Approximating Langevin Monte Carlo with ResNet-like Neural Network architectures
This work addresses sampling challenges in machine learning, offering a novel neural network-based approach, but it appears incremental as it builds on existing Langevin Monte Carlo methods.
The paper tackles the problem of sampling from smooth, log-concave target distributions by constructing a neural network that maps samples from a reference distribution to the target, inspired by the Langevin Monte Carlo algorithm. It shows approximation rates in Wasserstein-2 distance, deriving bounds on variance proxies under perturbation assumptions.
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.