Score-based Metropolis-Hastings for Fractional Langevin Algorithms

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.