对比两种NUTS采样器的收敛性，发现其效率在高维下接近但有差异。
A Theoretical Comparison of No-U-Turn Sampler Variants: Necessary and Sufficient Convergence Conditions and Mixing Time Analysis under Gaussian Targets
- 首次推导出两种NUTS变体的几何遍历必要条件
- 证明在标准高斯分布下，混合时间均为O(d^{1/4})阶
- 揭示NUTS-BPS收敛常数更优，适合高维贝叶斯推断
No-U-Turn Sampler (NUTS) 是现代贝叶斯软件库的核心计算工具，但其定性和定量收敛保证仅近期才建立。本文填补了其两大变体——使用多项式采样的NUTS-mul与使用有偏渐进采样的NUTS-BPS——之间的理论比较空白。首先，我们推导出两者几何遍历性的首个必要条件；其次，建立了NUTS-mul几何遍历性和遍历性的首个充分条件；第三，首次获得NUTS-BPS在标准高斯分布下的混合时间结果。结果表明，两者定性行为几乎相同，几何遍历性取决于目标分布尾部性质。然而在收敛速率上存在定量差异：当初始化于标准高斯测度的典型集时，两者的混合时间均以O(d^{1/4})阶增长（对数因子内），但NUTS-BPS的常数更小。
原文摘要 · Abstract (English)
The No-U-Turn Sampler (NUTS) is the computational workhorse of modern Bayesian software libraries, yet its qualitative and quantitative convergence guarantees were established only recently. A significant gap remains in the theoretical comparison of its two main variants: NUTS-mul and NUTS-BPS, which use multinomial sampling and biased progressive sampling, respectively, for index selection. In this paper, we address this gap in three contributions. First, we derive the first necessary conditions for geometric ergodicity for both variants. Second, we establish the first sufficient conditions for geometric ergodicity and ergodicity for NUTS-mul. Third, we obtain the first mixing time result for NUTS-BPS on a standard Gaussian distribution. Our results show that NUTS-mul and NUTS-BPS exhibit nearly identical qualitative behavior, with geometric ergodicity depending on the tail properties of the target distribution. However, they differ quantitatively in their convergence rates. More precisely, when initialized in the typical set of the canonical Gaussian measure, the mixing times of both NUTS-mul and NUTS-BPS scale as $O(d^{1/4})$ up to logarithmic factors, where $d$ denotes the dimension. Nevertheless, the associated constants are strictly smaller for NUTS-BPS.
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