Resonances in reflective Hamiltonian Monte Carlo

The challenge of efficiently sampling complex, high-dimensional probability distributions is a cornerstone of modern computational science, underpinning progress in fields from Bayesian inference in astrophysics to materials discovery. A new publication, “Resonances in reflective Hamiltonian Monte Carlo” (2504.12374), by lead author Namu Kroupa, alongside Gábor Csányi and Will Handley, delves into persistent issues plaguing a popular sampling algorithm: Reflective Hamiltonian Monte Carlo (RHMC). This work sheds light on subtle dynamical behaviors that can lead to slow mixing and systematic errors, particularly when RHMC is applied with inexact reflections—a common scenario when analytical boundaries are unknown or computationally prohibitive to determine.

Understanding Mixing Challenges in High Dimensions

Reflective Hamiltonian Monte Carlo, especially variants like Galilean Monte Carlo (GMC), is widely used within sophisticated frameworks such as nested sampling to compute Bayesian evidence or partition functions. The increasing availability of gradient information from automatic differentiation tools has further fueled interest in such gradient-based samplers. However, practitioners have long observed that RHMC can struggle in high-dimensional spaces ($\mathcal{O}(10)$ dimensions and beyond), often introducing systematic biases in crucial calculations, as noted in works like Lemos et al. (2312.03911). The featured paper tackles this problem head-on by investigating why these issues arise, particularly when particle ensembles are initialized from highly localized (Dirac delta) distributions to explore a target uniform distribution.

To dissect the mixing problems, the authors employ the Sinkhorn divergence as a metric to quantify the instantaneous non-uniformity of the particle distribution over time. This approach allows for a detailed characterization of how the particle ensemble evolves, and the study leverages computational tools such as the \textsc{OTT} library, as discussed in related optimal transport literature (e.g., 2201.12324).

The Core Problem: Resonances and Particle Unmixing

A key contribution of the paper is the identification and elucidation of “resonances” in the particle density. These are phenomena where the particle ensemble, contrary to the goal of uniform mixing, can spontaneously unmix or “bunch up.” This unmixing is a primary cause of the observed slow convergence and systematic errors.

The research reveals that the collective motion of particles under RHMC with inexact reflections can transition between two distinct regimes:

• Fluid-like behavior: At smaller step sizes, the particle ensemble spreads out more smoothly.
• Discretisation-dominated behavior: At larger step sizes, the discrete nature of the simulation steps significantly impacts the dynamics.

Crucially, the critical step size demarcating these regimes is shown to scale as a power law with the dimensionality of the space. In both regimes, however, the potential for spontaneous unmixing and resonance formation persists.

A Closer Look: Dynamics in Spheres and Cubes

The authors meticulously analyze RHMC dynamics within two canonical geometries—spheres and cubes—which serve as tractable models for more complex, real-world scenarios.

• In Spherical Geometries:
  • The high symmetry allows for an effectively lossless two-dimensional representation of particle trajectories.
  • A significant finding is that particles tend to get “stuck” near their initial radius, with motion predominantly occurring along the boundary of the 2D projected disc.
  • This leads to the emergence of a “supersonic” frequency in the particle density oscillations. Particles moving along the boundary, due to the mechanics of inexact reflections, can achieve a higher effective speed than those traversing the diameter, leading to faster (supersonic) arrival at antipodal points but poor exploration of the radial dimension.
• In Cubic Geometries:
  • The dynamics can transition to quasi-periodic behavior, where particles become trapped along effectively one-dimensional paths.
  • The “rejection branch” of the GMC algorithm (where a particle reverses momentum if it cannot be reflected back inside) becomes increasingly prevalent in higher dimensions due to the geometry of cube corners.
  • Inexact reflections on flat boundaries are shown to cause a focusing effect: a dispersing wave packet of particles can be refocused after reflection, leading to density bunching. This is a direct consequence of the inexact reflection mechanism preserving particle order, unlike exact billiard dynamics.

Implications for Sampling and Future Directions

The insights from this research have significant implications for the practical application of RHMC. The observed resonances and poor radial mixing in spherical scenarios directly contribute to the underestimation of volumes in nested sampling, leading to systematically negative errors in Bayesian model evidence. The study also highlights that current tuning heuristics for RHMC, such as monitoring trajectory-wise acceptance rates, are often uninformative about these detrimental resonance phenomena and the true mixing efficiency. Numerical intricacies of Hamiltonian Monte Carlo methods are indeed a subject of ongoing research (see, for instance, 2111.09995).

While the paper touches upon potential mitigation strategies, such as adding noise to particle momenta to damp resonances, it primarily underscores the need for a more fundamental understanding to guide the development of robust tuning practices and potentially new algorithms.

In conclusion, Kroupa, Csányi, and Handley provide a detailed and critical examination of RHMC dynamics, offering a clear explanation for its long-standing issues in high-dimensional settings. Their work serves as a crucial step towards developing more reliable and efficient sampling techniques for the complex scientific models of tomorrow.

Content generated by gemini-2.5-pro-preview-05-06 using this prompt.

Image generated by imagen-3.0-generate-002 using this prompt.