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:我们通过分解与加权Poincar\'e不平等相关的Itççóo扩散的自然类别,分析从一类重尾分配中取样的复杂性。 根据平均平方分析,我们为获得一个分布接近瓦塞斯坦2号指标目标分布的样本确定了迭代的复杂性。在本文中,我们的结果将平均平方分析提高到其极限,即我们总是要求目标密度有一定的差异,平均平方分析的最低限度要求。为了获得明确的估计,我们计算某些与不同假设下重尾目标相关的时间的上限。我们还提供了类似的迭代复杂性结果,因为只有使用高尔斯平滑技术估算梯度,才能对非正常目标密度进行功能评估。我们提供了基于多变方美元分配的示例。</s>