Abstract
We study the classical binary classification problem for hypothesis spaces of Deep Neural Networks (DNNs) with ReLU activation under Tsybakov's low-noise condition with exponent $q>0$, and its limit-case $q\to\infty$ which we refer to as the "hard-margin condition". We show that DNNs which minimize the empirical risk with square loss surrogate and $\ell_p$ penalty can achieve finite-sample excess risk bounds of order $\mathcal{O}\left(n^{-\alpha}\right)$ for arbitrarily large $\alpha>0$ under the hard-margin condition, provided that the regression function $\eta$ is sufficiently smooth. The proof relies on a novel decomposition of the excess risk which might be of independent interest.