of
the data, with an extra factor of at least $\mathcal{O}(\sqrt{m})$ or
$\mathcal{O}(\sqrt{d})$.
  This paper presents upper bounds for adversarial Rademacher complexity of
DNNs that match the best-known upper bounds in standard settings, as
established in the work of Bartlett et al. (2017), with the dependency on width
and dimension being $\mathcal{O}(\ln(dm))$. The central challenge addressed is
calculating the covering number of adversarial function classes. We aim to
construct a new cover that possesses two properties: 1) compatibility with
adversarial examples, and 2) precision comparable to covers used in standard
settings. To this end, we introduce a new variant of covering number called the
\emph{uniform covering number}, specifically designed and proven to reconcile
these two properties. Consequently, our method effectively bridges the gap
between Rademacher complexity in robust and standard generalization.
