Batch, match, and patch: low-rank approximations for score-based variational inference

TL;DR

This paper introduces a low-rank approximation method for score-based variational inference that improves scalability by efficiently parameterizing covariance matrices, enabling high-dimensional Bayesian inference.

Contribution

It extends the batch-and-match framework with a low-rank covariance parameterization and a patch step, enhancing scalability for high-dimensional problems.

Findings

Effective on synthetic distributions

Applicable to real-world high-dimensional problems

Improves computational efficiency of BBVI

Abstract

Black-box variational inference (BBVI) scales poorly to high-dimensional problems when it is used to estimate a multivariate Gaussian approximation with a full covariance matrix. In this paper, we extend the batch-and-match (BaM) framework for score-based BBVI to problems where it is prohibitively expensive to store such covariance matrices, let alone to estimate them. Unlike classical algorithms for BBVI, which use stochastic gradient descent to minimize the reverse Kullback-Leibler divergence, BaM uses more specialized updates to match the scores of the target density and its Gaussian approximation. We extend the updates for BaM by integrating them with a more compact parameterization of full covariance matrices. In particular, borrowing ideas from factor analysis, we add an extra step to each iteration of BaM--a patch--that projects each newly updated covariance matrix into a more efficiently parameterized family of diagonal plus low rank matrices. We evaluate this approach on a variety of synthetic target distributions and real-world problems in high-dimensional inference.