# Path-Gradient Estimators for Continuous Normalizing Flows

arXiv: https://arxiv.org/abs/2206.09016 · HTML: https://mlsanity.com/paper/2206.09016/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 30 references that are in the map.
- First posted on arXiv 17 Jun 2022.

## Builds on (most cited first, 21 of 21)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Neural Importance Sampling](https://mlsanity.com/paper/1808.03856/) (2018)
- [Equivariant flow-based sampling for lattice gauge theory](https://mlsanity.com/paper/2003.06413/) (2020)
- [Solving Statistical Mechanics Using Variational Autoregressive Networks](https://mlsanity.com/paper/1809.10606/) (2018)
- [Sticking the Landing: Simple, Lower-Variance Gradient Estimators for Variational Inference](https://mlsanity.com/paper/1703.09194/) (2017)
- [Sylvester Normalizing Flows for Variational Inference](https://mlsanity.com/paper/1803.05649/) (2018)
- [Reweighted Wake-Sleep](https://mlsanity.com/paper/1406.2751/) (2014)
- [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)
- [Introduction to Normalizing Flows for Lattice Field Theory](https://mlsanity.com/paper/2101.08176/) (2021)
- [Scaling Up Machine Learning For Quantum Field Theory with Equivariant Continuous Flows](https://mlsanity.com/paper/2110.02673/) (2021)
- [On the Difficulty of Unbiased Alpha Divergence Minimization](https://mlsanity.com/paper/2010.09541/) (2020)
- [Generalized Doubly Reparameterized Gradient Estimators](https://mlsanity.com/paper/2101.11046/) (2021)
- [Empirical Evaluation of Biased Methods for Alpha Divergence Minimization](https://mlsanity.com/paper/2105.06587/) (2021)
- [Machine Learning Trivializing Maps: A First Step Towards Understanding How Flow-Based Samplers Scale Up](https://mlsanity.com/paper/2112.15532/) (2021)
- [Machine Learning and Variational Algorithms for Lattice Field Theory](https://mlsanity.com/paper/2106.01975/) (2021)

## Built on by (most cited first, 11 of 11)

- [Diffusion Generative Flow Samplers: Improving learning signals through partial trajectory optimization](https://mlsanity.com/paper/2310.02679/) (2023)
- [Learning Lattice Quantum Field Theories with Equivariant Continuous Flows](https://mlsanity.com/paper/2207.00283/) (2022)
- [Gradients should stay on Path: Better Estimators of the Reverse- and Forward KL Divergence for Normalizing Flows](https://mlsanity.com/paper/2207.08219/) (2022)
- [Advances in machine-learning-based sampling motivated by lattice quantum chromodynamics](https://mlsanity.com/paper/2309.01156/) (2023)
- [Sampling the lattice Nambu-Goto string using Continuous Normalizing Flows](https://mlsanity.com/paper/2307.01107/) (2023)
- [Bridging the Gap Between Variational Inference and Wasserstein Gradient Flows](https://mlsanity.com/paper/2310.20090/) (2023)
- [Fast and Unified Path Gradient Estimators for Normalizing Flows](https://mlsanity.com/paper/2403.15881/) (2024)
- [Path Gradients after Flow Matching](https://mlsanity.com/paper/2505.10139/) (2025)
- [SESaMo: Symmetry-Enforcing Stochastic Modulation for Normalizing Flows](https://mlsanity.com/paper/2505.19619/) (2025)
- [Multi-Lattice Sampling of Quantum Field Theories via Neural Operator-based Flows](https://mlsanity.com/paper/2401.00828/) (2024)
- [Scaling Online Optimal Transport Minibatches for Flow Matching](https://mlsanity.com/paper/2610.18838/) (2026)

## Abstract (arXiv)

Recent work has established a path-gradient estimator for simple variational Gaussian distributions and has argued that the path-gradient is particularly beneficial in the regime in which the variational distribution approaches the exact target distribution. In many applications, this regime can however not be reached by a simple Gaussian variational distribution. In this work, we overcome this crucial limitation by proposing a path-gradient estimator for the considerably more expressive variational family of continuous normalizing flows. We outline an efficient algorithm to calculate this estimator and establish its superior performance empirically.