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：本文考虑基于未归一化密度查询的非对数凹分布采样问题。首先描述了一个称为扩散蒙特卡洛（Diffusion Monte Carlo, DMC）的框架，该框架通过模拟去噪扩散过程，并使用通用蒙特卡洛估计器近似其得分函数。DMC是一种基于元算法的黑箱方法，其黑箱假设为能够获取生成蒙特卡洛得分估计器的样本。随后我们提供了一种基于拒绝采样的该黑箱实现，将DMC转化为实际算法，称为零阶扩散蒙特卡洛（Zeroth-Order Diffusion Monte Carlo, ZOD-MC）。我们首先构建通用框架，即在不假设目标分布为对数凹或满足任何等周不等式的前提下，给出DMC的性能保证，并以此进行收敛性分析。接着证明ZOD-MC对期望采样精度的依赖性为逆多项式形式，尽管仍受维度灾难影响。因此对于低维分布，ZOD-MC是一种高效采样器，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。最后通过实验证明ZOD-MC对于模式间递增高阶势垒或非凸势函数的不连续性具有不敏感性。