# Sticking the Landing: Simple, Lower-Variance Gradient Estimators for Variational Inference

arXiv: https://arxiv.org/abs/1703.09194 · HTML: https://mlsanity.com/paper/1703.09194/

## Key facts

- Cited by 98 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Mar 2017.

## Built on by (most cited first, 30 of 98)

- [DreamFusion: Text-to-3D using 2D Diffusion](https://mlsanity.com/paper/2209.14988/) (2022)
- [Variational Mixture-of-Experts Autoencoders for Multi-Modal Deep Generative Models](https://mlsanity.com/paper/1911.03393/) (2019)
- [Align Your Gaussians: Text-to-4D with Dynamic 3D Gaussians and Composed Diffusion Models](https://mlsanity.com/paper/2312.13763/) (2023)
- [An optimal control perspective on diffusion-based generative modeling](https://mlsanity.com/paper/2211.01364/) (2022)
- [Improved sampling via learned diffusions](https://mlsanity.com/paper/2307.01198/) (2023)
- [Generative Time Series Forecasting with Diffusion, Denoise, and Disentanglement](https://mlsanity.com/paper/2301.03028/) (2023)
- [TSD-SR: One-Step Diffusion with Target Score Distillation for Real-World Image Super-Resolution](https://mlsanity.com/paper/2411.18263/) (2024)
- [Score Regularized Policy Optimization through Diffusion Behavior](https://mlsanity.com/paper/2310.07297/) (2023)
- [BCD Nets: Scalable Variational Approaches for Bayesian Causal Discovery](https://mlsanity.com/paper/2112.02761/) (2021)
- [Diffusion Generative Flow Samplers: Improving learning signals through partial trajectory optimization](https://mlsanity.com/paper/2310.02679/) (2023)
- [Bayesian Learning via Neural Schr"odinger-F"ollmer Flows](https://mlsanity.com/paper/2111.10510/) (2021)
- [Training Chain-of-Thought via Latent-Variable Inference](https://mlsanity.com/paper/2312.02179/) (2023)
- [Variational methods for simulation-based inference](https://mlsanity.com/paper/2203.04176/) (2022)
- [Score Distillation via Reparametrized DDIM](https://mlsanity.com/paper/2405.15891/) (2024)
- [Stochastic Optimal Control Matching](https://mlsanity.com/paper/2312.02027/) (2023)
- [Infinitely Deep Bayesian Neural Networks with Stochastic Differential Equations](https://mlsanity.com/paper/2102.06559/) (2021)
- [Challenges and Opportunities in High-dimensional Variational Inference](https://mlsanity.com/paper/2103.01085/) (2021)
- [Mitigating Modality Collapse in Multimodal VAEs via Impartial Optimization](https://mlsanity.com/paper/2206.04496/) (2022)
- [Provable convergence guarantees for black-box variational inference](https://mlsanity.com/paper/2306.03638/) (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)
- [SteinDreamer: Variance Reduction for Text-to-3D Score Distillation via Stein Identity](https://mlsanity.com/paper/2401.00604/) (2023)
- [A Unified Perspective on Natural Gradient Variational Inference with Gaussian Mixture Models](https://mlsanity.com/paper/2209.11533/) (2022)
- [On the Convergence of Black-Box Variational Inference](https://mlsanity.com/paper/2305.15349/) (2023)
- [Transport meets Variational Inference: Controlled Monte Carlo Diffusions](https://mlsanity.com/paper/2307.01050/) (2023)
- [Amortized Variational Inference for Simple Hierarchical Models](https://mlsanity.com/paper/2111.03144/) (2021)
- [Generalized Doubly Reparameterized Gradient Estimators](https://mlsanity.com/paper/2101.11046/) (2021)
- [Nested Variational Inference](https://mlsanity.com/paper/2106.11302/) (2021)
- [MonoFlow: Rethinking Divergence GANs via the Perspective of Wasserstein Gradient Flows](https://mlsanity.com/paper/2302.01075/) (2023)
- [Scalable Bayesian Learning with posteriors](https://mlsanity.com/paper/2406.00104/) (2024)
- [A Taxonomy of Loss Functions for Stochastic Optimal Control](https://mlsanity.com/paper/2410.00345/) (2024)

## Abstract (arXiv)

We propose a simple and general variant of the standard reparameterized gradient estimator for the variational evidence lower bound. Specifically, we remove a part of the total derivative with respect to the variational parameters that corresponds to the score function. Removing this term produces an unbiased gradient estimator whose variance approaches zero as the approximate posterior approaches the exact posterior. We analyze the behavior of this gradient estimator theoretically and empirically, and generalize it to more complex variational distributions such as mixtures and importance-weighted posteriors.