# Quantifying the mini-batching error in Bayesian inference for Adaptive Langevin dynamics

arXiv: https://arxiv.org/abs/2105.10347 · HTML: https://mlsanity.com/paper/2105.10347/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 76 references that are in the map.
- First posted on arXiv 21 May 2021.

## Builds on (most cited first, 18 of 18)

- [Stochastic Gradient Hamiltonian Monte Carlo](https://mlsanity.com/paper/1402.4102/) (2014)
- [What Are Bayesian Neural Network Posteriors Really Like?](https://mlsanity.com/paper/2104.14421/) (2021)
- [Stochastic gradient descent performs variational inference, converges to limit cycles for deep networks](https://mlsanity.com/paper/1710.11029/) (2017)
- [Bridging the Gap between Constant Step Size Stochastic Gradient Descent and Markov Chains](https://mlsanity.com/paper/1707.06386/) (2017)
- [Sampling Can Be Faster Than Optimization](https://mlsanity.com/paper/1811.08413/) (2018)
- [Consistency and fluctuations for stochastic gradient Langevin dynamics](https://mlsanity.com/paper/1409.0578/) (2014)
- [On sampling from a log-concave density using kinetic Langevin diffusions](https://mlsanity.com/paper/1807.09382/) (2018)
- [The promises and pitfalls of Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/1811.10072/) (2018)
- [High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm](https://mlsanity.com/paper/1908.10859/) (2019)
- [Control Variates for Stochastic Gradient MCMC](https://mlsanity.com/paper/1706.05439/) (2017)
- [Austerity in MCMC Land: Cutting the Metropolis-Hastings Budget](https://mlsanity.com/paper/1304.5299/) (2013)
- [Decentralized Stochastic Gradient Langevin Dynamics and Hamiltonian Monte Carlo](https://mlsanity.com/paper/2007.00590/) (2020)
- [Speeding Up MCMC by Efficient Data Subsampling](https://mlsanity.com/paper/1404.4178/) (2014)
- [The True Cost of Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/1706.02692/) (2017)
- [Covariance-Controlled Adaptive Langevin Thermostat for Large-Scale Bayesian Sampling](https://mlsanity.com/paper/1510.08692/) (2015)
- [Partitioned integrators for thermodynamic parameterization of neural networks](https://mlsanity.com/paper/1908.11843/) (2019)
- [Stochastic Bouncy Particle Sampler](https://mlsanity.com/paper/1609.00770/) (2016)
- [Langevin Markov Chain Monte Carlo with stochastic gradients](https://mlsanity.com/paper/1805.08863/) (2018)

## Built on by (most cited first, 1 of 1)

- [Sampling from Bayesian Neural Network Posteriors with Symmetric Minibatch Splitting Langevin Dynamics](https://mlsanity.com/paper/2410.19780/) (2024)

## Abstract (arXiv)

Bayesian inference allows to obtain useful information on the parameters of models, either in computational statistics or more recently in the context of Bayesian Neural Networks. The computational cost of usual Monte Carlo methods for sampling posterior laws in Bayesian inference scales linearly with the number of data points. One option to reduce it to a fraction of this cost is to resort to mini-batching in conjunction with unadjusted discretizations of Langevin dynamics, in which case only a random fraction of the data is used to estimate the gradient. However, this leads to an additional noise in the dynamics and hence a bias on the invariant measure which is sampled by the Markov chain. We advocate using the so-called Adaptive Langevin dynamics, which is a modification of standard inertial Langevin dynamics with a dynamical friction which automatically corrects for the increased noise arising from mini-batching. We investigate the practical relevance of the assumptions underpinning Adaptive Langevin (constant covariance for the estimation of the gradient, Gaussian minibatching noise), which are not satisfied in typical models of Bayesian inference, and quantify the bias induced by minibatching in this case. We also suggest a possible extension of AdL to further reduce the bias on the posterior distribution, by considering a dynamical friction depending on the current value of the parameter to sample.