提出新采样算法，突破对数凹性限制，计算更高效且误差有保障。
When Langevin Monte Carlo Meets Randomization: New Sampling Algorithms with Non-asymptotic Error Bounds beyond Log-Concavity and Gradient Lipschitzness
- 引入随机分裂策略，降低梯度计算次数，提升效率。
- 在非对数凹分布下仍保持 $O(\ ext{\sqrt{d}}h)$ 的非渐近误差界。
- 适用于梯度超线性增长场景，适合高维复杂分布采样研究者。
从复杂高维目标分布中高效采样是科学计算、统计和机器学习中的基础任务。本文提出一种新型随机分裂朗之万蒙特卡洛（RSLMC）算法，用于在无对数凹性假设下进行高维采样。相比现有随机朗之万蒙特卡洛（RLMC）算法，RSLMC所需梯度评估次数更少，计算成本更低。在梯度利普希茨条件与对数索博列夫不等式下，证明了RLMC与RSLMC在$\mathcal{W}_2$距离上的均匀时间误差界为$O(\sqrt{d}h)$，该结果在对数凹性条件下达到文献最优水平。当势能函数$U$的梯度非全局利普希茨且具有超线性增长时，进一步提出并分析了改进的R(S)LMC算法，建立了非渐近误差界。数值实验验证了理论结论的有效性。
原文摘要 · Abstract (English)
Efficient sampling from complex and high dimensional target distributions turns out to be a fundamental task in diverse disciplines such as scientific computing, statistics and machine learning. In this paper, we propose a new kind of randomized splitting Langevin Monte Carlo (RSLMC) algorithm for sampling from high dimensional distributions without log-concavity. Compared with the existing randomized Langevin Monte Carlo (RLMC), the newly proposed RSLMC algorithm requires less evaluations of gradients and is thus computationally cheaper. Under the gradient Lipschitz condition and the log-Sobolev inequality, we prove a uniform-in-time error bound in $\mathcal{W}_2$-distance of order $O(\sqrt{d}h)$ for both RLMC and RSLMC sampling algorithms, which matches the best one in the literature under the log-concavity condition. Moreover, when the gradient of the potential $U$ is non-globally Lipschitz with superlinear growth, new modified R(S)LMC algorithms are introduced and analyzed, with non-asymptotic error bounds established. Numerical examples are finally reported to corroborate the theoretical findings.
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