To sample from a general target distribution $p_*\propto e^{-f_*}$ beyond the isoperimetric condition, Huang et al. (2023) proposed to perform sampling through reverse diffusion, giving rise to Diffusion-based Monte Carlo (DMC). Specifically, DMC follows the reverse SDE of a diffusion process that transforms the target distribution to the standard Gaussian, utilizing a non-parametric score estimation. However, the original DMC algorithm encountered high gradient complexity, resulting in an exponential dependency on the error tolerance $\epsilon$ of the obtained samples. In this paper, we demonstrate that the high complexity of DMC originates from its redundant design of score estimation, and proposed a more efficient algorithm, called RS-DMC, based on a novel recursive score estimation method. In particular, we first divide the entire diffusion process into multiple segments and then formulate the score estimation step (at any time step) as a series of interconnected mean estimation and sampling subproblems accordingly, which are correlated in a recursive manner. Importantly, we show that with a proper design of the segment decomposition, all sampling subproblems will only need to tackle a strongly log-concave distribution, which can be very efficient to solve using the Langevin-based samplers with a provably rapid convergence rate. As a result, we prove that the gradient complexity of RS-DMC only has a quasi-polynomial dependency on $\epsilon$, which significantly improves exponential gradient complexity in Huang et al. (2023). Furthermore, under commonly used dissipative conditions, our algorithm is provably much faster than the popular Langevin-based algorithms. Our algorithm design and theoretical framework illuminate a novel direction for addressing sampling problems, which could be of broader applicability in the community.

翻译：为对满足一般目标分布 $p_*\propto e^{-f_*}$ 且不满足等周条件的情况进行采样，Huang等人(2023)提出通过反向扩散过程执行采样，由此发展出基于扩散的蒙特卡洛方法(DMC)。具体而言，DMC遵循将目标分布转化为标准高斯分布的扩散过程的反向随机微分方程，并采用非参数得分估计。然而，原始DMC算法存在高梯度复杂度问题，导致所得样本对误差容限$\epsilon$的依赖呈指数级。本文证明DMC的高复杂度源于其得分估计的冗余设计，并提出一种基于新型递归得分估计方法的更高效算法——RS-DMC。我们首先将整个扩散过程划分为多个片段，然后相应地将任意时间步的得分估计步骤构建为一系列相互关联的均值估计与采样子问题，这些子问题以递归方式相互关联。重要的是，我们证明通过合理设计片段分解，所有采样子问题仅需处理强对数凹分布，而此类分布可通过具有可证明快速收敛速率的朗之万采样器高效求解。基于此，我们证明RS-DMC的梯度复杂度仅关于$\epsilon$具有拟多项式依赖，这显著改进了Huang等人(2023)中的指数梯度复杂度。此外，在常用耗散条件下，本文算法被证明明显快于流行的朗之万类算法。我们的算法设计与理论框架为解决采样问题开辟了新方向，有望在相关领域具有更广泛的适用性。