A Multiscale Perspective on Maximum Marginal Likelihood Estimation

In this paper, we provide a multiscale perspective on the problem of maximum marginal likelihood estimation. We consider and analyse a diffusion-based maximum marginal likelihood estimation scheme using ideas from multiscale dynamics. Our perspective is based on stochastic averaging; we make an explicit connection between ideas in applied probability and parameter inference in computational statistics. In particular, we consider a general class of coupled Langevin diffusions for joint inference of latent variables and parameters in statistical models, where the latent variables are sampled from a fast Langevin process (which acts as a sampler), and the parameters are updated using a slow Langevin process (which acts as an optimiser). We show that the resulting system of stochastic differential equations (SDEs) can be viewed as a two-time scale system. To demonstrate the utility of such a perspective, we show that the averaged parameter dynamics obtained in the limit of scale separation can be used to estimate the optimal parameter, within the strongly convex setting. We do this by using recent uniform-in-time non-asymptotic averaging bounds. Finally, we conclude by showing that the slow-fast algorithm we consider here, termed Slow-Fast Langevin Algorithm, performs on par with state-of-the-art methods on a variety of examples. We believe that the stochastic averaging approach we provide in this paper enables us to look at these algorithms from a fresh angle, as well as unlocking the path to develop and analyse new methods using well-established averaging principles.