Deep frequency principle towards understanding why deeper learning is faster

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 deeper learning is faster. To this end, we separate a deep neural network into two parts, one is a pre-condition component and the other is a learning component, in which the output of the pre-condition one is the input of the learning one. 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 has a bias towards a function with more low 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.