Optimal Convergence Rate of Hamiltonian Monte Carlo for Strongly Logconcave Distributions

Abstract

We study the Hamiltonian Monte Carlo (HMC) algorithm for sampling from a strongly logconcave density proportional to e−f where f: ℝd → ℝ is μ-strongly convex and L-smooth (the condition number is κ = L/μ). We show that the relaxation time (inverse of the spectral gap) of ideal HMC is N(κ), improving on the previous best bound of N(κ1.5) (Lee et al., 2018); we complement this with an example where the relaxation time is Ω(κ), for any step-size. When implemented with an ODE solver, HMC returns an ε-approximate point in 2-Wasserstein distance using ˜N((κd)0.5 ε−1) gradient evaluations per step and ˜N((κd)1.5 ε−1) total time.