On the Convergence of Black-Box Variational Inference

TL;DR

This paper proves the first convergence guarantees for full black-box variational inference (BBVI) without simplifying assumptions, showing that certain design choices affect convergence and that proximal stochastic gradient descent improves it.

Contribution

It provides the first convergence analysis for general BBVI, identifies suboptimal design choices, and demonstrates the effectiveness of proximal SGD in practice.

Findings

Proximal SGD achieves the strongest convergence guarantees for BBVI.

Certain nonlinear parameterizations can slow convergence.

Empirical results favor proximal SGD over standard BBVI implementations.

Abstract

We provide the first convergence guarantee for full black-box variational inference (BBVI), also known as Monte Carlo variational inference. While preliminary investigations worked on simplified versions of BBVI (e.g., bounded domain, bounded support, only optimizing for the scale, and such), our setup does not need any such algorithmic modifications. Our results hold for log-smooth posterior densities with and without strong log-concavity and the location-scale variational family. Also, our analysis reveals that certain algorithm design choices commonly employed in practice, particularly, nonlinear parameterizations of the scale of the variational approximation, can result in suboptimal convergence rates. Fortunately, running BBVI with proximal stochastic gradient descent fixes these limitations, and thus achieves the strongest known convergence rate guarantees. We evaluate this theoretical insight by comparing proximal SGD against other standard implementations of BBVI on large-scale Bayesian inference problems.