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：我们考虑一族未调整的广义HMC采样器，其包括标准位置HMC采样器以及欠阻尼朗之万过程的离散化。在高斯情形下，我们对参数进行了详细分析与优化，结果表明：相对于经典的完全速度刷新，采用部分速度刷新可将条件数$\kappa$下的收敛速率从$1/\kappa$提升至$1/\sqrt{\kappa}$。对于两种相关算法（即Metropolis调整的gHMC与动力学分段确定性马尔可夫过程），在经验上也观察到类似效果。随后，我们研究了采样器的随机梯度版本，并针对对数凹光滑目标在大范围参数下建立了无维度收敛速率。这统一了位置HMC与欠阻尼朗之万的现有结果，并将其推广至具有惯性的HMC。