Provable convergence guarantees for black-box variational inference

TL;DR

This paper provides the first rigorous convergence guarantees for black-box variational inference methods, addressing theoretical gaps related to gradient noise bounds and non-smooth objectives in stochastic optimization.

Contribution

It introduces novel convergence guarantees for proximal and projected stochastic gradient descent in dense Gaussian variational inference, based on quadratic noise bounds.

Findings

Existing gradient estimators satisfy quadratic noise bounds.

Proximal and projected stochastic gradient descent converge under these bounds.

The results apply to realistic inference problems.

Abstract

Black-box variational inference is widely used in situations where there is no proof that its stochastic optimization succeeds. We suggest this is due to a theoretical gap in existing stochastic optimization proofs: namely the challenge of gradient estimators with unusual noise bounds, and a composite non-smooth objective. For dense Gaussian variational families, we observe that existing gradient estimators based on reparameterization satisfy a quadratic noise bound and give novel convergence guarantees for proximal and projected stochastic gradient descent using this bound. This provides rigorous guarantees that methods similar to those used in practice converge on realistic inference problems.