# Stochastic Gradient Hamiltonian Monte Carlo

arXiv: https://arxiv.org/abs/1402.4102 · HTML: https://mlsanity.com/paper/1402.4102/

## Key facts

- Cited by 343 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Feb 2014.

## Built on by (most cited first, 30 of 343)

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- [Heavy Ball Neural Ordinary Differential Equations](https://mlsanity.com/paper/2110.04840/) (2021)
- [MCMC Variational Inference via Uncorrected Hamiltonian Annealing](https://mlsanity.com/paper/2107.04150/) (2021)
- [Learning Functional Priors and Posteriors from Data and Physics](https://mlsanity.com/paper/2106.05863/) (2021)
- [Differentiable Annealed Importance Sampling and the Perils of Gradient Noise](https://mlsanity.com/paper/2107.10211/) (2021)
- [BayesDiff: Estimating Pixel-wise Uncertainty in Diffusion via Bayesian Inference](https://mlsanity.com/paper/2310.11142/) (2023)
- [BayesDAG: Gradient-Based Posterior Inference for Causal Discovery](https://mlsanity.com/paper/2307.13917/) (2023)
- [Improving Task-free Continual Learning by Distributionally Robust Memory Evolution](https://mlsanity.com/paper/2207.07256/) (2022)
- [Exact Langevin Dynamics with Stochastic Gradients](https://mlsanity.com/paper/2102.01691/) (2021)
- [DeepONet-Grid-UQ: A Trustworthy Deep Operator Framework for Predicting the Power Grid's Post-Fault Trajectories](https://mlsanity.com/paper/2202.07176/) (2022)
- [Adapting the Linearised Laplace Model Evidence for Modern Deep Learning](https://mlsanity.com/paper/2206.08900/) (2022)
- [Copula Conformal Prediction for Multi-step Time Series Forecasting](https://mlsanity.com/paper/2212.03281/) (2022)
- [Accelerated Linearized Laplace Approximation for Bayesian Deep Learning](https://mlsanity.com/paper/2210.12642/) (2022)
- [Quantification of Uncertainty with Adversarial Models](https://mlsanity.com/paper/2307.03217/) (2023)
- [Conformalized-DeepONet: A Distribution-Free Framework for Uncertainty Quantification in Deep Operator Networks](https://mlsanity.com/paper/2402.15406/) (2024)
- [RegMixup: Mixup as a Regularizer Can Surprisingly Improve Accuracy and Out Distribution Robustness](https://mlsanity.com/paper/2206.14502/) (2022)
- [Sparsity Agnostic Depth Completion](https://mlsanity.com/paper/2212.00790/) (2022)
- [Transfer Learning for Bayesian Optimization: A Survey](https://mlsanity.com/paper/2302.05927/) (2023)
- [A Primer on Bayesian Neural Networks: Review and Debates](https://mlsanity.com/paper/2309.16314/) (2023)

## Abstract (arXiv)

Hamiltonian Monte Carlo (HMC) sampling methods provide a mechanism for defining distant proposals with high acceptance probabilities in a Metropolis-Hastings framework, enabling more efficient exploration of the state space than standard random-walk proposals. The popularity of such methods has grown significantly in recent years. However, a limitation of HMC methods is the required gradient computation for simulation of the Hamiltonian dynamical system-such computation is infeasible in problems involving a large sample size or streaming data. Instead, we must rely on a noisy gradient estimate computed from a subset of the data. In this paper, we explore the properties of such a stochastic gradient HMC approach. Surprisingly, the natural implementation of the stochastic approximation can be arbitrarily bad. To address this problem we introduce a variant that uses second-order Langevin dynamics with a friction term that counteracts the effects of the noisy gradient, maintaining the desired target distribution as the invariant distribution. Results on simulated data validate our theory. We also provide an application of our methods to a classification task using neural networks and to online Bayesian matrix factorization.