# Bridging the Gap Between Variational Inference and Wasserstein Gradient Flows

arXiv: https://arxiv.org/abs/2310.20090 · HTML: https://mlsanity.com/paper/2310.20090/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 51 references that are in the map.
- First posted on arXiv 31 Oct 2023.

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

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems](https://mlsanity.com/paper/1603.04467/) (2016)
- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [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)
- [Expectation Propagation for approximate Bayesian inference](https://mlsanity.com/paper/1301.2294/) (2013)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [R'enyi Divergence Variational Inference](https://mlsanity.com/paper/1602.02311/) (2016)
- [Sticking the Landing: Simple, Lower-Variance Gradient Estimators for Variational Inference](https://mlsanity.com/paper/1703.09194/) (2017)
- [Convergence of Langevin MCMC in KL-divergence](https://mlsanity.com/paper/1705.09048/) (2017)
- [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022)
- [VarGrad: A Low-Variance Gradient Estimator for Variational Inference](https://mlsanity.com/paper/2010.10436/) (2020)
- [SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergence](https://mlsanity.com/paper/2006.02509/) (2020)
- [Markovian Score Climbing: Variational Inference with KL(p||q)](https://mlsanity.com/paper/2003.10374/) (2020)
- [Refining Deep Generative Models via Discriminator Gradient Flow](https://mlsanity.com/paper/2012.00780/) (2020)
- [Doubly Reparameterized Gradient Estimators for Monte Carlo Objectives](https://mlsanity.com/paper/1810.04152/) (2018)
- [Advances in Black-Box VI: Normalizing Flows, Importance Weighting, and Optimization](https://mlsanity.com/paper/2006.10343/) (2020)
- [KALE Flow: A Relaxed KL Gradient Flow for Probabilities with Disjoint Support](https://mlsanity.com/paper/2106.08929/) (2021)
- [Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Space](https://mlsanity.com/paper/2304.05398/) (2023)
- [Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent](https://mlsanity.com/paper/2106.08502/) (2021)
- [Variational Refinement for Importance Sampling Using the Forward Kullback-Leibler Divergence](https://mlsanity.com/paper/2106.15980/) (2021)
- [Gradients should stay on Path: Better Estimators of the Reverse- and Forward KL Divergence for Normalizing Flows](https://mlsanity.com/paper/2207.08219/) (2022)
- [Learning to Draw Samples: With Application to Amortized MLE for Generative Adversarial Learning](https://mlsanity.com/paper/1611.01722/) (2016)
- [On the Difficulty of Unbiased Alpha Divergence Minimization](https://mlsanity.com/paper/2010.09541/) (2020)
- [Training Neural Networks for Likelihood/Density Ratio Estimation](https://mlsanity.com/paper/1911.00405/) (2019)
- [Generalized Doubly Reparameterized Gradient Estimators](https://mlsanity.com/paper/2101.11046/) (2021)
- [MonoFlow: Rethinking Divergence GANs via the Perspective of Wasserstein Gradient Flows](https://mlsanity.com/paper/2302.01075/) (2023)
- [Path-Gradient Estimators for Continuous Normalizing Flows](https://mlsanity.com/paper/2206.09016/) (2022)

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

- [Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space](https://mlsanity.com/paper/2312.02849/) (2023)
- [Theoretical Guarantees for Variational Inference with Fixed-Variance Mixture of Gaussians](https://mlsanity.com/paper/2406.04012/) (2024)
- [Disentangling impact of capacity, objective, batchsize, estimators, and step-size on flow VI](https://mlsanity.com/paper/2412.08824/) (2024)
- [Wasserstein Gradient Boosting: A Framework for Distribution-Valued Supervised Learning](https://mlsanity.com/paper/2405.09536/) (2024)
- [Inclusive KL Gradient Flows: Otto-Wasserstein, Fisher-Rao-Gaussian, and Local-Estimator Dynamics](https://mlsanity.com/paper/2411.00214/) (2024)
- [A Particle Algorithm for Mean-Field Variational Inference](https://mlsanity.com/paper/2412.20385/) (2024)
- [Stochastic variance-reduced Gaussian variational inference on the Bures-Wasserstein manifold](https://mlsanity.com/paper/2410.02490/) (2024)
- [Stochastic Gradient Variational Inference with Price's Gradient Estimator from Bures-Wasserstein to Parameter Space](https://mlsanity.com/paper/2602.18718/) (2026)

## Abstract (arXiv)

Variational inference is a technique that approximates a target distribution by optimizing within the parameter space of variational families. On the other hand, Wasserstein gradient flows describe optimization within the space of probability measures where they do not necessarily admit a parametric density function. In this paper, we bridge the gap between these two methods. We demonstrate that, under certain conditions, the Bures-Wasserstein gradient flow can be recast as the Euclidean gradient flow where its forward Euler scheme is the standard black-box variational inference algorithm. Specifically, the vector field of the gradient flow is generated via the path-derivative gradient estimator. We also offer an alternative perspective on the path-derivative gradient, framing it as a distillation procedure to the Wasserstein gradient flow. Distillations can be extended to encompass f-divergences and non-Gaussian variational families. This extension yields a new gradient estimator for f-divergences, readily implementable using contemporary machine learning libraries like PyTorch or TensorFlow.