Information distance for neural network functions

We provide a practical distance measure in the space of functions parameterized by neural networks. It is based on the classical information distance, and we propose to replace the uncomputable Kolmogorov complexity with information measured by codelength of prequential coding. We also provide a method for directly estimating the expectation of such codelength with limited examples. Empirically, we show that information distance is invariant with respect to different parameterization of the neural networks. We also verify that information distance can faithfully reflect similarities of neural network functions. Finally, we applied information distance to investigate the relationship between neural network models, and demonstrate the connection between information distance and multiple characteristics and behaviors of neural networks.