Momentum Particle Maximum Likelihood
This work addresses the challenge of fitting latent variable models to data, offering an incremental improvement over prior particle-based algorithms.
The authors tackled the problem of maximum likelihood estimation in latent variable models by proposing a dynamical-systems-inspired approach that blends Nesterov's Accelerated Gradient, underdamped Langevin diffusion, and particle methods, resulting in an algorithm that outperforms existing particle methods in numerical experiments.
Maximum likelihood estimation (MLE) of latent variable models is often recast as the minimization of a free energy functional over an extended space of parameters and probability distributions. This perspective was recently combined with insights from optimal transport to obtain novel particle-based algorithms for fitting latent variable models to data. Drawing inspiration from prior works which interpret `momentum-enriched' optimization algorithms as discretizations of ordinary differential equations, we propose an analogous dynamical-systems-inspired approach to minimizing the free energy functional. The result is a dynamical system that blends elements of Nesterov's Accelerated Gradient method, the underdamped Langevin diffusion, and particle methods. Under suitable assumptions, we prove that the continuous-time system minimizes the functional. By discretizing the system, we obtain a practical algorithm for MLE in latent variable models. The algorithm outperforms existing particle methods in numerical experiments and compares favourably with other MLE algorithms.