提出新型采样算法MAFLA，解决重尾多峰分布采样难题。
Score-based Metropolis-Hastings for Fractional Langevin Algorithms
- 基于得分的马尔可夫链修正机制，无需评估目标或提议密度。
- 在组合优化任务中显著提升有限时间采样精度，误差更小。
- 适合需要精准控制尾部行为的复杂分布采样场景。
从重尾和多模态分布中采样在目标密度与提议密度均不可计算时极具挑战性，如α-稳定莱维驱动的分数阶朗之万算法。尽管可通过得分模型或能量模型从数据中估计目标分布，但α-稳定提议密度及其得分通常无法获取，导致经典密度依赖的马尔可夫链蒙特卡洛（MH）校正难以应用。因此，现有分数阶朗之万方法处于未校正状态，表现出显著的有限时间误差且难以控制尾部行为。本文提出马尔可夫链调整分数阶朗之万算法（MAFLA），一种受MH启发、完全基于得分的校正机制。MAFLA在各向同性对称α-稳定噪声下设计提议得分梯度的代理，并通过得分平衡匹配学习接受函数。实验证明，MAFLA在一系列任务中表现优异，尤其在组合优化问题上显著优于未校正的分数阶朗之万动力学。
原文摘要 · Abstract (English)
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.
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