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 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.

💡 Research Summary

The paper addresses the problem of sampling from a target Gibbs distribution π(x) ∝ exp(−U(x)) in high‑dimensional, multimodal, or nonsmooth settings where direct sampling is infeasible. Classical overdamped Langevin dynamics dXₜ = −∇U(Xₜ)dt + √2 dWₜ converges to π under strong convexity and smoothness, but its performance deteriorates on complex landscapes. To mitigate this, practitioners often introduce an annealing or tempering schedule τ(t) that gradually transforms an easy‑to‑sample intermediate distribution π_{τ(t)} into the target. The resulting time‑inhomogeneous Langevin diffusion is

 dXₜ = −∇U_{τ(t)}(Xₜ)dt + √2 dWₜ, (4)

where τ :