This paper introduces and analyses interacting underdamped Langevin algorithms, termed Kinetic Interacting Particle Langevin Monte Carlo (KIPLMC) methods, for statistical inference in latent variable models. We propose a diffusion process that evolves jointly in the space of parameters and latent variables and show that the stationary distribution of this diffusion concentrates around the maximum marginal likelihood estimate of the parameters. We then provide two explicit discretisations of this diffusion as practical algorithms to estimate parameters of statistical models. For each algorithm, we obtain nonasymptotic rates of convergence in Wasserstein-2 distance for the case where the joint log-likelihood is strongly concave with respect to latent variables and parameters. We achieve accelerated convergence rates clearly demonstrating improvement in dimension dependence. To demonstrate the utility of the introduced methodology, we provide numerical experiments that illustrate the effectiveness of the proposed diffusion for statistical inference. Our setting covers a broad number of applications, including unsupervised learning, statistical inference, and inverse problems.

翻译：本文提出并分析了交互式欠阻尼朗之万算法，称为动力学相互作用粒子朗之万蒙特卡洛（KIPLMC）方法，用于潜变量模型中的统计推断。我们提出一种在参数与潜变量空间中联合演化的扩散过程，并证明该扩散的平稳分布集中于参数的边际最大似然估计附近。随后，我们给出该扩散过程的两种显式离散化方案，作为估计统计模型参数的实用算法。对于每个算法，我们推导了在联合对数似然关于潜变量和参数强凹情况下的瓦瑟斯坦-2距离非渐近收敛速率，并实现了加速收敛速度，清晰展示了维度依赖性的改善。为验证所提方法的实用性，我们通过数值实验展示了该扩散过程在统计推断中的有效性。本文研究框架涵盖无监督学习、统计推断及反问题等广泛应用场景。