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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 familyspecification. In particular, we prove a quadratic bound on the gradientvariance of the STL estimator, one which encompasses misspecified variationalfamilies. Combined with previous works on the quadratic variance condition,this directly implies convergence of BBVI with the use of projected stochasticgradient descent. We also improve existing analysis on the regular closed-formentropy gradient estimators, which enables comparison against the STL estimatorand provides explicit non-asymptotic complexity guarantees for both.

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