We propose a Monte Carlo sampler from the reverse diffusion process. Unlike the practice of diffusion models, where the intermediary updates -- the score functions -- are learned with a neural network, we transform the score matching problem into a mean estimation one. By estimating the means of the regularized posterior distributions, we derive a novel Monte Carlo sampling algorithm called reverse diffusion Monte Carlo (rdMC), which is distinct from the Markov chain Monte Carlo (MCMC) methods. We determine the sample size from the error tolerance and the properties of the posterior distribution to yield an algorithm that can approximately sample the target distribution with any desired accuracy. Additionally, we demonstrate and prove under suitable conditions that sampling with rdMC can be significantly faster than that with MCMC. For multi-modal target distributions such as those in Gaussian mixture models, rdMC greatly improves over the Langevin-style MCMC sampling methods both theoretically and in practice. The proposed rdMC method offers a new perspective and solution beyond classical MCMC algorithms for the challenging complex distributions.

翻译：我们提出一种基于反向扩散过程的蒙特卡洛采样器。与扩散模型的常规做法不同——其通过神经网络学习中间更新量（即得分函数）——我们将得分匹配问题转化为均值估计问题。通过估计正则化后验分布的均值，我们推导出一种新颖的蒙特卡洛采样算法，称为反向扩散蒙特卡洛（rdMC），该算法区别于马尔可夫链蒙特卡洛（MCMC）方法。根据误差容限和后验分布的特性确定样本量，我们得到一种算法，能够以任意期望精度近似采样目标分布。此外，我们证明并在适当条件下验证，采用rdMC进行采样可以显著快于MCMC方法。对于多模态目标分布（如高斯混合模型中的分布），rdMC在理论和实践中均大幅优于基于Langevin动力学的MCMC采样方法。所提出的rdMC方法为经典MCMC算法难以处理的复杂分布提供了全新的视角和解决方案。