  Recently, Zhang et al. (2021) developed a new neural network architecture
based on $\ell_\infty$-distance functions, which naturally possesses certified
$\ell_\infty$ robustness by its construction. Despite the novel design and
theoretical foundation, so far the model only achieved comparable performance
to conventional networks. In this paper, we make the following two
contributions: $\mathrm{(i)}$ We demonstrate that $\ell_\infty$-distance nets
enjoy a fundamental advantage in certified robustness over conventional
networks (under typical certification approaches); $\mathrm{(ii)}$ With an
improved training process we are able to significantly boost the certified
accuracy of $\ell_\infty$-distance nets. Our training approach largely
alleviates the optimization problem that arose in the previous training scheme,
in particular, the unexpected large Lipschitz constant due to the use of a
crucial trick called $\ell_p$-relaxation. The core of our training approach is
a novel objective function that combines scaled cross-entropy loss and clipped
hinge loss with a decaying mixing coefficient. Experiments show that using the
proposed training strategy, the certified accuracy of $\ell_\infty$-distance
net can be dramatically improved from 33.30% to 40.06% on CIFAR-10
($\epsilon=8/255$), meanwhile outperforming other approaches in this area by a
large margin. Our results clearly demonstrate the effectiveness and potential
of $\ell_\infty$-distance net for certified robustness. Codes are available at
https://github.com/zbh2047/L_inf-dist-net-v2.
