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.

通过构建一个神经网络，我们从一个简单的参考分布（如标准正态分布）映射到目标分布的样本，从而从目标分布中采样。通过Langevin Monte Carlo算法，我们提出了一个神经网络架构， 并在Wasserstein-2距离下的光滑、对数凹目标分布的逼近速率进行了分析，其中分析主要依赖于扰动的中间测量的次高斯性质，并在不同假设下导出了中间方差代理的增长上界，同时我们还提出了一种类似于深度残差神经网络的架构，并推导了用于逼近样本到目标分布映射的表达能力结果。

利用类ResNet神经网络架构近似Langevin Monte Carlo