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.

翻译：我们证明了在完美变分族设定的条件下，采用控制变量的黑箱变分推理（BBVI），特别是“保持着陆”（STL）估计器，具有几何（传统上称为“线性”）收敛速度。具体而言，我们证明了STL估计器梯度方差的一个二次界，该界涵盖了错误指定的变分族。结合先前关于二次方差条件的研究，这直接表明使用投影随机梯度下降的BBVI具有收敛性。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，并为两者提供明确的非渐近复杂度保证。