  Understanding the effect of depth in deep learning is a critical problem. In
this work, we utilize the Fourier analysis to empirically provide a promising
mechanism to understand why feedforward deeper learning is faster. To this end,
we separate a deep neural network, trained by normal stochastic gradient
descent, into two parts during analysis, i.e., a pre-condition component and a
learning component, in which the output of the pre-condition one is the input
of the learning one. We use a filtering method to characterize the frequency
distribution of a high-dimensional function. Based on experiments of deep
networks and real dataset, we propose a deep frequency principle, that is, the
effective target function for a deeper hidden layer biases towards lower
frequency during the training. Therefore, the learning component effectively
learns a lower frequency function if the pre-condition component has more
layers. Due to the well-studied frequency principle, i.e., deep neural networks
learn lower frequency functions faster, the deep frequency principle provides a
reasonable explanation to why deeper learning is faster. We believe these
empirical studies would be valuable for future theoretical studies of the
effect of depth in deep learning.
