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 Lévy-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.

翻译：从重尾和多峰分布中采样具有挑战性，尤其是在目标密度和提议密度均无法评估的情况下，例如在$α$稳定Lévy驱动的分数阶Langevin算法中。虽然可以通过基于分数或基于能量的模型从数据中估计目标分布，但$α$稳定的提议密度及其分数通常无法获取，这使得经典的基于密度的Metropolis–Hastings（MH）校正方法难以实施。因此，现有的分数阶Langevin方法在未校正的机制下运行，可能表现出显著的有限时间误差以及对尾部行为的经验控制不佳。我们提出了Metropolis校正分数阶Langevin算法（MAFLA），这是一种受MH启发、完全基于分数的校正机制。MAFLA在各向同性对称$α$稳定噪声下使用设计的分数提议分数梯度代理，并通过分数平衡匹配学习接受函数。我们通过实验展示了MAFLA在一系列任务上的优异性能，包括组合优化问题，其中该方法相较于未校正的分数阶Langevin动力学显著提高了有限时间采样的准确性。