# Fast and Unified Path Gradient Estimators for Normalizing Flows

arXiv: https://arxiv.org/abs/2403.15881 · HTML: https://mlsanity.com/paper/2403.15881/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 44 references that are in the map.
- First posted on arXiv 23 Mar 2024.

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

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [Monte Carlo Gradient Estimation in Machine Learning](https://mlsanity.com/paper/1906.10652/) (2019)
- [The Reversible Residual Network: Backpropagation Without Storing Activations](https://mlsanity.com/paper/1707.04585/) (2017)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [Equivariant flow-based sampling for lattice gauge theory](https://mlsanity.com/paper/2003.06413/) (2020)
- [Sticking the Landing: Simple, Lower-Variance Gradient Estimators for Variational Inference](https://mlsanity.com/paper/1703.09194/) (2017)
- [Augmented Normalizing Flows: Bridging the Gap Between Generative Flows and Latent Variable Models](https://mlsanity.com/paper/2002.07101/) (2020)
- [Continual Repeated Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2201.13117/) (2022)
- [Sum-of-Squares Polynomial Flow](https://mlsanity.com/paper/1905.02325/) (2019)
- [Smooth Normalizing Flows](https://mlsanity.com/paper/2110.00351/) (2021)
- [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)
- [SE(3) Equivariant Augmented Coupling Flows](https://mlsanity.com/paper/2308.10364/) (2023)
- [Pathwise Derivatives Beyond the Reparameterization Trick](https://mlsanity.com/paper/1806.01851/) (2018)
- [Introduction to Normalizing Flows for Lattice Field Theory](https://mlsanity.com/paper/2101.08176/) (2021)
- [Challenges and Opportunities in High-dimensional Variational Inference](https://mlsanity.com/paper/2103.01085/) (2021)
- [Reducing Reparameterization Gradient Variance](https://mlsanity.com/paper/1705.07880/) (2017)
- [Detecting and Mitigating Mode-Collapse for Flow-based Sampling of Lattice Field Theories](https://mlsanity.com/paper/2302.14082/) (2023)
- [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)
- [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)
- [A unified view of likelihood ratio and reparameterization gradients](https://mlsanity.com/paper/2105.14900/) (2021)

## Built on by (most cited first, 5 of 5)

- [Disentangling impact of capacity, objective, batchsize, estimators, and step-size on flow VI](https://mlsanity.com/paper/2412.08824/) (2024)
- [Path Gradients after Flow Matching](https://mlsanity.com/paper/2505.10139/) (2025)
- [Efficient Training of Boltzmann Generators Using Off-Policy Log-Dispersion Regularization](https://mlsanity.com/paper/2602.03729/) (2026)
- [SESaMo: Symmetry-Enforcing Stochastic Modulation for Normalizing Flows](https://mlsanity.com/paper/2505.19619/) (2025)
- [How I learned to stop worrying and love StopGrads: Stationarity, Convergence, and a case study on Flow Map Learning](https://mlsanity.com/paper/2609.16222/) (2026)

## Abstract (arXiv)

Recent work shows that path gradient estimators for normalizing flows have lower variance compared to standard estimators for variational inference, resulting in improved training. However, they are often prohibitively more expensive from a computational point of view and cannot be applied to maximum likelihood training in a scalable manner, which severely hinders their widespread adoption. In this work, we overcome these crucial limitations. Specifically, we propose a fast path gradient estimator which improves computational efficiency significantly and works for all normalizing flow architectures of practical relevance. We then show that this estimator can also be applied to maximum likelihood training for which it has a regularizing effect as it can take the form of a given target energy function into account. We empirically establish its superior performance and reduced variance for several natural sciences applications.