Linear Convergence of Black-Box Variational Inference: Should We Stick the Landing?

TL;DR

This paper proves that black-box variational inference with control variates, especially the STL estimator, converges geometrically under perfect model specification, providing new theoretical guarantees and complexity bounds.

Contribution

It establishes the linear convergence of BBVI with STL estimator, including under misspecified models, and offers explicit complexity bounds for entropy gradient estimators.

Findings

BBVI with STL converges at a geometric rate under perfect model specification.

A quadratic bound on the gradient variance of STL is derived, applicable to misspecified models.

Explicit non-asymptotic complexity guarantees are provided for entropy gradient estimators.

Abstract

We prove that black-box variational inference (BBVI) with control variates, particularly the sticking-the-landing (STL) estimator, converges at a geometric (traditionally called "linear") rate under perfect variational family specification. In particular, we prove a quadratic bound on the gradient variance of the STL estimator, one which encompasses misspecified variational families. Combined with previous works on the quadratic variance condition, this directly implies convergence of BBVI with the use of projected stochastic gradient descent. For the projection operator, we consider a domain with triangular scale matrices, which the projection onto is computable in $\Theta(d)$ time, where $d$ is the dimensionality of the target posterior. We also improve existing analysis on the regular closed-form entropy gradient estimators, which enables comparison against the STL estimator, providing explicit non-asymptotic complexity guarantees for both.