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.

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.