# Variance reduction properties of the reparameterization trick

arXiv: https://arxiv.org/abs/1809.10330 · HTML: https://mlsanity.com/paper/1809.10330/

## Key facts

- Cited by 41 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Sep 2018.

## Built on by (most cited first, 30 of 41)

- [Hierarchical Model-Based Imitation Learning for Planning in Autonomous Driving](https://mlsanity.com/paper/2210.09539/) (2022)
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- [Embedding Synthetic Off-Policy Experience for Autonomous Driving via Zero-Shot Curricula](https://mlsanity.com/paper/2212.01375/) (2022)
- [A unified view of likelihood ratio and reparameterization gradients](https://mlsanity.com/paper/2105.14900/) (2021)
- [Bayesian Experimental Design via Contrastive Diffusions](https://mlsanity.com/paper/2410.11826/) (2024)
- [Linear Convergence of Black-Box Variational Inference: Should We Stick the Landing?](https://mlsanity.com/paper/2307.14642/) (2023)
- [Model-free Policy Learning with Reward Gradients](https://mlsanity.com/paper/2103.05147/) (2021)
- [Practical and Matching Gradient Variance Bounds for Black-Box Variational Bayesian Inference](https://mlsanity.com/paper/2303.10472/) (2023)
- [Understanding What Affects the Generalization Gap in Visual Reinforcement Learning: Theory and Empirical Evidence](https://mlsanity.com/paper/2402.02701/) (2024)
- [Hierarchical Multi-Agent Reinforcement Learning with Control Barrier Functions for Safety-Critical Autonomous Systems](https://mlsanity.com/paper/2507.14850/) (2025)
- [Provably Scalable Black-Box Variational Inference with Structured Variational Families](https://mlsanity.com/paper/2401.10989/) (2024)
- [Fast post-process Bayesian inference with Variational Sparse Bayesian Quadrature](https://mlsanity.com/paper/2303.05263/) (2023)
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- [Demystifying SGD with Doubly Stochastic Gradients](https://mlsanity.com/paper/2406.00920/) (2024)
- [Learning Hyperparameters via a Data-Emphasized Variational Objective](https://mlsanity.com/paper/2502.01861/) (2025)
- [Improving Controller Generalization with Dimensionless Markov Decision Processes](https://mlsanity.com/paper/2504.10006/) (2025)
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- [Parameter Estimation in DAGs from Incomplete Data via Optimal Transport](https://mlsanity.com/paper/2305.15927/) (2023)
- [Variance Control for Black Box Variational Inference Using The James-Stein Estimator](https://mlsanity.com/paper/2405.05485/) (2024)
- [Learning the Regularization Strength for Deep Fine-Tuning via a Data-Emphasized Variational Objective](https://mlsanity.com/paper/2410.19675/) (2024)
- [Nearly Dimension-Independent Convergence of Mean-Field Black-Box Variational Inference](https://mlsanity.com/paper/2505.21721/) (2025)
- [Rao-Blackwellised Reparameterisation Gradients](https://mlsanity.com/paper/2506.07687/) (2025)
- [MixDPO: Modeling Preference Strength for Pluralistic Alignment](https://mlsanity.com/paper/2601.06180/) (2026)
- [Revisiting Mixture Policies in Entropy-Regularized Actor-Critic](https://mlsanity.com/paper/2605.09157/) (2026)
- [Joint control variate for faster black-box variational inference](https://mlsanity.com/paper/2210.07290/) (2022)
- [Approaching Globally Optimal Energy Efficiency in Interference Networks via Machine Learning](https://mlsanity.com/paper/2212.12329/) (2022)
- [Large-scale global optimization of ultra-high dimensional non-convex landscapes based on generative neural networks](https://mlsanity.com/paper/2307.04065/) (2023)

## Abstract (arXiv)

The reparameterization trick is widely used in variational inference as it yields more accurate estimates of the gradient of the variational objective than alternative approaches such as the score function method. Although there is overwhelming empirical evidence in the literature showing its success, there is relatively little research exploring why the reparameterization trick is so effective. We explore this under the idealized assumptions that the variational approximation is a mean-field Gaussian density and that the log of the joint density of the model parameters and the data is a quadratic function that depends on the variational mean. From this, we show that the marginal variances of the reparameterization gradient estimator are smaller than those of the score function gradient estimator. We apply the result of our idealized analysis to real-world examples.