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

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. We also improve existing analysis on the regular closed-form entropy gradient estimators, which enables comparison against the STL estimator and provides explicit non-asymptotic complexity guarantees for both.