Proximal Gradient Algorithms for Gaussian Variational Inference:Optimization in the Bures–Wasserstein Space

Variational inference (VI) seeks to approximate a target distribution π by an element of a tractable family of distributions. Of key interest in statistics and machine learning is Gaussian VI, which approximates π by minimizing the Kullback–Leibler (KL) divergence to π 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 π is log-smooth and log-concave, as well as the first convergence guarantees to first-order stationary solutions when π is only log-smooth. Additionally, in the setting where the potential admits a representation as the average of many smooth component functionals, we develop and analyze a variance-reduced extension to (Stochastic) FB–GVI with improved complexity guarantees.

Name thesis:degree_name

Master

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2023

Author dc:creator

• Diao, Michael Ziyang

Advisors dc:contributor.advisor

• Moitra, Ankur
• Chewi, Sinho

Statement dc:rights

• In Copyright - Educational Use Permitted
• Copyright retained by author(s)

Licence dc:rights.uri

• https://rightsstatements.org/page/InC-EDU/1.0/

Handle dc:identifier.uri

https://hdl.handle.net/1721.1/151664

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/151664