Provable convergence guarantees for black-box variational inference

Justin Domke, Robert M. Gower, Guillaume Garrigos

Keywords: optimization, variational inference

TL;DR: Black-box variational inference with projected or proximal SGD converges at a 1/√T rate for log-smooth and log-concave targets, and 1/T if strongly log-concave.

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.

Submission Number: 3741