Abstract: We prove a \emph{multi-channel telescoping sum boosting} theory for the ResNet architectures which simultaneously creates a new technique for boosting over features (in contrast with labels) and provides a new algorithm for ResNet-style architectures. Our proposed training algorithm, \emph{BoostResNet}, is particularly suitable in non-differentiable architectures. Our method only requires the relatively inexpensive sequential training of $T$ ``shallow ResNets''. We prove that the training error decays exponentially with the depth $T$ if the 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. A generalization error bound based on margin theory is proved and suggests that ResNet could be resistant to overfitting using a network with $l_1$ norm bounded weights.