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. With a properly chosen smoothing intensity, the accuracy of the algorithm is guaranteed. 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：本文研究从$\mathbb{R}^d$中势能函数非光滑的目标分布中采样的问题。与非光滑采样问题相比，光滑势能下的采样问题由于缺乏光滑性而远未得到充分理解。本文针对一类非光滑势能提出一种新颖的采样算法，该算法首先通过类似Nesterov平滑的技术用光滑势能逼近非光滑势能，随后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。通过合理选择平滑强度，算法精度得到保证。因此我们基于现有光滑采样分析获得了非渐近收敛结果。我们在合成示例上验证了收敛性结论，并通过实际案例应用该方法改善了贝叶斯推断的最差性能。