Forward-KL Convergence of Time-Inhomogeneous Langevin Diffusions

TL;DR

This paper provides a unified non-asymptotic convergence analysis for time-inhomogeneous Langevin diffusions and their discretizations, applicable to various annealing schemes, with theoretical bounds and numerical experiments.

Contribution

It introduces a single set of conditions for analyzing convergence of time-dependent Langevin diffusions and their discretizations, covering many practical annealing methods.

Findings

Non-asymptotic bounds for continuous-time diffusion in forward-KL divergence.

Convergence results for Euler--Maruyama discretization under abstract conditions.

Numerical experiments comparing annealing schemes in different dimensions.

Abstract

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 diffusions 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- and high-dimensional settings.