# Decentralized Bayesian Learning with Metropolis-Adjusted Hamiltonian Monte Carlo

arXiv: https://arxiv.org/abs/2107.07211 · HTML: https://mlsanity.com/paper/2107.07211/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 33 references that are in the map.
- First posted on arXiv 15 Jul 2021.

## Builds on (most cited first, 22 of 22)

- [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)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [Overcoming Forgetting in Federated Learning on Non-IID Data](https://mlsanity.com/paper/1910.07796/) (2019)
- [NEXT: In-Network Nonconvex Optimization](https://mlsanity.com/paper/1602.00591/) (2016)
- [Distributed Stochastic Gradient Tracking Methods](https://mlsanity.com/paper/1805.11454/) (2018)
- [A Hitting Time Analysis of Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/1702.05575/) (2017)
- [Consistency and fluctuations for stochastic gradient Langevin dynamics](https://mlsanity.com/paper/1409.0578/) (2014)
- [Scaling Hamiltonian Monte Carlo Inference for Bayesian Neural Networks with Symmetric Splitting](https://mlsanity.com/paper/2010.06772/) (2020)
- [Coupling and Convergence for Hamiltonian Monte Carlo](https://mlsanity.com/paper/1805.00452/) (2018)
- [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)
- [Balancing Communication and Computation in Distributed Optimization](https://mlsanity.com/paper/1709.02999/) (2017)
- [Decentralized Stochastic Gradient Langevin Dynamics and Hamiltonian Monte Carlo](https://mlsanity.com/paper/2007.00590/) (2020)
- [On Stationary-Point Hitting Time and Ergodicity of Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/1904.13016/) (2019)
- [Nonasymptotic analysis of Stochastic Gradient Hamiltonian Monte Carlo under local conditions for nonconvex optimization](https://mlsanity.com/paper/2002.05465/) (2020)
- [Embarrassingly parallel MCMC using deep invertible transformations](https://mlsanity.com/paper/1903.04556/) (2019)
- [Decentralized Bayesian Learning over Graphs](https://mlsanity.com/paper/1905.10466/) (2019)
- [A Divide-and-Conquer Solver for Kernel Support Vector Machines](https://mlsanity.com/paper/1311.0914/) (2013)
- [Stochastic Gradient Hamiltonian Monte Carlo for Non-Convex Learning](https://mlsanity.com/paper/1903.10328/) (2019)
- [The Fundamental Incompatibility of Hamiltonian Monte Carlo and Data Subsampling](https://mlsanity.com/paper/1502.01510/) (2015)
- [Decentralized Langevin Dynamics over a Directed Graph](https://mlsanity.com/paper/2103.05444/) (2021)
- [Decentralized Langevin Dynamics](https://mlsanity.com/paper/2001.00665/) (2020)

## Built on by (most cited first, 4 of 4)

- [DIGing--SGLD: Decentralized and Scalable Langevin Sampling over Time--Varying Networks](https://mlsanity.com/paper/2511.12836/) (2025)
- [ISR: Invertible Symbolic Regression](https://mlsanity.com/paper/2405.06848/) (2024)
- [Asynchronous Bayesian Learning over a Network](https://mlsanity.com/paper/2211.08603/) (2022)
- [Distributed Markov Chain Monte Carlo Sampling based on the Alternating Direction Method of Multipliers](https://mlsanity.com/paper/2401.15838/) (2024)

## Abstract (arXiv)

Federated learning performed by a decentralized networks of agents is becoming increasingly important with the prevalence of embedded software on autonomous devices. Bayesian approaches to learning benefit from offering more information as to the uncertainty of a random quantity, and Langevin and Hamiltonian methods are effective at realizing sampling from an uncertain distribution with large parameter dimensions. Such methods have only recently appeared in the decentralized setting, and either exclusively use stochastic gradient Langevin and Hamiltonian Monte Carlo approaches that require a diminishing stepsize to asymptotically sample from the posterior and are known in practice to characterize uncertainty less faithfully than constant step-size methods with a Metropolis adjustment, or assume strong convexity properties of the potential function. We present the first approach to incorporating constant stepsize Metropolis-adjusted HMC in the decentralized sampling framework, show theoretical guarantees for consensus and probability distance to the posterior stationary distribution, and demonstrate their effectiveness numerically on standard real world problems, including decentralized learning of neural networks which is known to be highly non-convex.