# Revisiting the Effects of Stochasticity for Hamiltonian Samplers

arXiv: https://arxiv.org/abs/2106.16200 · HTML: https://mlsanity.com/paper/2106.16200/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 72 references that are in the map.
- First posted on arXiv 30 Jun 2021.

## Builds on (most cited first, 20 of 20)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Stochastic Gradient Hamiltonian Monte Carlo](https://mlsanity.com/paper/1402.4102/) (2014)
- [Stochastic Gradient Descent as Approximate Bayesian Inference](https://mlsanity.com/paper/1704.04289/) (2017)
- [Finite Versus Infinite Neural Networks: an Empirical Study](https://mlsanity.com/paper/2007.15801/) (2020)
- [A Complete Recipe for Stochastic Gradient MCMC](https://mlsanity.com/paper/1506.04696/) (2015)
- [Stochastic gradient descent performs variational inference, converges to limit cycles for deep networks](https://mlsanity.com/paper/1710.11029/) (2017)
- [Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimization](https://mlsanity.com/paper/1707.06618/) (2017)
- [Random Shuffling Beats SGD after Finite Epochs](https://mlsanity.com/paper/1806.10077/) (2018)
- [All You Need is a Good Functional Prior for Bayesian Deep Learning](https://mlsanity.com/paper/2011.12829/) (2020)
- [Scaling Hamiltonian Monte Carlo Inference for Bayesian Neural Networks with Symmetric Splitting](https://mlsanity.com/paper/2010.06772/) (2020)
- [Fluctuation-dissipation relations for stochastic gradient descent](https://mlsanity.com/paper/1810.00004/) (2018)
- [On the Convergence of Stochastic Gradient MCMC Algorithms with High-Order Integrators](https://mlsanity.com/paper/1610.06665/) (2016)
- [On Learning Rates and Schr"odinger Operators](https://mlsanity.com/paper/2004.06977/) (2020)
- [Global Convergence of Stochastic Gradient Hamiltonian Monte Carlo for Non-Convex Stochastic Optimization: Non-Asymptotic Performance Bounds and Momentum-Based Acceleration](https://mlsanity.com/paper/1809.04618/) (2018)
- [Conformal Symplectic and Relativistic Optimization](https://mlsanity.com/paper/1903.04100/) (2019)
- [On dissipative symplectic integration with applications to gradient-based optimization](https://mlsanity.com/paper/2004.06840/) (2020)
- [On the Importance of Strong Baselines in Bayesian Deep Learning](https://mlsanity.com/paper/1811.09385/) (2018)
- [Breaking Reversibility Accelerates Langevin Dynamics for Global Non-Convex Optimization](https://mlsanity.com/paper/1812.07725/) (2018)
- [Covariance-Controlled Adaptive Langevin Thermostat for Large-Scale Bayesian Sampling](https://mlsanity.com/paper/1510.08692/) (2015)
- [Pseudo-Marginal Hamiltonian Monte Carlo](https://mlsanity.com/paper/1607.02516/) (2016)

## Built on by (most cited first, 3 of 3)

- [Sampling from Bayesian Neural Network Posteriors with Symmetric Minibatch Splitting Langevin Dynamics](https://mlsanity.com/paper/2410.19780/) (2024)
- [On permutation symmetries in Bayesian neural network posteriors: a variational perspective](https://mlsanity.com/paper/2310.10171/) (2023)
- [Randomised Splitting Methods and Stochastic Gradient Descent](https://mlsanity.com/paper/2504.04274/) (2025)

## Abstract (arXiv)

We revisit the theoretical properties of Hamiltonian stochastic differential equations (SDES) for Bayesian posterior sampling, and we study the two types of errors that arise from numerical SDE simulation: the discretization error and the error due to noisy gradient estimates in the context of data subsampling. Our main result is a novel analysis for the effect of mini-batches through the lens of differential operator splitting, revising previous literature results. The stochastic component of a Hamiltonian SDE is decoupled from the gradient noise, for which we make no normality assumptions. This leads to the identification of a convergence bottleneck: when considering mini-batches, the best achievable error rate is O(η2), with η being the integrator step size. Our theoretical results are supported by an empirical study on a variety of regression and classification tasks for Bayesian neural networks.