We sample from a given target distribution by constructing a neural network which maps samples from a simple reference, e.g. the standard normal distribution, to samples from the target. To that end, we propose using a neural network architecture inspired by the Langevin Monte Carlo (LMC) algorithm. Based on LMC perturbation results, we show approximation rates of the proposed architecture for smooth, log-concave target distributions measured in the Wasserstein-$2$ distance. The analysis heavily relies on the notion of sub-Gaussianity of the intermediate measures of the perturbed LMC process. In particular, we derive bounds on the growth of the intermediate variance proxies under different assumptions on the perturbations. Moreover, we propose an architecture similar to deep residual neural networks and derive expressivity results for approximating the sample to target distribution map.

翻译：我们通过构建一个神经网络来从给定目标分布中采样，该网络将简单参考分布（如标准正态分布）的样本映射为目标分布的样本。为此，我们提出了一种受Langevin Monte Carlo (LMC)算法启发的神经网络架构。基于LMC扰动结果，我们展示了该架构在Wasserstein-2距离度量下对光滑对数凹目标分布的逼近速率。分析很大程度上依赖于扰动LMC过程中间测度的次高斯性概念。特别地，我们推导了在不同扰动假设下中间方差代理的增长界。此外，我们提出了一种类似于深度残差神经网络的架构，并推导了逼近样本到目标分布映射的表达能力结果。