Hamiltonian Monte Carlo (HMC) is a popular method in sampling. While there are quite a few works of studying this method on various aspects, an interesting question is how to choose its integration time to achieve acceleration. In this work, we consider accelerating the process of sampling from a distribution $\pi(x) \propto \exp(-f(x))$ via HMC via time-varying integration time. When the potential $f$ is $L$-smooth and $m$-strongly convex, i.e.\ for sampling from a log-smooth and strongly log-concave target distribution $\pi$, it is known that under a constant integration time, the number of iterations that ideal HMC takes to get an $\epsilon$ Wasserstein-2 distance to the target $\pi$ is $O( \kappa \log \frac{1}{\epsilon} )$, where $\kappa := \frac{L}{m}$ is the condition number. We propose a scheme of time-varying integration time based on the roots of Chebyshev polynomials. We show that in the case of quadratic potential $f$, i.e., when the target $\pi$ is a Gaussian distribution, ideal HMC with this choice of integration time only takes $O( \sqrt{\kappa} \log \frac{1}{\epsilon} )$ number of iterations to reach Wasserstein-2 distance less than $\epsilon$; this improvement on the dependence on condition number is akin to acceleration in optimization. The design and analysis of HMC with the proposed integration time is built on the tools of Chebyshev polynomials. Experiments find the advantage of adopting our scheme of time-varying integration time even for sampling from distributions with smooth strongly convex potentials that are not quadratic.

翻译：哈密顿蒙特卡洛（HMC）是一种流行的采样方法。虽然已有不少工作从不同方面研究了该方法，但一个有趣的问题是如何选择其积分时间以实现加速。在这项工作中，我们考虑通过时变积分时间，利用HMC加速从分布$\pi(x) \propto \exp(-f(x))$中采样的过程。当势函数$f$是$L$-光滑且$m$-强凸时，即对于从对数光滑且强对数凹的目标分布$\pi$中采样，已知在恒定积分时间下，理想HMC达到与目标$\pi$的$\epsilon$ Wasserstein-2距离所需的迭代次数为$O( \kappa \log \frac{1}{\epsilon} )$，其中$\kappa := \frac{L}{m}$是条件数。我们提出了一种基于切比雪夫多项式根的时变积分时间方案。我们证明，在二次势函数$f$的情况下，即当目标$\pi$是高斯分布时，采用这种积分时间选择的理想HMC仅需$O( \sqrt{\kappa} \log \frac{1}{\epsilon} )$次迭代即可使Wasserstein-2距离小于$\epsilon$；这种对条件数依赖性的改进类似于优化中的加速。所提出的积分时间HMC的设计和分析基于切比雪夫多项式工具。实验发现，即使对于从非二次的光滑强凸势函数分布中采样，采用我们的时变积分时间方案也具有优势。