Langevin Dynamics is a Stochastic Differential Equation (SDE) central to sampling and generative modeling and is implemented via time discretization. Langevin Monte Carlo (LMC), based on the Euler-Maruyama discretization, is the simplest and most studied algorithm. LMC can suffer from slow convergence - requiring a large number of steps of small step-size to obtain good quality samples. This becomes stark in the case of diffusion models where a large number of steps gives the best samples, but the quality degrades rapidly with smaller number of steps. Randomized Midpoint Method has been recently proposed as a better discretization of Langevin dynamics for sampling from strongly log-concave distributions. However, important applications such as diffusion models involve non-log concave densities and contain time varying drift. We propose its variant, the Poisson Midpoint Method, which approximates a small step-size LMC with large step-sizes. We prove that this can obtain a quadratic speed up of LMC under very weak assumptions. We apply our method to diffusion models for image generation and show that it maintains the quality of DDPM with 1000 neural network calls with just 50-80 neural network calls and outperforms ODE based methods with similar compute.

翻译：朗格文动力学是一种随机微分方程（SDE），在采样和生成建模中处于核心地位，通常通过时间离散化实现。基于欧拉-丸山离散化的朗格文蒙特卡洛（LMC）是最简单且研究最广泛的算法。LMC可能收敛缓慢——需要大量小步长步骤才能获得高质量样本。这在扩散模型的情况下尤为突出：大量步骤能产生最佳样本，但步骤数减少时样本质量会迅速下降。最近提出的随机中点法被证明是从强对数凹分布采样时朗格文动力学的一种更优离散化方案。然而，扩散模型等重要应用涉及非对数凹密度且包含时变漂移项。我们提出其变体——泊松中点法，该方法能以大步长逼近小步长LMC。我们证明在极弱假设下，该方法可实现LMC的二次加速。我们将该方法应用于图像生成的扩散模型，结果表明仅需50-80次神经网络调用即可保持DDPM使用1000次神经网络调用时的质量，且优于计算量相似的基于ODE的方法。