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.

翻译：Langevin蒙特卡洛（LMC）及其随机梯度版本是从复杂高维分布中采样的强大算法。为了从密度为$\pi(\theta)\propto \exp(-U(\theta)) $的分布中采样，LMC通过沿梯度方向$\nabla U$迈出一步并添加高斯扰动来迭代生成下一个样本。目标分布$\pi$的期望通过对LMC样本取平均来估计。在普通蒙特卡洛中，众所周知，用低偏差点序列等拟随机样本替代独立随机样本可以显著降低估计误差。在本工作中，我们证明LMC的估计误差也可通过使用拟随机样本降低。具体而言，我们提出使用具有特定低偏差性质的完全均匀分布（CUD）序列生成高斯扰动。在光滑性和凸性条件下，我们证明采用低偏差CUD序列的LMC比标准LMC能达到更小的误差。该理论分析得到了具有说服力的数值实验支持，这些实验展示了我们方法的有效性。