Approximating Langevin Monte Carlo with ResNet-like Neural Network Architectures

Abstract

Abstract We analyse a method to sample from a given target distribution by constructing a neural network which maps samples from a simple reference distribution, e.g. the standard normal, to samples from the target distribution. For this, we propose using a neural network architecture inspired by the Langevin Monte Carlo (LMC) algorithm. Based on LMC perturbation results, approximation rates of the proposed architecture for smooth, log-concave target distributions measured in the Wasserstein-2 distance are shown. 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 (ResNets) and derive expressivity results for approximating the sample to target distribution map.