# Multilevel Monte Carlo Variational Inference

arXiv: https://arxiv.org/abs/1902.00468 · HTML: https://mlsanity.com/paper/1902.00468/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Feb 2019.

## Built on by (most cited first, 15 of 15)

- [Provable convergence guarantees for black-box variational inference](https://mlsanity.com/paper/2306.03638/) (2023)
- [On the Convergence of Black-Box Variational Inference](https://mlsanity.com/paper/2305.15349/) (2023)
- [Linear Convergence of Black-Box Variational Inference: Should We Stick the Landing?](https://mlsanity.com/paper/2307.14642/) (2023)
- [Practical and Matching Gradient Variance Bounds for Black-Box Variational Bayesian Inference](https://mlsanity.com/paper/2303.10472/) (2023)
- [Bayesian Deep Learning with Multilevel Trace-class Neural Networks](https://mlsanity.com/paper/2203.12961/) (2022)
- [Multi-level Monte-Carlo Gradient Methods for Stochastic Optimization with Biased Oracles](https://mlsanity.com/paper/2408.11084/) (2024)
- [Disentangling impact of capacity, objective, batchsize, estimators, and step-size on flow VI](https://mlsanity.com/paper/2412.08824/) (2024)
- [Multilevel neural simulation-based inference](https://mlsanity.com/paper/2506.06087/) (2025)
- [Accelerating Look-ahead in Bayesian Optimization: Multilevel Monte Carlo is All you Need](https://mlsanity.com/paper/2402.02111/) (2024)
- [Multifidelity linear regression for scientific machine learning from scarce data](https://mlsanity.com/paper/2403.08627/) (2024)
- [Nearly Dimension-Independent Convergence of Mean-Field Black-Box Variational Inference](https://mlsanity.com/paper/2505.21721/) (2025)
- [Multi-level Monte Carlo Dropout for Efficient Uncertainty Quantification](https://mlsanity.com/paper/2601.13272/) (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)
- [Theoretical and Practical Advances in Hierarchical Bayesian Deep Learning Methodology](https://mlsanity.com/paper/2610.19074/) (2026)

## Abstract (arXiv)

We propose a variance reduction framework for variational inference using the Multilevel Monte Carlo (MLMC) method. Our framework is built on reparameterized gradient estimators and "recycles" parameters obtained from past update history in optimization. In addition, our framework provides a new optimization algorithm based on stochastic gradient descent (SGD) that adaptively estimates the sample size used for gradient estimation according to the ratio of the gradient variance. We theoretically show that, with our method, the variance of the gradient estimator decreases as optimization proceeds and that a learning rate scheduler function helps improve the convergence. We also show that, in terms of the textitsignal-to-noise ratio, our method can improve the quality of gradient estimation by the learning rate scheduler function without increasing the initial sample size. Finally, we confirm that our method achieves faster convergence and reduces the variance of the gradient estimator compared with other methods through experimental comparisons with baseline methods using several benchmark datasets.