  Deep neural networks are known to be difficult to train due to the
instability of back-propagation. A deep \emph{residual network} (ResNet) with
identity loops remedies this by stabilizing gradient computations. We prove a
boosting theory for the ResNet architecture. We construct $T$ weak module
classifiers, each contains two of the $T$ layers, such that the combined strong
learner is a ResNet. Therefore, we introduce an alternative Deep ResNet
training algorithm, \emph{BoostResNet}, which is particularly suitable in
non-differentiable architectures. Our proposed algorithm merely requires a
sequential training of $T$ "shallow ResNets" which are inexpensive. We prove
that the training error decays exponentially with the depth $T$ if the
\emph{weak module classifiers} that we train perform slightly better than some
weak baseline. In other words, we propose a weak learning condition and prove a
boosting theory for ResNet under the weak learning condition. Our results apply
to general multi-class ResNets. A generalization error bound based on margin
theory is proved and suggests ResNet's resistant to overfitting under network
with $l_1$ norm bounded weights.
