# Understanding Stochastic Natural Gradient Variational Inference

arXiv: https://arxiv.org/abs/2406.01870 · HTML: https://mlsanity.com/paper/2406.01870/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 60 references that are in the map.
- First posted on arXiv 4 Jun 2024.

## Builds on (most cited first, 30 of 34)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [New insights and perspectives on the natural gradient method](https://mlsanity.com/paper/1412.1193/) (2014)
- [Gaussian Processes for Big Data](https://mlsanity.com/paper/1309.6835/) (2013)
- [Black Box Variational Inference](https://mlsanity.com/paper/1401.0118/) (2013)
- [A Theory of Regularized Markov Decision Processes](https://mlsanity.com/paper/1901.11275/) (2019)
- [Scalable Variational Gaussian Process Classification](https://mlsanity.com/paper/1411.2005/) (2014)
- [Practical Deep Learning with Bayesian Principles](https://mlsanity.com/paper/1906.02506/) (2019)
- [Fast and Scalable Bayesian Deep Learning by Weight-Perturbation in Adam](https://mlsanity.com/paper/1806.04854/) (2018)
- [On the Convergence Rates of Policy Gradient Methods](https://mlsanity.com/paper/2201.07443/) (2022)
- [Adaptive Trust Region Policy Optimization: Global Convergence and Faster Rates for Regularized MDPs](https://mlsanity.com/paper/1909.02769/) (2019)
- [Relatively-Smooth Convex Optimization by First-Order Methods, and Applications](https://mlsanity.com/paper/1610.05708/) (2016)
- [Noisy Natural Gradient as Variational Inference](https://mlsanity.com/paper/1712.02390/) (2017)
- [Fast Convergence of Natural Gradient Descent for Overparameterized Neural Networks](https://mlsanity.com/paper/1905.10961/) (2019)
- [Conjugate-Computation Variational Inference : Converting Variational Inference in Non-Conjugate Models to Inferences in Conjugate Models](https://mlsanity.com/paper/1703.04265/) (2017)
- [The Information Geometry of Mirror Descent](https://mlsanity.com/paper/1310.7780/) (2013)
- [Fast and Simple Natural-Gradient Variational Inference with Mixture of Exponential-family Approximations](https://mlsanity.com/paper/1906.02914/) (2019)
- [High Probability Convergence of Stochastic Gradient Methods](https://mlsanity.com/paper/2302.14843/) (2023)
- [On the Linear convergence of Natural Policy Gradient Algorithm](https://mlsanity.com/paper/2105.01424/) (2021)
- [Linear Convergence of Natural Policy Gradient Methods with Log-Linear Policies](https://mlsanity.com/paper/2210.01400/) (2022)
- [Natural Gradients in Practice: Non-Conjugate Variational Inference in Gaussian Process Models](https://mlsanity.com/paper/1803.09151/) (2018)
- [Handling the Positive-Definite Constraint in the Bayesian Learning Rule](https://mlsanity.com/paper/2002.10060/) (2020)
- [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)
- [Tractable structured natural gradient descent using local parameterizations](https://mlsanity.com/paper/2102.07405/) (2021)
- [A Unified Perspective on Natural Gradient Variational Inference with Gaussian Mixture Models](https://mlsanity.com/paper/2209.11533/) (2022)
- [On the Convergence of Black-Box Variational Inference](https://mlsanity.com/paper/2305.15349/) (2023)
- [Faster Stochastic Variational Inference using Proximal-Gradient Methods with General Divergence Functions](https://mlsanity.com/paper/1511.00146/) (2015)
- [Accelerating Natural Gradient with Higher-Order Invariance](https://mlsanity.com/paper/1803.01273/) (2018)

## Built on by (most cited first, 8 of 8)

- [Optimization Guarantees for Square-Root Natural-Gradient Variational Inference](https://mlsanity.com/paper/2507.07853/) (2025)
- [Natural Gradient VI: Guarantees for Non-Conjugate Models](https://mlsanity.com/paper/2510.19163/) (2025)
- [Scalable Mean-Field Variational Inference via Preconditioned Primal-Dual Optimization](https://mlsanity.com/paper/2602.07632/) (2026)
- [Convergence of projected stochastic natural gradient variational inference for various step size and sample or batch size schedules](https://mlsanity.com/paper/2604.00683/) (2026)
- [ONG: Orthogonal Natural Gradient Descent](https://mlsanity.com/paper/2508.17169/) (2025)
- [A Flexible Empirical Bayes Approach to Generalized Linear Models, with Applications to Sparse Logistic Regression](https://mlsanity.com/paper/2601.21217/) (2026)
- [Stochastic Gradient Variational Inference with Price's Gradient Estimator from Bures-Wasserstein to Parameter Space](https://mlsanity.com/paper/2602.18718/) (2026)
- [SGD for Variational Inference: Tackling Unbounded Variance via Preconditioning and Dynamic Batching](https://mlsanity.com/paper/2605.07531/) (2026)

## Abstract (arXiv)

Stochastic natural gradient variational inference (NGVI) is a popular posterior inference method with applications in various probabilistic models. Despite its wide usage, little is known about the non-asymptotic convergence rate in the stochastic setting. We aim to lessen this gap and provide a better understanding. For conjugate likelihoods, we prove the first O(frac1T) non-asymptotic convergence rate of stochastic NGVI. The complexity is no worse than stochastic gradient descent (aka black-box variational inference) and the rate likely has better constant dependency that leads to faster convergence in practice. For non-conjugate likelihoods, we show that stochastic NGVI with the canonical parameterization implicitly optimizes a non-convex objective. Thus, a global convergence rate of O(frac1T) is unlikely without some significant new understanding of optimizing the ELBO using natural gradients.