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.

在完美变分族规范下，证明了带有控制变量的黑盒变分推断（BBVI），特别是附着着陆（STL）估计器，在几何（传统称为“线性”）速率下收敛。我们证明了STL估计器的梯度方差的二次界限，由此可以直接推出使用投影随机梯度下降的BBVI的收敛性。我们还改进了现有的关于常规闭式熵梯度估计器的分析，从而使其可以与STL估计器进行比较，并为两者提供明确的非渐近复杂性保证。

黑盒变分推断的线性收敛性：我们应该坚持完成吗？