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.

研究了用于采样强对数凹密度的哈密顿蒙特卡罗方法在实现时，理想状态下的弛豫时间是 O (κ) 时，其松弛时间（谱间隙的倒数）是 O (κ)，这比之前最好的上限 O (κ^1.5) 更优。当使用近乎最优的 ODE 求解器实现时，每一步需要进行 O ((κd)^0.5 (ε^-1)) 个梯度评估，总时间为 O ((κd)^1.5 (ε^-1))，并返回在 2-Wasserstein 距离内的一个 ε- 近似点。