  Recently, we proposed to transform the outputs of each hidden neuron in a
multi-layer perceptron network to have zero output and zero slope on average,
and use separate shortcut connections to model the linear dependencies instead.
We continue the work by firstly introducing a third transformation to normalize
the scale of the outputs of each hidden neuron, and secondly by analyzing the
connections to second order optimization methods. We show that the
transformations make a simple stochastic gradient behave closer to second-order
optimization methods and thus speed up learning. This is shown both in theory
and with experiments. The experiments on the third transformation show that
while it further increases the speed of learning, it can also hurt performance
by converging to a worse local optimum, where both the inputs and outputs of
many hidden neurons are close to zero.
