# Particle-based Variational Inference with Preconditioned Functional Gradient Flow

arXiv: https://arxiv.org/abs/2211.13954 · HTML: https://mlsanity.com/paper/2211.13954/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 62 references that are in the map.
- First posted on arXiv 25 Nov 2022.

## Builds on (most cited first, 30 of 32)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Stochastic Gradient Hamiltonian Monte Carlo](https://mlsanity.com/paper/1402.4102/) (2014)
- [Black Box Variational Inference](https://mlsanity.com/paper/1401.0118/) (2013)
- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Stein Variational Gradient Descent as Gradient Flow](https://mlsanity.com/paper/1704.07520/) (2017)
- [A Kernel Test of Goodness of Fit](https://mlsanity.com/paper/1602.02964/) (2016)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [Preconditioned Stochastic Gradient Langevin Dynamics for Deep Neural Networks](https://mlsanity.com/paper/1512.07666/) (2015)
- [On the geometry of Stein variational gradient descent](https://mlsanity.com/paper/1912.00894/) (2019)
- [Large-Scale Wasserstein Gradient Flows](https://mlsanity.com/paper/2106.00736/) (2021)
- [SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergence](https://mlsanity.com/paper/2006.02509/) (2020)
- [Optimizing Functionals on the Space of Probabilities with Input Convex Neural Networks](https://mlsanity.com/paper/2106.00774/) (2021)
- [Variational Wasserstein gradient flow](https://mlsanity.com/paper/2112.02424/) (2021)
- [A Stein variational Newton method](https://mlsanity.com/paper/1806.03085/) (2018)
- [Message Passing Stein Variational Gradient Descent](https://mlsanity.com/paper/1711.04425/) (2017)
- [Incremental Gradient, Subgradient, and Proximal Methods for Convex Optimization: A Survey](https://mlsanity.com/paper/1507.01030/) (2015)
- [Accelerated Information Gradient flow](https://mlsanity.com/paper/1909.02102/) (2019)
- [Stein Variational Gradient Descent With Matrix-Valued Kernels](https://mlsanity.com/paper/1910.12794/) (2019)
- [Learning to Draw Samples with Amortized Stein Variational Gradient Descent](https://mlsanity.com/paper/1707.06626/) (2017)
- [Information Newton's flow: second-order optimization method in probability space](https://mlsanity.com/paper/2001.04341/) (2020)
- [Projected Wasserstein gradient descent for high-dimensional Bayesian inference](https://mlsanity.com/paper/2102.06350/) (2021)
- [Sliced Kernelized Stein Discrepancy](https://mlsanity.com/paper/2006.16531/) (2020)
- [Learning to Draw Samples: With Application to Amortized MLE for Generative Adversarial Learning](https://mlsanity.com/paper/1611.01722/) (2016)
- [Projected Stein Variational Newton: A Fast and Scalable Bayesian Inference Method in High Dimensions](https://mlsanity.com/paper/1901.08659/) (2019)
- [Stein Neural Sampler](https://mlsanity.com/paper/1810.03545/) (2018)
- [Neural Variational Gradient Descent](https://mlsanity.com/paper/2107.10731/) (2021)
- [Wasserstein Proximal of GANs](https://mlsanity.com/paper/2102.06862/) (2021)
- [Grassmann Stein Variational Gradient Descent](https://mlsanity.com/paper/2202.03297/) (2022)

## Built on by (most cited first, 20 of 20)

- [ProlificDreamer: High-Fidelity and Diverse Text-to-3D Generation with Variational Score Distillation](https://mlsanity.com/paper/2305.16213/) (2023)
- [Reverse Diffusion Monte Carlo](https://mlsanity.com/paper/2307.02037/) (2023)
- [Generative Sliced MMD Flows with Riesz Kernels](https://mlsanity.com/paper/2305.11463/) (2023)
- [Retrieval-Augmented Score Distillation for Text-to-3D Generation](https://mlsanity.com/paper/2402.02972/) (2024)
- [Particle-based Variational Inference with Generalized Wasserstein Gradient Flow](https://mlsanity.com/paper/2310.16516/) (2023)
- [Neural Wasserstein Gradient Flows for Maximum Mean Discrepancies with Riesz Kernels](https://mlsanity.com/paper/2301.11624/) (2023)
- [Kernel Semi-Implicit Variational Inference](https://mlsanity.com/paper/2405.18997/) (2024)
- [Path-Guided Particle-based Sampling](https://mlsanity.com/paper/2412.03312/) (2024)
- [Kernel Approximation of Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2410.20622/) (2024)
- [Wasserstein Gradient Flow over Variational Parameter Space for Variational Inference](https://mlsanity.com/paper/2310.16705/) (2023)
- [Faster Sampling via Stochastic Gradient Proximal Sampler](https://mlsanity.com/paper/2405.16734/) (2024)
- [GradeADreamer: Enhanced Text-to-3D Generation Using Gaussian Splatting and Multi-View Diffusion](https://mlsanity.com/paper/2406.09850/) (2024)
- [Rethinking the Diffusion Models for Numerical Tabular Data Imputation from the Perspective of Wasserstein Gradient Flow](https://mlsanity.com/paper/2406.15762/) (2024)
- [Perturb-and-Revise: Flexible 3D Editing with Generative Trajectories](https://mlsanity.com/paper/2412.05279/) (2024)
- [Semi-Implicit Functional Gradient Flow for Efficient Sampling](https://mlsanity.com/paper/2410.17935/) (2024)
- [ELBOing Stein: Variational Bayes with Stein Mixture Inference](https://mlsanity.com/paper/2410.22948/) (2024)
- [A Gradient Flow Perspective on Minimum MMD Estimation](https://mlsanity.com/paper/2607.03871/) (2026)
- [Functional Gradient Flows for Constrained Sampling](https://mlsanity.com/paper/2410.23170/) (2024)
- [Analyzing and Improving Diffusion Models for Time-Series Data Imputation: A Proximal Recursion Perspective](https://mlsanity.com/paper/2602.01182/) (2026)
- [Slack More, Predict Better: Proximal Relaxation for Probabilistic Latent Variable Model-based Soft Sensors](https://mlsanity.com/paper/2603.11473/) (2026)

## Abstract (arXiv)

Particle-based variational inference (VI) minimizes the KL divergence between model samples and the target posterior with gradient flow estimates. With the popularity of Stein variational gradient descent (SVGD), the focus of particle-based VI algorithms has been on the properties of functions in Reproducing Kernel Hilbert Space (RKHS) to approximate the gradient flow. However, the requirement of RKHS restricts the function class and algorithmic flexibility. This paper offers a general solution to this problem by introducing a functional regularization term that encompasses the RKHS norm as a special case. This allows us to propose a new particle-based VI algorithm called preconditioned functional gradient flow (PFG). Compared to SVGD, PFG has several advantages. It has a larger function class, improved scalability in large particle-size scenarios, better adaptation to ill-conditioned distributions, and provable continuous-time convergence in KL divergence. Additionally, non-linear function classes such as neural networks can be incorporated to estimate the gradient flow. Our theory and experiments demonstrate the effectiveness of the proposed framework.