Motivated by applications to deep learning which often fail standard Lipschitz smoothness requirements, we examine the problem of sampling from distributions that are not log-concave and are only weakly dissipative, with log-gradients allowed to grow superlinearly at infinity. In terms of structure, we only assume that the target distribution satisfies either a log-Sobolev or a Poincar\'e inequality and a local Lipschitz smoothness assumption with modulus growing possibly polynomially at infinity. This set of assumptions greatly exceeds the operational limits of the "vanilla" unadjusted Langevin algorithm (ULA), making sampling from such distributions a highly involved affair. To account for this, we introduce a taming scheme which is tailored to the growth and decay properties of the target distribution, and we provide explicit non-asymptotic guarantees for the proposed sampler in terms of the Kullback-Leibler (KL) divergence, total variation, and Wasserstein distance to the target distribution.

翻译：受深度学习应用的启发（这些应用通常不满足标准的Lipschitz光滑性要求），我们研究了从非对数凹且仅具有弱耗散性的分布中采样的问题，其中对数梯度在无穷远处允许超线性增长。在结构方面，我们仅假设目标分布满足对数Sobolev不等式或Poincaré不等式，以及一个局部Lipschitz光滑性假设，其模在无穷远处可能以多项式方式增长。这组假设大大超出了"原始"未调整朗之万算法（ULA）的操作范围，使得从此类分布中采样变得非常复杂。为此，我们引入了一种根据目标分布的增长和衰减特性定制的驯服方案，并就所提出采样器与目标分布之间的Kullback-Leibler（KL）散度、总变差距离和Wasserstein距离，给出了明确的非渐近保证。