标题： Picard-Lagrange 框架用于高阶朗之万蒙特卡罗 显示英文标题

标题： The Picard-Lagrange Framework for Higher-Order Langevin Monte Carlo

摘要： 从对数凹分布中采样是统计学和机器学习中的核心问题。 先前的工作为基于过阻尼和欠阻尼朗之万动力学的朗之万蒙特卡洛算法建立了理论保证，最近一些三阶变体也得到了研究。 在本文中，我们引入了一个新的采样算法，该算法基于一般的$K$阶朗之万动力学，超越了二阶和三阶方法。 为了离散化$K$阶动力学，我们通过拉格朗日插值近似势能引起的漂移，并使用皮卡德迭代校正来细化插值点处的节点值，从而得到一个灵活的方案，充分利用了高阶朗之万动力学的加速效果。 对于具有平滑、强对数凹密度的目标，我们证明了 Wasserstein 距离下的维度相关收敛性：该采样器在$\varepsilon$精度下，对于$K \ge 3$，在$\widetilde O(d^{\frac{K-1}{2K-3}}\varepsilon^{-\frac{2}{2K-3}})$次梯度评估内达到。 据我们所知，这是第一个实现这种查询复杂度的采样算法。 随着阶数$K$的增加，该速率得到改善，优于现有的一阶到三阶方法。 显示英文摘要

摘要： Sampling from log-concave distributions is a central problem in statistics and machine learning. Prior work establishes theoretical guarantees for Langevin Monte Carlo algorithm based on overdamped and underdamped Langevin dynamics and, more recently, some third-order variants. In this paper, we introduce a new sampling algorithm built on a general $K$th-order Langevin dynamics, extending beyond second- and third-order methods. To discretize the $K$th-order dynamics, we approximate the drift induced by the potential via Lagrange interpolation and refine the node values at the interpolation points using Picard-iteration corrections, yielding a flexible scheme that fully utilizes the acceleration of higher-order Langevin dynamics. For targets with smooth, strongly log-concave densities, we prove dimension-dependent convergence in Wasserstein distance: the sampler achieves $\varepsilon$-accuracy within $\widetilde O(d^{\frac{K-1}{2K-3}}\varepsilon^{-\frac{2}{2K-3}})$ gradient evaluations for $K \ge 3$. To our best knowledge, this is the first sampling algorithm achieving such query complexity. The rate improves with the order $K$ increases, yielding better rates than existing first to third-order approaches.