Degrees of Freedom in Deep Neural Networks

In this paper, we explore degrees of freedom in deep sigmoidal neural networks. We show that the degrees of freedom in these models are related to the expected optimism, which is the expected difference between test error and training error. We provide an efficient Monte-Carlo method to estimate the degrees of freedom for multi-class classification methods. We show that the degrees of freedom is less than the parameter count in a simple XOR network. We extend these results to neural nets trained on synthetic and real data and investigate the impact of network’s architecture and different regularization choices. The degrees of freedom in deep networks is dramatically less than the number of parameters. In some real datasets, the number of parameters is several orders of magnitude larger than the degrees of freedom. Further, we observe that for fixed number of parameters, deeper networks have less degrees of freedom exhibiting a regularization-by-depth. Finally, we show that the degrees of freedom of deep neural networks can be used in a model selection criterion. This criterion has comparable performance to crossvalidation with lower computational cost.