# Score-based Metropolis-Hastings for Fractional Langevin Algorithms

arXiv: https://arxiv.org/abs/2602.00835 · HTML: https://mlsanity.com/paper/2602.00835/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 30 references that are in the map.
- First posted on arXiv 31 Jan 2026.

## Builds on (most cited first, 10 of 10)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [Estimating High Order Gradients of the Data Distribution by Denoising](https://mlsanity.com/paper/2111.04726/) (2021)
- [Fractional Underdamped Langevin Dynamics: Retargeting SGD with Momentum under Heavy-Tailed Gradient Noise](https://mlsanity.com/paper/2002.05685/) (2020)
- [Heavy-Tailed Diffusion with Denoising L'evy Probabilistic Models](https://mlsanity.com/paper/2407.18609/) (2024)
- [Fractional Langevin Monte Carlo: Exploring L'evy Driven Stochastic Differential Equations for Markov Chain Monte Carlo](https://mlsanity.com/paper/1706.03649/) (2017)
- [MCMC-Correction of Score-Based Diffusion Models for Model Composition](https://mlsanity.com/paper/2307.14012/) (2023)
- [Implicit Langevin Algorithms for Sampling From Log-concave Densities](https://mlsanity.com/paper/1903.12322/) (2019)
- [Score-Based Metropolis-Hastings Algorithms](https://mlsanity.com/paper/2501.00467/) (2024)

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

- [Diffusion-Based Data-Driven Assortment Optimization](https://mlsanity.com/paper/2608.11419/) (2026)

## Abstract (arXiv)

Sampling from heavy-tailed and multimodal distributions is challenging when neither the target density nor the proposal density can be evaluated, as in α-stable L'evy-driven fractional Langevin algorithms. While the target distribution can be estimated from data via score-based or energy-based models, the α-stable proposal density and its score are generally unavailable, rendering classical density-based Metropolis--Hastings (MH) corrections impractical. Consequently, existing fractional Langevin methods operate in an unadjusted regime and can exhibit substantial finite-time errors and poor empirical control of tail behavior. We introduce the Metropolis-Adjusted Fractional Langevin Algorithm (MAFLA), an MH-inspired, fully score-based correction mechanism. MAFLA employs designed proxies for fractional proposal score gradients under isotropic symmetric α-stable noise and learns an acceptance function via Score Balance Matching. We empirically illustrate the strong performance of MAFLA on a series of tasks including combinatorial optimization problems where the method significantly improves finite time sampling accuracy over unadjusted fractional Langevin dynamics.