A splitting Hamiltonian Monte Carlo method for efficient sampling

Abstract

We propose a splitting Hamiltonian Monte Carlo (SHMC) algorithm, which can be computationally efficient when combined with the random mini-batch strategy. By splitting the potential energy into numerically nonstiff and stiff parts, one makes a proposal using the nonstiff part of $U$, followed by a Metropolis rejection step using the stiff part that is often easy to compute. The splitting allows efficient sampling from systems with singular potentials (or distributions with degenerate points) and/or with multiple potential barriers. In our SHMC algorithm, the proposal only based on the nonstiff part in the splitting is generated by the Hamiltonian dynamics, which can be potentially more efficient than the overdamped Langevin dynamics. We also use random batch strategies to reduce the computational cost to $\mathcal{O}(1)$ per time step in generating the proposals for problems arising from many-body systems and Bayesian inference, and prove that the errors of the Hamiltonian induced by the random batch approximation is $\mathcal{O}(\sqrt{\Delta t})$ in the strong and $\mathcal{O}(\Delta t)$ in the weak sense, where $\Delta t$ is the time step. Numerical experiments are conducted to verify the theoretical results and the computational efficiency of the proposed algorithms in practice.

References in corpus (14)

• Escaping free-energy minima
• An efficient, multiple range random walk algorithm to calculate the density of states
• Simulated Tempering: A New Monte Carlo Scheme
• Determining the density of states for classical statistical models: A random walk algorithm to produce a flat histogram
• General state space Markov chains and MCMC algorithms
• A Conceptual Introduction to Hamiltonian Monte Carlo
• User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient
• On the Random Batch Method for second order interacting particle systems
• A stochastic version of Stein Variational Gradient Descent for efficient sampling
• Does Hamiltonian Monte Carlo mix faster than a random walk on multimodal densities?
• On polynomial-time computation of high-dimensional posterior measures by Langevin-type algorithms
• The Wang-Landau Algorithm as Stochastic Optimization and Its Acceleration
• A random batch Ewald method for particle systems with Coulomb interactions
• Random Batch Algorithms for Quantum Monte Carlo simulations