Generating samples from a continuous probability density is a central algorithmic problem across statistics, engineering, and the sciences. For high-dimensional settings, Hamiltonian Monte Carlo (HMC) is the default algorithm across mainstream software packages. However, despite the extensive line of work on HMC and its widespread empirical success, it remains unclear how many iterations of HMC are required as a function of the dimension $d$. On one hand, a variety of results show that Metropolized HMC converges in $O(d^{1/4})$ iterations from a warm start close to stationarity. On the other hand, Metropolized HMC is significantly slower without a warm start, e.g., requiring $Ω(d^{1/2})$ iterations even for simple target distributions such as isotropic Gaussians. Finding a warm start is therefore the computational bottleneck for HMC. We resolve this issue for the well-studied setting of sampling from a probability distribution satisfying strong log-concavity (or isoperimetry) and third-order derivative bounds. We prove that \emph{non-Metropolized} HMC generates a warm start in $\tilde{O}(d^{1/4})$ iterations, after which we can exploit the warm start using Metropolized HMC. Our final complexity of $\tilde{O}(d^{1/4})$ is the fastest algorithm for high-accuracy sampling under these assumptions, improving over the prior best of $\tilde{O}(d^{1/2})$. This closes the long line of work on the dimensional complexity of MHMC for such settings, and also provides a simple warm-start prescription for practical implementations.

翻译：从连续概率密度中生成样本是统计学、工程学和科学领域的核心算法问题。在高维场景下，哈密顿蒙特卡洛（HMC）已成为主流软件包中的默认算法。然而，尽管关于HMC的研究工作层出不穷且其经验性成功广泛存在，但关于HMC所需迭代次数作为维度$d$的函数仍不明确。一方面，多项结果表明，Metropolis-HMC算法从接近平稳状态的热启动出发，可在$O(d^{1/4})$次迭代内收敛。另一方面，若无热启动，Metropolis-HMC的收敛速度显著变慢，即使用于各向同性高斯分布这类简单目标分布，也需$Ω(d^{1/2})$次迭代。因此，寻找热启动已成为HMC的计算瓶颈。针对满足强对数凹性（或等周不等式）及三阶导数界条件的概率分布采样这一经典设定，我们解决了该问题。我们证明，非Metropolis化的HMC可在$\tilde{O}(d^{1/4})$次迭代内生成热启动，随后可利用该热启动通过Metropolis-HMC加速采样。在此假设下，我们最终得到的$\tilde{O}(d^{1/4})$复杂度是高精度采样的最快算法，相较于此前最优的$\tilde{O}(d^{1/2})$实现了突破。这一结果不仅完善了该设定下MHMC维度复杂性的长期研究，也为实际实现提供了简单的热启动方案。