Markov Chain Monte Carlo from Lagrangian Dynamics.

Area of Science:

• Computational Statistics
• Machine Learning
• Numerical Analysis

Background:

• Hamiltonian Monte Carlo (HMC) enhances the Metropolis-Hastings algorithm by minimizing random walk behavior.
• Riemannian HMC (RHMC) leverages geometric properties of parameter spaces for improved computational efficiency.
• RHMC's reliance on implicit equations and fixed-point iterations can introduce significant computational overhead, potentially negating its benefits.

Purpose of the Study:

• To develop a more computationally efficient variant of Riemannian HMC.
• To address the computational bottleneck caused by implicit equations in standard RHMC.
• To propose an explicit integration method that maintains or improves upon RHMC's performance.

Main Methods:

• An explicit integrator was developed, replacing the momentum variable in RHMC with velocity.
• This transformation was shown to be equivalent to converting Riemannian Hamiltonian dynamics to Lagrangian dynamics.
• The proposed method avoids the need for fixed-point iterations required by implicit integrators.

Main Results:

• Experimental results indicate that the proposed explicit integrator improves the overall computational efficiency of RHMC.
• The method is particularly effective in cases where implicit equation solving is computationally intensive.
• The study provides open-source code and datasets for result replication.

Conclusions:

• The proposed explicit integration method offers a computationally advantageous alternative to standard RHMC.
• This approach effectively circumvents the overhead associated with implicit solvers in RHMC.
• The findings suggest a practical improvement for applying advanced Monte Carlo methods in statistical inference and machine learning.