未调整凸光滑情形下统一的HMC与欠阻尼朗之万方法
HMC and underdamped Langevin united in the unadjusted convex smooth case
AI总结：
该研究提出一类未调整的广义HMC采样器，统一了位置HMC与欠阻尼朗之万方法，通过部分速度刷新提升收敛速率，还建立了其随机梯度版本的无维度收敛率，拓展了相关理论结果。
摘要：
我们研究一类未调整的广义HMC（哈密顿蒙特卡洛）采样器，其涵盖标准位置HMC采样器以及欠阻尼朗之万过程的离散化形式。我们在高斯情形下开展了详细的参数分析与优化，结果表明，相较于经典的全刷新策略，采用部分速度刷新时，收敛速率关于条件数κ的依赖从1/κ提升至1/√κ。我们在两种相关算法——Metropolis调整的gHMC（广义哈密顿蒙特卡洛）和动力学分段确定性马尔可夫过程——上通过实验观测到了类似效应。随后我们研究了该类采样器的随机梯度版本，在大参数范围内针对对数凹光滑目标建立了无维度收敛速率，将位置HMC与欠阻尼朗之万的已有结果纳入统一框架，并将其推广至带惯性的HMC。
英文摘要：
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 $1/κ$ to $1/\sqrtκ$ for the convergence rate in terms of the condition number $κ$ 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.
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