  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.
