A splitting Hamiltonian Monte Carlo method for efficient sampling

Abstract

We propose a splitting Hamiltonian Monte Carlo (SHMC) algorithm, which can be numerically and computationally efficient when combined with the random mini-batch strategy. By splitting the "effective potential energy" U ∝ −β −1 log ρ into two parts U = U1 + U2, one makes a proposal using the "easy-to-sample" part U1, followed by probabilistically accepting that proposal by a Metropolis rejection step using U2. 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 using U1 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 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 O( √ ∆t) in the strong and O(∆t) in the weak sense, where ∆t is the time step. Numerical experiments are conducted to verify the theoretical results and the computational efficiency of the proposed algorithms in practice.

A sharp uniform-in-time error estimate for Stochastic Gradient Langevin Dynamics

“…First, we consider the special case η i = (ℓ + i) −θ , i = 0, 1, 2, ..., with θ ∈ (0, 1) being the decaying coefficient, which is common in many practical tasks [29]. In Section 3, we analyze the asymptotic convergence rate of D KL (ρ T k ||ρ T k ) as k → +∞ for different θ.…”