Markov Chain Monte Carlo for Gaussian: A Linear Control Perspective

Drawing samples from a given target probability distribution is a fundamental task in many science and engineering applications. A commonly used method for sampling is the Markov chain Monte Carlo (MCMC) which simulates a Markov chain whose stationary distribution coincides with the target one. In this letter, we study the convergence and complexity of MCMC algorithms from a dynamic system point of view. We focus on the special cases with Gaussian target distributions and provide a Lyapunov perspective to them using tools from linear control theory. In particular, we systematically analyze two popular MCMC algorithms: Langevin Monte Carlo (LMC) and kinetic Langevin Monte Carlo (KLMC). By applying Lyapunov theory we derive impressive complexity bounds to these algorithms: for LMC, our result is better than all existing results, and for KLMC, ours matches the best known bound. Our analysis also highlights subtle differences between sampling and optimization that could inform the more challenging task to sample from general distributions. Overall, our findings offer valuable insights for improving MCMC algorithms.