改进随机梯度算法，让其能处理非光滑和重尾分布的采样问题。
Anchored Langevin Algorithms
- 用平滑参考势替代原势，通过乘法缩放修正扩散过程。
- 在2-Wasserstein距离下给出非渐近收敛保证，适用于重尾分布。
- 适合处理不可导目标或重尾分布的采样任务，如稀疏建模、鲁棒学习。
标准一阶Langevin算法（如未调整Langevin算法ULA）通过离散化Langevin扩散实现，广泛用于机器学习中的采样，因其可扩展至高维和大规模数据集。然而，这类方法存在两大局限：(i) 要求对数密度可微，无法处理含不可导成分的目标；(ii) 通常无法有效采样重尾分布。本文提出锚定Langevin动力学，一种统一方法，可处理不可导目标及某些重尾分布。该方法将原始势能替换为平滑参考势，并通过乘法缩放修改Langevin扩散。我们建立了在2-Wasserstein距离下的非渐近收敛保证，并通过随机时间变换给出了等价形式。数值实验验证了理论分析与实际性能。
原文摘要 · Abstract (English)
Standard first-order Langevin algorithms such as the unadjusted Langevin algorithm (ULA) are obtained by discretizing the Langevin diffusion and are widely used for sampling in machine learning because they scale to high dimensions and large datasets. However, they face two key limitations: (i) they require differentiable log-densities, excluding targets with non-differentiable components; and (ii) they generally fail to sample heavy-tailed targets. We propose anchored Langevin dynamics, a unified approach that accommodates non-differentiable targets and certain classes of heavy-tailed distributions. The method replaces the original potential with a smooth reference potential and modifies the Langevin diffusion via multiplicative scaling. We establish non-asymptotic guarantees in the 2-Wasserstein distance to the target distribution and provide an equivalent formulation derived via a random time change of the Langevin diffusion. We provide numerical experiments to illustrate the theory and practical performance of our proposed approach.
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