Forward-Backward Gaussian Variational Inference via JKO in the Bures-Wasserstein Space

Michael Ziyang Diao, Krishna Balasubramanian, Sinho Chewi, Adil Salim
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:7960-7991, 2023.

Abstract

Variational inference (VI) seeks to approximate a target distribution $\pi$ by an element of a tractable family of distributions. Of key interest in statistics and machine learning is Gaussian VI, which approximates $\pi$ by minimizing the Kullback-Leibler (KL) divergence to $\pi$ over the space of Gaussians. In this work, we develop the (Stochastic) Forward-Backward Gaussian Variational Inference (FB-GVI) algorithm to solve Gaussian VI. Our approach exploits the composite structure of the KL divergence, which can be written as the sum of a smooth term (the potential) and a non-smooth term (the entropy) over the Bures-Wasserstein (BW) space of Gaussians endowed with the Wasserstein distance. For our proposed algorithm, we obtain state-of-the-art convergence guarantees when $\pi$ is log-smooth and log-concave, as well as the first convergence guarantees to first-order stationary solutions when $\pi$ is only log-smooth.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-diao23a, title = {Forward-Backward {G}aussian Variational Inference via {JKO} in the Bures-{W}asserstein Space}, author = {Diao, Michael Ziyang and Balasubramanian, Krishna and Chewi, Sinho and Salim, Adil}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {7960--7991}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/diao23a/diao23a.pdf}, url = {https://proceedings.mlr.press/v202/diao23a.html}, abstract = {Variational inference (VI) seeks to approximate a target distribution $\pi$ by an element of a tractable family of distributions. Of key interest in statistics and machine learning is Gaussian VI, which approximates $\pi$ by minimizing the Kullback-Leibler (KL) divergence to $\pi$ over the space of Gaussians. In this work, we develop the (Stochastic) Forward-Backward Gaussian Variational Inference (FB-GVI) algorithm to solve Gaussian VI. Our approach exploits the composite structure of the KL divergence, which can be written as the sum of a smooth term (the potential) and a non-smooth term (the entropy) over the Bures-Wasserstein (BW) space of Gaussians endowed with the Wasserstein distance. For our proposed algorithm, we obtain state-of-the-art convergence guarantees when $\pi$ is log-smooth and log-concave, as well as the first convergence guarantees to first-order stationary solutions when $\pi$ is only log-smooth.} }
Endnote
%0 Conference Paper %T Forward-Backward Gaussian Variational Inference via JKO in the Bures-Wasserstein Space %A Michael Ziyang Diao %A Krishna Balasubramanian %A Sinho Chewi %A Adil Salim %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-diao23a %I PMLR %P 7960--7991 %U https://proceedings.mlr.press/v202/diao23a.html %V 202 %X Variational inference (VI) seeks to approximate a target distribution $\pi$ by an element of a tractable family of distributions. Of key interest in statistics and machine learning is Gaussian VI, which approximates $\pi$ by minimizing the Kullback-Leibler (KL) divergence to $\pi$ over the space of Gaussians. In this work, we develop the (Stochastic) Forward-Backward Gaussian Variational Inference (FB-GVI) algorithm to solve Gaussian VI. Our approach exploits the composite structure of the KL divergence, which can be written as the sum of a smooth term (the potential) and a non-smooth term (the entropy) over the Bures-Wasserstein (BW) space of Gaussians endowed with the Wasserstein distance. For our proposed algorithm, we obtain state-of-the-art convergence guarantees when $\pi$ is log-smooth and log-concave, as well as the first convergence guarantees to first-order stationary solutions when $\pi$ is only log-smooth.
APA
Diao, M.Z., Balasubramanian, K., Chewi, S. & Salim, A.. (2023). Forward-Backward Gaussian Variational Inference via JKO in the Bures-Wasserstein Space. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:7960-7991 Available from https://proceedings.mlr.press/v202/diao23a.html.

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