Forward-KL Convergence of Time-Inhomogeneous Langevin Diffusions

Many practical samplers rely on time-dependent drifts---often induced by annealing or tempering schedules---to improve exploration and stability. This motivates a unified non-asymptotic analysis of the corresponding Langevin diffusion and their discretizations. We provide a convergence analysis that includes non-asymptotic bounds for the continuous-time diffusion and its Euler--Maruyama discretization in the forward-Kullback--Leibler divergence under a single set of abstract conditions on the time-dependent drift. The results apply to many practically-relevant annealing schemes, including geometric tempering and annealed Langevin sampling. In addition, we provide numerical experiments comparing the annealing schemes covered by our theory in low- as well as high-dimensional settings.

In modern machine learning and related fields drawing random samples from complex, high-dimensional distributions is a frequently occurring task. In this article, we investigate methods for drawing such samples, in which the samples follow first a very simple distribution and are successively guided towards the complicated actual target, commonly referred to as annealing. In particular, we provide a mathematical analysis of such approaches as well as numerical experiments. With this paper, we theoretically unify approaches that have been used for some time in the literature.