  We study the stability and convergence of training deep ResNets with gradient
descent. Specifically, we show that the parametric branch in the residual block
should be scaled down by a factor $\tau =O(1/\sqrt{L})$ to guarantee stable
forward/backward process, where $L$ is the number of residual blocks. Moreover,
we establish a converse result that the forward process is unbounded when
$\tau>L^{-\frac{1}{2}+c}$, for any positive constant $c$. The above two results
together establish a sharp value of the scaling factor in determining the
stability of deep ResNet. Based on the stability result, we further show that
gradient descent finds the global minima if the ResNet is properly
over-parameterized, which significantly improves over the previous work with a
much larger range of $\tau$ that admits global convergence. Moreover, we show
that the convergence rate is independent of the depth, theoretically justifying
the advantage of ResNet over vanilla feedforward network. Empirically, with
such a factor $\tau$, one can train deep ResNet without normalization layer.
Moreover, for ResNets with normalization layer, adding such a factor $\tau$
also stabilizes the training and obtains significant performance gain for deep
ResNet.
