Score-based Metropolis-Hastings for Fractional Langevin Algorithms

2026-01-31

AI Summary

This work addresses the limitations of existing fractional Langevin algorithms in sampling heavy-tailed or multimodal distributions when the target and proposal densities are intractable, particularly their lack of finite-time error control and poor tail behavior. To overcome these challenges, we propose the Metropolis-Adjusted Fractional Langevin Algorithm (MAFLA), which for the first time integrates the Metropolis–Hastings correction into fractional Langevin dynamics. MAFLA constructs a surrogate fractional gradient using isotropic symmetric α-stable noise and learns an acceptance probability function via Score Balance Matching that does not require explicit knowledge of the proposal density, enabling a purely score-based correction. Empirical results demonstrate that MAFLA significantly improves sampling accuracy and tail control within finite time, outperforming unadjusted methods on both multimodal distributions and combinatorial optimization tasks.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Active Learning
• Multiagent Systems: Multiagent Learning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Sampling from heavy-tailed and multimodal distributions is challenging when neither the target density nor the proposal density can be evaluated, as in $\alpha$-stable L\'evy-driven fractional Langevin algorithms. While the target distribution can be estimated from data via score-based or energy-based models, the $\alpha$-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 $\alpha$-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.

Problem

• heavy-tailed distributions
• multimodal distributions
• fractional Langevin algorithms
• Metropolis-Hastings
• score-based sampling

Innovation

• score-based sampling
• Metropolis-Hastings correction
• fractional Langevin dynamics
• α-stable noise
• Score Balance Matching