Understanding Priors in Bayesian Neural Networks at the Unit Level

Mariia Vladimirova, Jakob Verbeek, Pablo Mesejo, Julyan Arbel
Proceedings of the 36th International Conference on Machine Learning, PMLR 97:6458-6467, 2019.

Abstract

We investigate deep Bayesian neural networks with Gaussian priors on the weights and a class of ReLU-like nonlinearities. Bayesian neural networks with Gaussian priors are well known to induce an L2, “weight decay”, regularization. Our results indicate a more intricate regularization effect at the level of the unit activations. Our main result establishes that the induced prior distribution on the units before and after activation becomes increasingly heavy-tailed with the depth of the layer. We show that first layer units are Gaussian, second layer units are sub-exponential, and units in deeper layers are characterized by sub-Weibull distributions. Our results provide new theoretical insight on deep Bayesian neural networks, which we corroborate with simulation experiments.

Cite this Paper

BibTeX
@InProceedings{pmlr-v97-vladimirova19a, title = {Understanding Priors in {B}ayesian Neural Networks at the Unit Level}, author = {Vladimirova, Mariia and Verbeek, Jakob and Mesejo, Pablo and Arbel, Julyan}, booktitle = {Proceedings of the 36th International Conference on Machine Learning}, pages = {6458--6467}, year = {2019}, editor = {Chaudhuri, Kamalika and Salakhutdinov, Ruslan}, volume = {97}, series = {Proceedings of Machine Learning Research}, month = {09--15 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v97/vladimirova19a/vladimirova19a.pdf}, url = {https://proceedings.mlr.press/v97/vladimirova19a.html}, abstract = {We investigate deep Bayesian neural networks with Gaussian priors on the weights and a class of ReLU-like nonlinearities. Bayesian neural networks with Gaussian priors are well known to induce an L2, “weight decay”, regularization. Our results indicate a more intricate regularization effect at the level of the unit activations. Our main result establishes that the induced prior distribution on the units before and after activation becomes increasingly heavy-tailed with the depth of the layer. We show that first layer units are Gaussian, second layer units are sub-exponential, and units in deeper layers are characterized by sub-Weibull distributions. Our results provide new theoretical insight on deep Bayesian neural networks, which we corroborate with simulation experiments.} }
Endnote
%0 Conference Paper %T Understanding Priors in Bayesian Neural Networks at the Unit Level %A Mariia Vladimirova %A Jakob Verbeek %A Pablo Mesejo %A Julyan Arbel %B Proceedings of the 36th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2019 %E Kamalika Chaudhuri %E Ruslan Salakhutdinov %F pmlr-v97-vladimirova19a %I PMLR %P 6458--6467 %U https://proceedings.mlr.press/v97/vladimirova19a.html %V 97 %X We investigate deep Bayesian neural networks with Gaussian priors on the weights and a class of ReLU-like nonlinearities. Bayesian neural networks with Gaussian priors are well known to induce an L2, “weight decay”, regularization. Our results indicate a more intricate regularization effect at the level of the unit activations. Our main result establishes that the induced prior distribution on the units before and after activation becomes increasingly heavy-tailed with the depth of the layer. We show that first layer units are Gaussian, second layer units are sub-exponential, and units in deeper layers are characterized by sub-Weibull distributions. Our results provide new theoretical insight on deep Bayesian neural networks, which we corroborate with simulation experiments.
APA
Vladimirova, M., Verbeek, J., Mesejo, P. & Arbel, J.. (2019). Understanding Priors in Bayesian Neural Networks at the Unit Level. Proceedings of the 36th International Conference on Machine Learning, in Proceedings of Machine Learning Research 97:6458-6467 Available from https://proceedings.mlr.press/v97/vladimirova19a.html.

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