In this article we propose a novel taming Langevin-based scheme called $\mathbf{sTULA}$ to sample from distributions with superlinearly growing log-gradient which also satisfy a Log-Sobolev inequality. We derive non-asymptotic convergence bounds in $KL$ and consequently total variation and Wasserstein-$2$ distance from the target measure. Non-asymptotic convergence guarantees are provided for the performance of the new algorithm as an optimizer. Finally, some theoretical results on isoperimertic inequalities for distributions with superlinearly growing gradients are provided. Key findings are a Log-Sobolev inequality with constant independent of the dimension, in the presence of a higher order regularization and a Poincare inequality with constant independent of temperature and dimension under a novel non-convex theoretical framework.

翻译：本文提出了一种基于驯服化Langevin算法的新方案，称为$\mathbf{sTULA}$，用于从具有超线性增长对数梯度且满足对数索伯列夫不等式的分布中进行采样。我们推导了与目标测度之间$KL$散度（进而总变差距离和Wasserstein-2距离）的非渐近收敛界，并给出了该新算法作为优化器的非渐近收敛保证。最后，针对具有超线性增长梯度的分布，提供了关于等周不等式的若干理论结果。关键发现包括：在高阶正则化存在下，对数索伯列夫不等式的常数与维度无关；以及在新型非凸理论框架下，庞加莱不等式的常数独立于温度和维度。