Optimal Convergence Rate of Hamiltonian Monte Carlo for Strongly Logconcave Distributions

05/07/2019

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by   Zongchen Chen, et al.

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We study Hamiltonian Monte Carlo (HMC) for sampling from a strongly logconcave density proportional to e^-f where f:R^d →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 O(κ), improving on the previous best bound of O(κ^1.5); we complement this with an example where the relaxation time is Ω(κ). When implemented using a nearly optimal ODE solver, HMC returns an ε-approximate point in 2-Wasserstein distance using O((κ d)^0.5ε^-1) gradient evaluations per step and O((κ d)^1.5ε^-1) total time.