# Pathwise Derivatives Beyond the Reparameterization Trick

arXiv: https://arxiv.org/abs/1806.01851 · HTML: https://mlsanity.com/paper/1806.01851/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jun 2018.

## Built on by (most cited first, 30 of 36)

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- [Variational Sparse Coding with Learned Thresholding](https://mlsanity.com/paper/2205.03665/) (2022)
- [Sample Average Approximation for Black-Box VI](https://mlsanity.com/paper/2304.06803/) (2023)
- [BaLeNAS: Differentiable Architecture Search via the Bayesian Learning Rule](https://mlsanity.com/paper/2111.13204/) (2021)
- [Fast and Unified Path Gradient Estimators for Normalizing Flows](https://mlsanity.com/paper/2403.15881/) (2024)
- [Sparse Conditional Hidden Markov Model for Weakly Supervised Named Entity Recognition](https://mlsanity.com/paper/2205.14228/) (2022)
- [Hierarchical Variational Auto-Encoding for Unsupervised Domain Generalization](https://mlsanity.com/paper/2101.09436/) (2021)
- [Scalable Marked Point Processes for Exchangeable and Non-Exchangeable Event Sequences](https://mlsanity.com/paper/2105.14574/) (2021)
- [Reparameterized Variational Rejection Sampling](https://mlsanity.com/paper/2309.14612/) (2023)
- [Learning Interpretable Deep Disentangled Neural Networks for Hyperspectral Unmixing](https://mlsanity.com/paper/2310.02340/) (2023)
- [Differentiable and Stable Long-Range Tracking of Multiple Posterior Modes](https://mlsanity.com/paper/2404.08789/) (2024)
- [BN-Pool: Bayesian Nonparametric Pooling for Graphs](https://mlsanity.com/paper/2501.09821/) (2025)
- [Large Language Models Struggle to Describe the Haystack without Human Help: Human-in-the-loop Evaluation of Topic Models](https://mlsanity.com/paper/2502.14748/) (2025)
- [Bayesian sparsification for deep neural networks with Bayesian model reduction](https://mlsanity.com/paper/2309.12095/) (2023)
- [Variational Autoencoders for Feature Exploration and Malignancy Prediction of Lung Lesions](https://mlsanity.com/paper/2311.15719/) (2023)
- [Rao-Blackwellised Reparameterisation Gradients](https://mlsanity.com/paper/2506.07687/) (2025)
- [Retry Policy Gradients in Continuous Action Spaces](https://mlsanity.com/paper/2606.05888/) (2026)

## Abstract (arXiv)

We observe that gradients computed via the reparameterization trick are in direct correspondence with solutions of the transport equation in the formalism of optimal transport. We use this perspective to compute (approximate) pathwise gradients for probability distributions not directly amenable to the reparameterization trick: Gamma, Beta, and Dirichlet. We further observe that when the reparameterization trick is applied to the Cholesky-factorized multivariate Normal distribution, the resulting gradients are suboptimal in the sense of optimal transport. We derive the optimal gradients and show that they have reduced variance in a Gaussian Process regression task. We demonstrate with a variety of synthetic experiments and stochastic variational inference tasks that our pathwise gradients are competitive with other methods.