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Understanding the gradient variance of black-box variational inference (BBVI)is a crucial step for establishing its convergence and developing algorithmicimprovements. However, existing studies have yet to show that the gradientvariance of BBVI satisfies the conditions used to study the convergence ofstochastic gradient descent (SGD), the workhorse of BBVI. In this work, we showthat BBVI satisfies a matching bound corresponding to the $ABC$ condition usedin the SGD literature when applied to smooth and quadratically-growinglog-likelihoods. Our results generalize to nonlinear covarianceparameterizations widely used in the practice of BBVI. Furthermore, we showthat the variance of the mean-field parameterization has provably superiordimensional dependence.

Practical and Matching Gradient Variance Bounds for Black-Box Variational Bayesian Inference