# Stochastic Gradient Variational Inference with Price's Gradient Estimator from Bures-Wasserstein to Parameter Space

arXiv: https://arxiv.org/abs/2602.18718 · HTML: https://mlsanity.com/paper/2602.18718/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 47 of its 111 references that are in the map.
- First posted on arXiv 21 Feb 2026.

## Builds on (most cited first, 30 of 47)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Pyro: Deep Universal Probabilistic Programming](https://mlsanity.com/paper/1810.09538/) (2018)
- [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)
- [SGD: General Analysis and Improved Rates](https://mlsanity.com/paper/1901.09401/) (2019)
- [Automatic Differentiation Variational Inference](https://mlsanity.com/paper/1603.00788/) (2016)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [Fast and Faster Convergence of SGD for Over-Parameterized Models and an Accelerated Perceptron](https://mlsanity.com/paper/1810.07288/) (2018)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [The Bayesian Learning Rule](https://mlsanity.com/paper/2107.04562/) (2021)
- [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022)
- [A Unified Theory of SGD: Variance Reduction, Sampling, Quantization and Coordinate Descent](https://mlsanity.com/paper/1905.11261/) (2019)
- [Unified Optimal Analysis of the (Stochastic) Gradient Method](https://mlsanity.com/paper/1907.04232/) (2019)
- [Fast Convergence of Stochastic Gradient Descent under a Strong Growth Condition](https://mlsanity.com/paper/1308.6370/) (2013)
- [Fast and Simple Natural-Gradient Variational Inference with Mixture of Exponential-family Approximations](https://mlsanity.com/paper/1906.02914/) (2019)
- [Variance reduction properties of the reparameterization trick](https://mlsanity.com/paper/1809.10330/) (2018)
- [Fast yet Simple Natural-Gradient Descent for Variational Inference in Complex Models](https://mlsanity.com/paper/1807.04489/) (2018)
- [Advances in Black-Box VI: Normalizing Flows, Importance Weighting, and Optimization](https://mlsanity.com/paper/2006.10343/) (2020)
- [Unified Analysis of Stochastic Gradient Methods for Composite Convex and Smooth Optimization](https://mlsanity.com/paper/2006.11573/) (2020)
- [Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Space](https://mlsanity.com/paper/2304.05398/) (2023)
- [Automated Variational Inference in Probabilistic Programming](https://mlsanity.com/paper/1301.1299/) (2013)
- [Stochastic Quasi-Gradient Methods: Variance Reduction via Jacobian Sketching](https://mlsanity.com/paper/1805.02632/) (2018)
- [Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent](https://mlsanity.com/paper/2106.08502/) (2021)
- [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)

## Abstract (arXiv)

For approximating a target distribution given only its unnormalized log-density, stochastic gradient-based variational inference (VI) algorithms are a popular approach. For example, Wasserstein VI (WVI) and black-box VI (BBVI) perform gradient descent in measure space (Bures-Wasserstein space) and parameter space, respectively. Previously, for the Gaussian variational family, convergence guarantees for WVI have shown superiority over existing results for black-box VI with the reparametrization gradient, suggesting the measure space approach might provide some unique benefits. In this work, however, we close this gap by obtaining identical state-of-the-art iteration complexity guarantees for both. In particular, we identify that WVI's superiority stems from the specific gradient estimator it uses, which BBVI can also leverage with minor modifications. The estimator in question is usually associated with Price's theorem and utilizes second-order information (Hessians) of the target log-density. We will refer to this as Price's gradient. On the flip side, WVI can be made more widely applicable by using the reparametrization gradient, which requires only gradients of the log-density. We empirically demonstrate that the use of Price's gradient is the major source of performance improvement.