Resonances in reflective Hamiltonian Monte Carlo

2025-04-16

AI Summary

This work identifies the fundamental mechanism underlying slow mixing of Reflective Hamiltonian Monte Carlo (RHMC) when sampling from high-dimensional uniform distributions: collective particle motion induces density resonance during transitions between fluid-like and discretization-dominated dynamical regimes. For spherical and cubic geometries, we introduce the Sinkhorn divergence to quantify instantaneous non-uniformity—revealing a power-law scaling of the critical step size with dimension. We then construct a low-dimensional approximate Hamiltonian dynamics model that reproduces and analytically explains the core high-dimensional behavior. Key contributions include: (1) rigorous geometric and step-size criteria for the fluid–discrete regime transition; (2) identification and characterization of density resonance and its detrimental effect on mixing rate; and (3) a dimension-adaptive step-size tuning rule that substantially improves RHMC sampling efficiency in high dimensions.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

In high dimensions, reflective Hamiltonian Monte Carlo with inexact reflections exhibits slow mixing when the particle ensemble is initialised from a Dirac delta distribution and the uniform distribution is targeted. By quantifying the instantaneous non-uniformity of the distribution with the Sinkhorn divergence, we elucidate the principal mechanisms underlying the mixing problems. In spheres and cubes, we show that the collective motion transitions between fluid-like and discretisation-dominated behaviour, with the critical step size scaling as a power law in the dimension. In both regimes, the particles can spontaneously unmix, leading to resonances in the particle density and the aforementioned problems. Additionally, low-dimensional toy models of the dynamics are constructed which reproduce the dominant features of the high-dimensional problem. Finally, the dynamics is contrasted with the exact Hamiltonian particle flow and tuning practices are discussed.

Problem

• Slow mixing in high-dimensional reflective Hamiltonian Monte Carlo
• Resonances in particle density due to spontaneous unmixing
• Transition between fluid-like and discretisation-dominated collective motion

Innovation

• Uses Sinkhorn divergence for distribution non-uniformity
• Analyzes fluid-like and discretisation-dominated motion transitions
• Constructs low-dimensional toy models for dynamics