  Whereas it is believed that techniques such as Adam, batch normalization and,
more recently, SeLU nonlinearities "solve" the exploding gradient problem, we
show that this is not the case in general and that in a range of popular MLP
architectures, exploding gradients exist and that they limit the depth to which
networks can be effectively trained, both in theory and in practice. We explain
why exploding gradients occur and highlight the *collapsing domain problem*,
which can arise in architectures that avoid exploding gradients.
  ResNets have significantly lower gradients and thus can circumvent the
exploding gradient problem, enabling the effective training of much deeper
networks. We show this is a direct consequence of the Pythagorean equation. By
noticing that *any neural network is a residual network*, we devise the
*residual trick*, which reveals that introducing skip connections simplifies
the network mathematically, and that this simplicity may be the major cause for
their success.
