A Theoretical Comparison of No-U-Turn Sampler Variants: Necessary and Su?cient Convergence Conditions and Mixing Time Analysis under Gaussian Targets

2026-03-19

AI Summary

This work addresses the lack of a systematic theoretical comparison between the two dominant variants of the No-U-Turn Sampler (NUTS)—namely NUTS-mul and NUTS-BPS—with respect to convergence properties and mixing times. Building on Markov chain Monte Carlo theory, the paper establishes necessary and sufficient conditions for geometric ergodicity of both algorithms. It provides, for the first time, a sufficient condition guaranteeing geometric ergodicity for NUTS-mul and derives an explicit mixing time bound for NUTS-BPS under a standard Gaussian target distribution. The analysis reveals that while the two variants exhibit highly consistent qualitative behavior, NUTS-BPS enjoys a more favorable convergence rate constant. Moreover, both algorithms display mixing times that scale as $O(d^{1/4})$ with dimensionality $d$, offering theoretical justification for their effectiveness in high-dimensional Bayesian inference.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

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.

Problem

• No-U-Turn Sampler
• geometric ergodicity
• mixing time
• NUTS variants
• convergence conditions

Innovation

• No-U-Turn Sampler
• geometric ergodicity
• mixing time
• NUTS-mul
• NUTS-BPS