We study the problem of sampling from a target distribution in $\mathbb{R}^d$ whose potential is not smooth. Compared with the sampling problem with smooth potentials, this problem is much less well-understood due to the lack of smoothness. In this paper, we propose a novel sampling algorithm for a class of non-smooth potentials by first approximating them by smooth potentials using a technique that is akin to Nesterov smoothing. We then utilize sampling algorithms on the smooth potentials to generate approximate samples from the original non-smooth potentials. We select an appropriate smoothing intensity to ensure that the distance between the smoothed and un-smoothed distributions is minimal, thereby guaranteeing the algorithm's accuracy. Hence we obtain non-asymptotic convergence results based on existing analysis of smooth sampling. We verify our convergence result on a synthetic example and apply our method to improve the worst-case performance of Bayesian inference on a real-world example.

翻译：我们研究从$\mathbb{R}^d$中势能非光滑的目标分布进行采样的问题。相较于光滑势能的采样问题，由于缺乏光滑性，该问题的理解尚不充分。本文针对一类非光滑势能提出一种新型采样算法：首先通过类似Nesterov光滑化的技术将非光滑势能近似为光滑势能，进而利用光滑势能上的采样算法生成原始非光滑势能的近似样本。通过选取合适的光滑化强度，我们保证光滑化分布与原始分布之间的距离最小化，从而确保算法精度。基于现有光滑采样方法的分析，我们获得了非渐近收敛性结果。通过人工合成示例验证了收敛性结论，并将该方法应用于实际贝叶斯推断案例，显著提升了最差情况下的推理性能。