We analyze the complexity of sampling from a class of heavy-tailed distributions by discretizing a natural class of It\^o diffusions associated with weighted Poincar\'e inequalities. Based on a mean-square analysis, we establish the iteration complexity for obtaining a sample whose distribution is $\epsilon$ close to the target distribution in the Wasserstein-2 metric. In this paper, our results take the mean-square analysis to its limits, i.e., we invariably only require that the target density has finite variance, the minimal requirement for a mean-square analysis. To obtain explicit estimates, we compute upper bounds on certain moments associated with heavy-tailed targets under various assumptions. We also provide similar iteration complexity results for the case where only function evaluations of the unnormalized target density are available by estimating the gradients using a Gaussian smoothing technique. We provide illustrative examples based on the multivariate $t$-distribution.

翻译：我们通过离散化与加权庞加莱不等式相关联的一类自然伊藤扩散，分析了从一类重尾分布中采样的复杂度。基于均方分析，我们建立了在Wasserstein-2度量下获得分布与目标分布相差$\epsilon$的样本所需的迭代复杂度。在本文中，我们的结果将均方分析推至其极限，即我们始终仅要求目标密度具有有限方差——这是均方分析的最低要求。为获得显式估计，我们计算了与重尾目标相关的特定矩在不同假设下的上界。我们还针对仅能获得非归一化目标密度函数评估值的情况，通过采用高斯平滑技术估计梯度，提供了类似的迭代复杂度结果。我们基于多元$t$分布给出了示例说明。