# Fractional Langevin Monte Carlo: Exploring L'evy Driven Stochastic Differential Equations for Markov Chain Monte Carlo

arXiv: https://arxiv.org/abs/1706.03649 · HTML: https://mlsanity.com/paper/1706.03649/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jun 2017.

## Built on by (most cited first, 13 of 13)

- [Revisiting the Noise Model of Stochastic Gradient Descent](https://mlsanity.com/paper/2303.02749/) (2023)
- [A Convergence Theory for SVGD in the Population Limit under Talagrand's Inequality T1](https://mlsanity.com/paper/2106.03076/) (2021)
- [Heavy-tailed denoising score matching](https://mlsanity.com/paper/2112.09788/) (2021)
- [Heavy-Tailed Diffusion with Denoising L'evy Probabilistic Models](https://mlsanity.com/paper/2407.18609/) (2024)
- [Convergence of Stein Variational Gradient Descent under a Weaker Smoothness Condition](https://mlsanity.com/paper/2206.00508/) (2022)
- [Asymmetric Heavy Tails and Implicit Bias in Gaussian Noise Injections](https://mlsanity.com/paper/2102.07006/) (2021)
- [Heavy-tailed Sampling via Transformed Unadjusted Langevin Algorithm](https://mlsanity.com/paper/2201.08349/) (2022)
- [Implicit Compressibility of Overparametrized Neural Networks Trained with Heavy-Tailed SGD](https://mlsanity.com/paper/2306.08125/) (2023)
- [Accelerating Langevin Monte Carlo Sampling: A Large Deviations Analysis](https://mlsanity.com/paper/2503.19066/) (2025)
- [Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Models](https://mlsanity.com/paper/2303.17109/) (2023)
- [Upper Bound of Bayesian Generalization Error in Partial Concept Bottleneck Model (CBM): Partial CBM outperforms naive CBM](https://mlsanity.com/paper/2403.09206/) (2024)
- [Score-based Metropolis-Hastings for Fractional Langevin Algorithms](https://mlsanity.com/paper/2602.00835/) (2026)
- [Nonlocal Hamiltonian Dynamics on Sparse L'evy Graphs: Spectral Analysis and Multimodal Sampling](https://mlsanity.com/paper/2610.06904/) (2026)

## Abstract (arXiv)

Along with the recent advances in scalable Markov Chain Monte Carlo methods, sampling techniques that are based on Langevin diffusions have started receiving increasing attention. These so called Langevin Monte Carlo (LMC) methods are based on diffusions driven by a Brownian motion, which gives rise to Gaussian proposal distributions in the resulting algorithms. Even though these approaches have proven successful in many applications, their performance can be limited by the light-tailed nature of the Gaussian proposals. In this study, we extend classical LMC and develop a novel Fractional LMC (FLMC) framework that is based on a family of heavy-tailed distributions, called α-stable L'evy distributions. As opposed to classical approaches, the proposed approach can possess large jumps while targeting the correct distribution, which would be beneficial for efficient exploration of the state space. We develop novel computational methods that can scale up to large-scale problems and we provide formal convergence analysis of the proposed scheme. Our experiments support our theory: FLMC can provide superior performance in multi-modal settings, improved convergence rates, and robustness to algorithm parameters.