Linear Convergence of Black-Box Variational Inference: Should We Stick the Landing?
This provides theoretical guarantees for variational inference methods, addressing convergence issues in probabilistic modeling, though it is incremental as it builds on prior work on variance conditions.
The paper proves that black-box variational inference with the sticking-the-landing estimator converges at a geometric rate under perfect variational family specification, achieving a quadratic bound on gradient variance and enabling linear convergence with projected stochastic gradient descent in Θ(d) time.
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 $Θ(d)$ 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.