Langevin Quasi-Monte Carlo

TL;DR

This paper introduces a novel approach combining Langevin Monte Carlo with quasi-random sequences to reduce sampling error in high-dimensional distributions, supported by theoretical proofs and numerical experiments.

Contribution

It proposes using completely uniformly distributed sequences in Langevin Monte Carlo to improve sampling accuracy over traditional methods.

Findings

LMC with CUD sequences achieves lower estimation error.

Theoretical proof of improved convergence under smoothness and convexity.

Numerical experiments confirm the effectiveness of the proposed method.

Abstract

Langevin Monte Carlo (LMC) and its stochastic gradient versions are powerful algorithms for sampling from complex high-dimensional distributions. To sample from a distribution with density $\pi(\theta)\propto \exp(-U(\theta)) $, LMC iteratively generates the next sample by taking a step in the gradient direction $\nabla U$ with added Gaussian perturbations. Expectations w.r.t. the target distribution $\pi$ are estimated by averaging over LMC samples. In ordinary Monte Carlo, it is well known that the estimation error can be substantially reduced by replacing independent random samples by quasi-random samples like low-discrepancy sequences. In this work, we show that the estimation error of LMC can also be reduced by using quasi-random samples. Specifically, we propose to use completely uniformly distributed (CUD) sequences with certain low-discrepancy property to generate the Gaussian perturbations. Under smoothness and convexity conditions, we prove that LMC with a low-discrepancy CUD sequence achieves smaller error than standard LMC. The theoretical analysis is supported by compelling numerical experiments, which demonstrate the effectiveness of our approach.