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.

翻译：我们证明，在完美变分族设定的条件下，采用控制变量（特别是“着陆固定”（STL）估计器）的黑箱变分推断（BBVI）以几何速率（传统上称为“线性”速率）收敛。具体而言，我们给出了STL估计器梯度方差的二次上界，该上界涵盖了设定错误的变分族。结合先前关于二次方差条件的研究，这直接推得采用投影随机梯度下降的BBVI的收敛性。针对投影算子，我们考虑了一个具有三角尺度矩阵的域，其投影可在$\Theta(d)$时间内计算，其中$d$是目标后验的维度。我们还改进了对常规闭式熵梯度估计器的现有分析，从而能够与STL估计器进行比较，并为两者提供了显式的非渐近复杂度保证。