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

arXiv: https://arxiv.org/abs/2307.14642 · HTML: https://mlsanity.com/paper/2307.14642/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 42 of its 82 references that are in the map.
- First posted on arXiv 27 Jul 2023.

## Builds on (most cited first, 30 of 42)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Black Box Variational Inference](https://mlsanity.com/paper/1401.0118/) (2013)
- [Monte Carlo Gradient Estimation in Machine Learning](https://mlsanity.com/paper/1906.10652/) (2019)
- [Advances in Variational Inference](https://mlsanity.com/paper/1711.05597/) (2017)
- [SGD: General Analysis and Improved Rates](https://mlsanity.com/paper/1901.09401/) (2019)
- [Minimizing Finite Sums with the Stochastic Average Gradient](https://mlsanity.com/paper/1309.2388/) (2013)
- [Handbook of Convergence Theorems for (Stochastic) Gradient Methods](https://mlsanity.com/paper/2301.11235/) (2023)
- [Automatic Differentiation Variational Inference](https://mlsanity.com/paper/1603.00788/) (2016)
- [The Power of Interpolation: Understanding the Effectiveness of SGD in Modern Over-parametrized Learning](https://mlsanity.com/paper/1712.06559/) (2017)
- [Fast and Faster Convergence of SGD for Over-Parameterized Models and an Accelerated Perceptron](https://mlsanity.com/paper/1810.07288/) (2018)
- [Log-concave sampling: Metropolis-Hastings algorithms are fast](https://mlsanity.com/paper/1801.02309/) (2018)
- [Sticking the Landing: Simple, Lower-Variance Gradient Estimators for Variational Inference](https://mlsanity.com/paper/1703.09194/) (2017)
- [Tight Analyses for Non-Smooth Stochastic Gradient Descent](https://mlsanity.com/paper/1812.05217/) (2018)
- [Bridging the Gap between Constant Step Size Stochastic Gradient Descent and Markov Chains](https://mlsanity.com/paper/1707.06386/) (2017)
- [Variance-Reduced Methods for Machine Learning](https://mlsanity.com/paper/2010.00892/) (2020)
- [Tighter Variational Bounds are Not Necessarily Better](https://mlsanity.com/paper/1802.04537/) (2018)
- [Concentration of tempered posteriors and of their variational approximations](https://mlsanity.com/paper/1706.09293/) (2017)
- [Fast Convergence of Stochastic Gradient Descent under a Strong Growth Condition](https://mlsanity.com/paper/1308.6370/) (2013)
- [Variance reduction properties of the reparameterization trick](https://mlsanity.com/paper/1809.10330/) (2018)
- [Advances in Black-Box VI: Normalizing Flows, Importance Weighting, and Optimization](https://mlsanity.com/paper/2006.10343/) (2020)
- [Importance Sampling for Minibatches](https://mlsanity.com/paper/1602.02283/) (2016)
- [Provable Smoothness Guarantees for Black-Box Variational Inference](https://mlsanity.com/paper/1901.08431/) (2019)
- [Provable convergence guarantees for black-box variational inference](https://mlsanity.com/paper/2306.03638/) (2023)
- [Reducing Reparameterization Gradient Variance](https://mlsanity.com/paper/1705.07880/) (2017)
- [Quasi-Monte Carlo Variational Inference](https://mlsanity.com/paper/1807.01604/) (2018)
- [On the Convergence of Black-Box Variational Inference](https://mlsanity.com/paper/2305.15349/) (2023)
- [Using Large Ensembles of Control Variates for Variational Inference](https://mlsanity.com/paper/1810.12482/) (2018)
- [Provable Gradient Variance Guarantees for Black-Box Variational Inference](https://mlsanity.com/paper/1906.08241/) (2019)
- [On the Difficulty of Unbiased Alpha Divergence Minimization](https://mlsanity.com/paper/2010.09541/) (2020)
- [Faster Stochastic Variational Inference using Proximal-Gradient Methods with General Divergence Functions](https://mlsanity.com/paper/1511.00146/) (2015)

## Built on by (most cited first, 10 of 10)

- [Kernel Semi-Implicit Variational Inference](https://mlsanity.com/paper/2405.18997/) (2024)
- [On Divergence Measures for Training GFlowNets](https://mlsanity.com/paper/2410.09355/) (2024)
- [Efficient Mixture Learning in Black-Box Variational Inference](https://mlsanity.com/paper/2406.07083/) (2024)
- [Natural Gradient VI: Guarantees for Non-Conjugate Models](https://mlsanity.com/paper/2510.19163/) (2025)
- [AutoSGD: Automatic Learning Rate Selection for Stochastic Gradient Descent](https://mlsanity.com/paper/2505.21651/) (2025)
- [Understanding and mitigating difficulties in posterior predictive evaluation](https://mlsanity.com/paper/2405.19747/) (2024)
- [Nearly Dimension-Independent Convergence of Mean-Field Black-Box Variational Inference](https://mlsanity.com/paper/2505.21721/) (2025)
- [A Kernel Approach for Semi-implicit Variational Inference](https://mlsanity.com/paper/2601.12023/) (2026)
- [Stochastic Gradient Variational Inference with Price's Gradient Estimator from Bures-Wasserstein to Parameter Space](https://mlsanity.com/paper/2602.18718/) (2026)
- [Large-scale empirical tuning and comparison of default optimizers for variational inference](https://mlsanity.com/paper/2606.07841/) (2026)

## Abstract (arXiv)

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.