高阶朗之万算法提升采样效率，降低维度与精度依赖。
High-Order Langevin Monte Carlo Algorithms
- 用高阶微分方程与分裂积分设计新型采样算法
- 采样混合时间随阶数提升，对维度和精度更友好
- 适合大规模数据采样，尤其高维场景
朗之万算法是大数据科学中常用的大规模采样方法。本文提出基于任意阶数 $P\geq3$ 朗之万动力学离散化的蒙特卡洛算法，通过分裂与精确积分方法实现。针对对数凹且光滑的分布，给出沃尔什收敛保证：$P$ 阶朗之万蒙特卡洛算法的混合时间量级为 $O\left(d^{\frac{1}{R}}/ε^{\frac{1}{2R}}\right)$，其中 $R=4\cdot 1_{\{ P=3\}}+(2P-1)\cdot 1_{\{ P\geq 4\}}$。随着 $P$ 增大，算法对维度 $d$ 和精度 $ε$ 的依赖显著改善。数值实验验证了算法高效性。
原文摘要 · Abstract (English)
Langevin algorithms are popular Markov chain Monte Carlo (MCMC) methods for large-scale sampling problems that often arise in data science. We propose Monte Carlo algorithms based on the discretizations of $P$-th order Langevin dynamics for any $P\geq 3$. Our design of $P$-th order Langevin Monte Carlo (LMC) algorithms is by combining splitting and accurate integration methods. We obtain Wasserstein convergence guarantees for sampling from distributions with log-concave and smooth densities. Specifically, the mixing time of the $P$-th order LMC algorithm scales as $O\left(d^{\frac{1}{R}}/ε^{\frac{1}{2R}}\right)$ for $R=4\cdot 1_{\{ P=3\}}+ (2P-1)\cdot 1_{\{ P\geq 4\}}$, which has a better dependence on the dimension $d$ and the accuracy level $ε$ as $P$ grows. Numerical experiments illustrate the efficiency of our proposed algorithms.
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