A Spectral Approach to Gradient Estimation for Implicit Distributions

Jiaxin Shi, Shengyang Sun, Jun Zhu
Proceedings of the 35th International Conference on Machine Learning, PMLR 80:4644-4653, 2018.

Abstract

Recently there have been increasing interests in learning and inference with implicit distributions (i.e., distributions without tractable densities). To this end, we develop a gradient estimator for implicit distributions based on Stein’s identity and a spectral decomposition of kernel operators, where the eigenfunctions are approximated by the Nystr{ö}m method. Unlike the previous works that only provide estimates at the sample points, our approach directly estimates the gradient function, thus allows for a simple and principled out-of-sample extension. We provide theoretical results on the error bound of the estimator and discuss the bias-variance tradeoff in practice. The effectiveness of our method is demonstrated by applications to gradient-free Hamiltonian Monte Carlo and variational inference with implicit distributions. Finally, we discuss the intuition behind the estimator by drawing connections between the Nystr{ö}m method and kernel PCA, which indicates that the estimator can automatically adapt to the geometry of the underlying distribution.

Cite this Paper

BibTeX
@InProceedings{pmlr-v80-shi18a, title = {A Spectral Approach to Gradient Estimation for Implicit Distributions}, author = {Shi, Jiaxin and Sun, Shengyang and Zhu, Jun}, booktitle = {Proceedings of the 35th International Conference on Machine Learning}, pages = {4644--4653}, year = {2018}, editor = {Dy, Jennifer and Krause, Andreas}, volume = {80}, series = {Proceedings of Machine Learning Research}, month = {10--15 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v80/shi18a/shi18a.pdf}, url = {https://proceedings.mlr.press/v80/shi18a.html}, abstract = {Recently there have been increasing interests in learning and inference with implicit distributions (i.e., distributions without tractable densities). To this end, we develop a gradient estimator for implicit distributions based on Stein’s identity and a spectral decomposition of kernel operators, where the eigenfunctions are approximated by the Nystr{ö}m method. Unlike the previous works that only provide estimates at the sample points, our approach directly estimates the gradient function, thus allows for a simple and principled out-of-sample extension. We provide theoretical results on the error bound of the estimator and discuss the bias-variance tradeoff in practice. The effectiveness of our method is demonstrated by applications to gradient-free Hamiltonian Monte Carlo and variational inference with implicit distributions. Finally, we discuss the intuition behind the estimator by drawing connections between the Nystr{ö}m method and kernel PCA, which indicates that the estimator can automatically adapt to the geometry of the underlying distribution.} }
Endnote
%0 Conference Paper %T A Spectral Approach to Gradient Estimation for Implicit Distributions %A Jiaxin Shi %A Shengyang Sun %A Jun Zhu %B Proceedings of the 35th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2018 %E Jennifer Dy %E Andreas Krause %F pmlr-v80-shi18a %I PMLR %P 4644--4653 %U https://proceedings.mlr.press/v80/shi18a.html %V 80 %X Recently there have been increasing interests in learning and inference with implicit distributions (i.e., distributions without tractable densities). To this end, we develop a gradient estimator for implicit distributions based on Stein’s identity and a spectral decomposition of kernel operators, where the eigenfunctions are approximated by the Nystr{ö}m method. Unlike the previous works that only provide estimates at the sample points, our approach directly estimates the gradient function, thus allows for a simple and principled out-of-sample extension. We provide theoretical results on the error bound of the estimator and discuss the bias-variance tradeoff in practice. The effectiveness of our method is demonstrated by applications to gradient-free Hamiltonian Monte Carlo and variational inference with implicit distributions. Finally, we discuss the intuition behind the estimator by drawing connections between the Nystr{ö}m method and kernel PCA, which indicates that the estimator can automatically adapt to the geometry of the underlying distribution.
APA
Shi, J., Sun, S. & Zhu, J.. (2018). A Spectral Approach to Gradient Estimation for Implicit Distributions. Proceedings of the 35th International Conference on Machine Learning, in Proceedings of Machine Learning Research 80:4644-4653 Available from https://proceedings.mlr.press/v80/shi18a.html.

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