Degrees of Freedom in Deep Neural Networks

Tianxiang Gao, Vladimir Jojic
Proceedings of the 32nd Conference on Uncertainty in Artificial Intelligence, PMLR R14:722-731, 2016.

Abstract

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 cross-validation with lower computational cost.

Cite this Paper

BibTeX
@InProceedings{pmlr-vR14-gao16a, title = {Degrees of Freedom in Deep Neural Networks}, author = {Gao, Tianxiang and Jojic, Vladimir}, booktitle = {Proceedings of the 32nd Conference on Uncertainty in Artificial Intelligence}, pages = {722--731}, year = {2016}, editor = {Ihler, Alexander and Janzing, Dominik}, volume = {R14}, series = {Proceedings of Machine Learning Research}, month = {25--29 Jun}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/r14/main/assets/gao16a/gao16a.pdf}, url = {https://proceedings.mlr.press/r14/gao16a.html}, abstract = {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 cross-validation with lower computational cost.}, note = {Reissued by PMLR on 04 October 2026.} }
Endnote
%0 Conference Paper %T Degrees of Freedom in Deep Neural Networks %A Tianxiang Gao %A Vladimir Jojic %B Proceedings of the 32nd Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2016 %E Alexander Ihler %E Dominik Janzing %F pmlr-vR14-gao16a %I PMLR %P 722--731 %U https://proceedings.mlr.press/r14/gao16a.html %V R14 %X 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 cross-validation with lower computational cost. %Z Reissued by PMLR on 04 October 2026.
APA
Gao, T. & Jojic, V.. (2016). Degrees of Freedom in Deep Neural Networks. Proceedings of the 32nd Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research R14:722-731 Available from https://proceedings.mlr.press/r14/gao16a.html. Reissued by PMLR on 04 October 2026.

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