  People believe that depth plays an important role in success of deep neural
networks (DNN). However, this belief lacks solid theoretical justifications as
far as we know. We investigate role of depth from perspective of margin bound.
In margin bound, expected error is upper bounded by empirical margin error plus
Rademacher Average (RA) based capacity term. First, we derive an upper bound
for RA of DNN, and show that it increases with increasing depth. This indicates
negative impact of depth on test performance. Second, we show that deeper
networks tend to have larger representation power (measured by Betti numbers
based complexity) than shallower networks in multi-class setting, and thus can
lead to smaller empirical margin error. This implies positive impact of depth.
The combination of these two results shows that for DNN with restricted number
of hidden units, increasing depth is not always good since there is a tradeoff
between positive and negative impacts. These results inspire us to seek
alternative ways to achieve positive impact of depth, e.g., imposing
margin-based penalty terms to cross entropy loss so as to reduce empirical
margin error without increasing depth. Our experiments show that in this way,
we achieve significantly better test performance.
