Variance reduction properties of the reparameterization trick

Ming Xu, Matias Quiroz, Robert Kohn, Scott A. Sisson
Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, PMLR 89:2711-2720, 2019.

Abstract

The reparameterization trick is widely used in variational inference as it yields more accurate estimates of the gradient of the variational objective than alternative approaches such as the score function method. Although there is overwhelming empirical evidence in the literature showing its success, there is relatively little research exploring why the reparameterization trick is so effective. We explore this under the idealized assumptions that the variational approximation is a mean-field Gaussian density and that the log of the joint density of the model parameters and the data is a quadratic function that depends on the variational mean. From this, we show that the marginal variances of the reparameterization gradient estimator are smaller than those of the score function gradient estimator. We apply the result of our idealized analysis to real-world examples.

Cite this Paper


BibTeX
@InProceedings{pmlr-v89-xu19a, title = {Variance reduction properties of the reparameterization trick}, author = {Xu, Ming and Quiroz, Matias and Kohn, Robert and Sisson, Scott A.}, booktitle = {Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics}, pages = {2711--2720}, year = {2019}, editor = {Chaudhuri, Kamalika and Sugiyama, Masashi}, volume = {89}, series = {Proceedings of Machine Learning Research}, month = {16--18 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v89/xu19a/xu19a.pdf}, url = {http://proceedings.mlr.press/v89/xu19a.html}, abstract = {The reparameterization trick is widely used in variational inference as it yields more accurate estimates of the gradient of the variational objective than alternative approaches such as the score function method. Although there is overwhelming empirical evidence in the literature showing its success, there is relatively little research exploring why the reparameterization trick is so effective. We explore this under the idealized assumptions that the variational approximation is a mean-field Gaussian density and that the log of the joint density of the model parameters and the data is a quadratic function that depends on the variational mean. From this, we show that the marginal variances of the reparameterization gradient estimator are smaller than those of the score function gradient estimator. We apply the result of our idealized analysis to real-world examples.} }
Endnote
%0 Conference Paper %T Variance reduction properties of the reparameterization trick %A Ming Xu %A Matias Quiroz %A Robert Kohn %A Scott A. Sisson %B Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2019 %E Kamalika Chaudhuri %E Masashi Sugiyama %F pmlr-v89-xu19a %I PMLR %P 2711--2720 %U http://proceedings.mlr.press/v89/xu19a.html %V 89 %X The reparameterization trick is widely used in variational inference as it yields more accurate estimates of the gradient of the variational objective than alternative approaches such as the score function method. Although there is overwhelming empirical evidence in the literature showing its success, there is relatively little research exploring why the reparameterization trick is so effective. We explore this under the idealized assumptions that the variational approximation is a mean-field Gaussian density and that the log of the joint density of the model parameters and the data is a quadratic function that depends on the variational mean. From this, we show that the marginal variances of the reparameterization gradient estimator are smaller than those of the score function gradient estimator. We apply the result of our idealized analysis to real-world examples.
APA
Xu, M., Quiroz, M., Kohn, R. & Sisson, S.A.. (2019). Variance reduction properties of the reparameterization trick. Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 89:2711-2720 Available from http://proceedings.mlr.press/v89/xu19a.html.

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