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arXiv · 1905.02313

Optimal Convergence Rate of Hamiltonian Monte Carlo for Strongly Logconcave Distributions

Zongchen Chen·Santosh S. Vempala
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Abstract

We study Hamiltonian Monte Carlo (HMC) for sampling from a strongly logconcave density proportional to $e^{-f}$ where $f:\mathbb{R}^d \to \mathbb{R}$ is $\mu$-strongly convex and $L$-smooth (the condition number is $\kappa = L/\mu$). We show that the relaxation time (inverse of the spectral gap) of ideal HMC is $O(\kappa)$, improving on the previous best bound of $O(\kappa^{1.5})$; we complement this with an example where the relaxation time is $\Omega(\kappa)$. When implemented using a nearly optimal ODE solver, HMC returns an $\varepsilon$-approximate point in $2$-Wasserstein distance using $\widetilde{O}((\kappa d)^{0.5} \varepsilon^{-1})$ gradient evaluations per step and $\widetilde{O}((\kappa d)^{1.5}\varepsilon^{-1})$ total time.

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BibTeXRIS

Zongchen Chen, Santosh S. Vempala. 2019-05-07. Optimal Convergence Rate of Hamiltonian Monte Carlo for Strongly Logconcave Distributions. https://arxiv.org/abs/1905.02313

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