Provably Scalable Black-Box Variational Inference with Structured Variational Families

Abstract

Variational families with full-rank covariance approximations are known not to work well in black-box variational inference (BBVI), both empirically and theoretically. In fact, recent computational complexity results for BBVI have established that full-rank variational families scale poorly with the dimensionality of the problem compared to e.g. mean-field families. This is particularly critical to hierarchical Bayesian models with local variables; their dimensionality increases with the size of the datasets. Consequently, one gets an iteration complexity with an explicit O(N^2) dependence on the dataset size N. In this paper, we explore a theoretical middle ground between mean-field variational families and full-rank families: structured variational families. We rigorously prove that certain scale matrix structures can achieve a better iteration complexity of O(N), implying better scaling with respect to N. We empirically verify our theoretical results on large-scale hierarchical models.

Tasks

Reproductions