We consider a family of unadjusted generalized HMC samplers, which includes standard position HMC samplers and discretizations of the underdamped Langevin process. A detailed analysis and optimization of the parameters is conducted in the Gaussian case, which shows an improvement from $1/\kappa$ to $1/\sqrt{\kappa}$ for the convergence rate in terms of the condition number $\kappa$ by using partial velocity refreshment, with respect to classical full refreshments. A similar effect is observed empirically for two related algorithms, namely Metropolis-adjusted gHMC and kinetic piecewise-deterministic Markov processes. Then, a stochastic gradient version of the samplers is considered, for which dimension-free convergence rates are established for log-concave smooth targets over a large range of parameters, gathering in a unified framework previous results on position HMC and underdamped Langevin and extending them to HMC with inertia.

翻译：我们研究了一类非调整广义哈密顿蒙特卡洛采样器，其包含标准位置哈密顿蒙特卡洛采样器及欠阻尼朗之万过程的离散化形式。在高斯情形下对参数进行了详细分析与优化，结果表明：相较于传统的完全速度重置，采用部分速度重置可使收敛速率在条件数 $\kappa$ 的意义下从 $1/\kappa$ 提升至 $1/\sqrt{\kappa}$。对于两种相关算法——Metropolis调整广义哈密顿蒙特卡洛与动能分段确定性马尔可夫过程——实证研究中也观测到类似效应。随后，我们考察了该采样器的随机梯度版本，在对数凹光滑目标函数的大范围参数设定下，建立了与维度无关的收敛速率。此项工作将先前关于位置哈密顿蒙特卡洛与欠阻尼朗之万的研究结果纳入统一框架，并将其推广至含惯性的哈密顿蒙特卡洛方法。