This paper considers the problem of sampling from non-logconcave distribution, based on queries of its unnormalized density. It first describes a framework, Diffusion Monte Carlo (DMC), based on the simulation of a denoising diffusion process with its score function approximated by a generic Monte Carlo estimator. DMC is an oracle-based meta-algorithm, where its oracle is the assumed access to samples that generate a Monte Carlo score estimator. Then we provide an implementation of this oracle, based on rejection sampling, and this turns DMC into a true algorithm, termed Zeroth-Order Diffusion Monte Carlo (ZOD-MC). We provide convergence analyses by first constructing a general framework, i.e. a performance guarantee for DMC, without assuming the target distribution to be log-concave or satisfying any isoperimetric inequality. Then we prove that ZOD-MC admits an inverse polynomial dependence on the desired sampling accuracy, albeit still suffering from the curse of dimensionality. Consequently, for low dimensional distributions, ZOD-MC is a very efficient sampler, with performance exceeding latest samplers, including also-denoising-diffusion-based RDMC and RS-DMC. Last, we experimentally demonstrate the insensitivity of ZOD-MC to increasingly higher barriers between modes or discontinuity in non-convex potential.

翻译：本文研究基于非归一化密度查询从非对数凹分布中采样的问题。首先提出一个基于去噪扩散过程模拟的框架——扩散蒙特卡洛（DMC），其评分函数通过通用蒙特卡洛估计器进行近似。DMC是一种基于预言机的元算法，其预言机被设定为可访问生成蒙特卡洛评分估计器的样本。随后我们基于拒绝采样实现了该预言机，从而将DMC转化为可执行算法，称为零阶扩散蒙特卡洛（ZOD-MC）。我们通过构建通用分析框架（即DMC的性能保证）进行收敛性分析，该框架不要求目标分布满足对数凹性或等周不等式。进而证明ZOD-MC对采样精度的依赖呈逆多项式关系，但仍受维度诅咒的影响。因此对于低维分布，ZOD-MC成为高效采样器，其性能超越包括同样基于去噪扩散的RDMC和RS-DMC在内的最新采样器。最后，我们通过实验验证了ZOD-MC对模态间势垒增强或非凸势函数不连续性的不敏感性。