This paper provides a convergence analysis for generalized Hamiltonian Monte Carlo samplers, a family of Markov Chain Monte Carlo methods based on leapfrog integration of Hamiltonian dynamics and kinetic Langevin diffusion, that encompasses the unadjusted Hamiltonian Monte Carlo method. Assuming that the target distribution $\pi$ satisfies a log-Sobolev inequality and mild conditions on the corresponding potential function, we establish quantitative bounds on the relative entropy of the iterates defined by the algorithm, with respect to $\pi$. Our approach is based on a perturbative and discrete version of the modified entropy method developed to establish hypocoercivity for the continuous-time kinetic Langevin process. As a corollary of our main result, we are able to derive complexity bounds for the class of algorithms at hand. In particular, we show that the total number of iterations to achieve a target accuracy $\varepsilon >0$ is of order $d/\varepsilon^{1/4}$, where $d$ is the dimension of the problem. This result can be further improved in the case of weakly interacting mean field potentials, for which we find a total number of iterations of order $(d/\varepsilon)^{1/4}$.

翻译：本文针对广义哈密顿蒙特卡洛采样器（基于哈密顿动力学蛙跳积分与动力学朗之万扩散的一类马尔可夫链蒙特卡洛方法，包含未调整的哈密顿蒙特卡洛方法）提供了收敛性分析。在假设目标分布$\pi$满足对数索博列夫不等式且对应势函数满足温和条件的前提下，我们建立了算法迭代序列相对于$\pi$的相对熵的定量界限。我们的方法基于改进熵的扰动离散形式，该形式源自为连续时间动力学朗之万过程建立弱耗散性（hypocoercivity）而发展的技术。作为主要结果的推论，我们推导了该类算法的复杂度界值。特别地，我们证明达到目标精度$\varepsilon >0$所需的总迭代次数为$d/\varepsilon^{1/4}$量级，其中$d$为问题维度。在弱相互作用平均场势函数情形下，该结果可进一步优化为$(d/\varepsilon)^{1/4}$量级的总迭代次数。