Given an unnormalized probability density $\pi\propto\mathrm{e}^{-V}$, estimating its normalizing constant $Z=\int_{\mathbb{R}^d}\mathrm{e}^{-V(x)}\mathrm{d}x$ or free energy $F=-\log Z$ is a crucial problem in Bayesian statistics, statistical mechanics, and machine learning. It is challenging especially in high dimensions or when $\pi$ is multimodal. To mitigate the high variance of conventional importance sampling estimators, annealing-based methods such as Jarzynski equality and annealed importance sampling are commonly adopted, yet their quantitative complexity guarantees remain largely unexplored. We take a first step toward a non-asymptotic analysis of annealed importance sampling. In particular, we derive an oracle complexity of $\widetilde{O}\left(\frac{d\beta^2{\mathcal{A}}^2}{\varepsilon^4}\right)$ for estimating $Z$ within $\varepsilon$ relative error with high probability, where $\beta$ is the smoothness of $V$ and $\mathcal{A}$ denotes the action of a curve of probability measures interpolating $\pi$ and a tractable reference distribution. Our analysis, leveraging Girsanov theorem and optimal transport, does not explicitly require isoperimetric assumptions on the target distribution. Finally, to tackle the large action of the widely used geometric interpolation, we propose a new algorithm based on reverse diffusion samplers, establish a framework for analyzing its complexity, and empirically demonstrate its efficiency in tackling multimodality.

翻译：给定未归一化的概率密度 $\pi\propto\mathrm{e}^{-V}$，估计其归一化常数 $Z=\int_{\mathbb{R}^d}\mathrm{e}^{-V(x)}\mathrm{d}x$ 或自由能 $F=-\log Z$ 是贝叶斯统计、统计力学和机器学习中的关键问题。该问题在高维情形或 $\pi$ 为多峰分布时尤为困难。为降低传统重要性采样估计量的高方差，通常采用基于退火的方法，如Jarzynski等式和退火重要性采样，但其定量复杂度保证仍鲜有研究。本文首次对退火重要性采样进行非渐近分析。具体而言，我们推导出以高概率在 $\varepsilon$ 相对误差内估计 $Z$ 的预言机复杂度为 $\widetilde{O}\left(\frac{d\beta^2{\mathcal{A}}^2}{\varepsilon^4}\right)$，其中 $\beta$ 为 $V$ 的光滑性，$\mathcal{A}$ 表示连接 $\pi$ 与易处理的参考分布的概率测度曲线的"作用量"。我们的分析借助Girsanov定理和最优传输理论，无需显式要求目标分布满足等周假设。最后，为应对广泛使用的几何插值所产生的大作用量，我们提出一种基于反向扩散采样器的新算法，建立其复杂度分析框架，并通过实验验证其在处理多峰问题上的有效性。