# Exact Langevin Dynamics with Stochastic Gradients

arXiv: https://arxiv.org/abs/2102.01691 · HTML: https://mlsanity.com/paper/2102.01691/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 26 references that are in the map.
- First posted on arXiv 2 Feb 2021.

## Builds on (most cited first, 10 of 10)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Stochastic Gradient Hamiltonian Monte Carlo](https://mlsanity.com/paper/1402.4102/) (2014)
- [Cyclical Stochastic Gradient MCMC for Bayesian Deep Learning](https://mlsanity.com/paper/1902.03932/) (2019)
- [Stochastic Normalizing Flows](https://mlsanity.com/paper/2002.06707/) (2020)
- [A Complete Recipe for Stochastic Gradient MCMC](https://mlsanity.com/paper/1506.04696/) (2015)
- [AMAGOLD: Amortized Metropolis Adjustment for Efficient Stochastic Gradient MCMC](https://mlsanity.com/paper/2003.00193/) (2020)
- [Hamiltonian Monte Carlo Without Detailed Balance](https://mlsanity.com/paper/1409.5191/) (2014)
- [Partitioned integrators for thermodynamic parameterization of neural networks](https://mlsanity.com/paper/1908.11843/) (2019)
- [Irreversible Samplers from Jump and Continuous Markov Processes](https://mlsanity.com/paper/1608.05973/) (2016)
- [Langevin Markov Chain Monte Carlo with stochastic gradients](https://mlsanity.com/paper/1805.08863/) (2018)

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

- [Laplace Redux -- Effortless Bayesian Deep Learning](https://mlsanity.com/paper/2106.14806/) (2021)
- [What Are Bayesian Neural Network Posteriors Really Like?](https://mlsanity.com/paper/2104.14421/) (2021)
- [Bayesian Neural Network Priors Revisited](https://mlsanity.com/paper/2102.06571/) (2021)
- [Repulsive Deep Ensembles are Bayesian](https://mlsanity.com/paper/2106.11642/) (2021)
- [Priors in Bayesian Deep Learning: A Review](https://mlsanity.com/paper/2105.06868/) (2021)
- [Position: Bayesian Deep Learning is Needed in the Age of Large-Scale AI](https://mlsanity.com/paper/2402.00809/) (2024)
- [Data augmentation in Bayesian neural networks and the cold posterior effect](https://mlsanity.com/paper/2106.05586/) (2021)
- [On Stein Variational Neural Network Ensembles](https://mlsanity.com/paper/2106.10760/) (2021)
- [BNNpriors: A library for Bayesian neural network inference with different prior distributions](https://mlsanity.com/paper/2105.06964/) (2021)
- [Gaussian Stochastic Weight Averaging for Bayesian Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2405.03425/) (2024)
- [Neural Variational Gradient Descent](https://mlsanity.com/paper/2107.10731/) (2021)
- [Improving Neural Additive Models with Bayesian Principles](https://mlsanity.com/paper/2305.16905/) (2023)
- [Pathologies in priors and inference for Bayesian transformers](https://mlsanity.com/paper/2110.04020/) (2021)
- [Scalable Bayesian Uncertainty Quantification for Neural Network Potentials: Promise and Pitfalls](https://mlsanity.com/paper/2212.07959/) (2022)
- [Promises and Pitfalls of the Linearized Laplace in Bayesian Optimization](https://mlsanity.com/paper/2304.08309/) (2023)
- [Deep Classifiers with Label Noise Modeling and Distance Awareness](https://mlsanity.com/paper/2110.02609/) (2021)
- [Can Microcanonical Langevin Dynamics Leverage Mini-Batch Gradient Noise?](https://mlsanity.com/paper/2602.06500/) (2026)
- [FedBIAD: Communication-Efficient and Accuracy-Guaranteed Federated Learning with Bayesian Inference-Based Adaptive Dropout](https://mlsanity.com/paper/2307.07172/) (2023)
- [Contextual Similarity Distillation: Ensemble Uncertainties with a Single Model](https://mlsanity.com/paper/2503.11339/) (2025)
- [Functional Stochastic Gradient MCMC for Bayesian Neural Networks](https://mlsanity.com/paper/2409.16632/) (2024)
- [Parameter Expanded Stochastic Gradient Markov Chain Monte Carlo](https://mlsanity.com/paper/2503.00699/) (2025)
- [Data Subsampling for Bayesian Neural Networks](https://mlsanity.com/paper/2210.09141/) (2022)
- [JaxSGMC: Modular stochastic gradient MCMC in JAX](https://mlsanity.com/paper/2505.11190/) (2025)
- [Accelerating Hamiltonian Monte Carlo for Bayesian Inference in Neural Networks and Neural Operators](https://mlsanity.com/paper/2507.14652/) (2025)
- [Position: The Time for Sampling Is Now! Charting a New Course for Bayesian Deep Learning](https://mlsanity.com/paper/2605.21765/) (2026)
- [Complexity from Adaptive-Symmetries Breaking: Global Minima in the Statistical Mechanics of Deep Neural Networks](https://mlsanity.com/paper/2201.07934/) (2022)
- [Uncertainty in Graph Contrastive Learning with Bayesian Neural Networks](https://mlsanity.com/paper/2312.00232/) (2023)
- [Stein Variational Newton Neural Network Ensembles](https://mlsanity.com/paper/2411.01887/) (2024)
- [Priors Matter: Addressing Misspecification in Bayesian Deep Q-Learning](https://mlsanity.com/paper/2508.21488/) (2025)
- [On the Equivalence of Random Network Distillation, Deep Ensembles, and Bayesian Inference](https://mlsanity.com/paper/2602.19964/) (2026)

## Abstract (arXiv)

Stochastic gradient Markov Chain Monte Carlo algorithms are popular samplers for approximate inference, but they are generally biased. We show that many recent versions of these methods (e.g. Chen et al. (2014)) cannot be corrected using Metropolis-Hastings rejection sampling, because their acceptance probability is always zero. We can fix this by employing a sampler with realizable backwards trajectories, such as Gradient-Guided Monte Carlo (Horowitz, 1991), which generalizes stochastic gradient Langevin dynamics (Welling and Teh, 2011) and Hamiltonian Monte Carlo. We show that this sampler can be used with stochastic gradients, yielding nonzero acceptance probabilities, which can be computed even across multiple steps.