Forward-KL Convergence of Time-Inhomogeneous Langevin Diffusions

TL;DR

This work provides a unified, non-asymptotic analysis of time-inhomogeneous Langevin diffusions with moving targets, establishing forward-KL convergence for both the continuous dynamics and their Euler–Maruyama discretizations under a common drift framework. It demonstrates that several practically important annealing paths—such as geometric tempering, dilation, Moreau-envelope-based diffusion, and convolutional paths—fit the abstract assumptions and thus inherit convergence guarantees. The results illuminate the trade-offs between mixing (via large initial $\tau$) and accurate tracking to the target (via rapid decay of $\tau$), and are complemented by numerical experiments on multimodal densities across dimensions. Practically, this provides guidance for designing annealing schedules in high-dimensional sampling and diffusion-model contexts, including Bayesian inverse problems, with bias-free convergence in the vanishing stepsize limit.

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.

Figures (5)

• Figure 1: Forward-Kullback--Leiber divergence between the sample distribution and the one-dimensional Gaussian-mixture target in terms of the iteration for different paths and traversal speeds controlled by the parameter $T$. Direct sample: A sample of the same size as used in the other schemes, directly drawn from the Gaussian mixture.
• Figure 2: Histogram of the sample distributions compared to the one-dimensional Gaussian-mixture target. Rows show different values of the annealing speed controlled by $T$. Columns show the empirical histograms after a different number of steps of the discretization.
• Figure 3: 2D Gaussian mixture. Convergence of different annealing schemes in $\mathop{\mathrm{KL}}\limits$. Different plots show results for different annealing speeds controlled by the parameter $T$. Direct sample refers to the $\mathop{\mathrm{KL}}\limits$ divergence obtained for a sample directly drawn from the Gaussian-mixture model and of the same sample size $N$ as simulated using the annealing schemes.
• Figure 4: 2D Gaussian mixture. Empirical distributions compared to the true density. Rows show different values of the annealing speed controlled by $T$ and different numbers of steps of the discretization. Columns show different paths $(p_\tau)_\tau$.
• Figure 5: 10D Gaussian mixture. Convergence of different annealing schemes in $\mathop{\mathrm{KL}}\limits$. We plot the $\mathop{\mathrm{KL}}\limits$ divergence $\mathop{\mathrm{KL}}\limits(\Pi_{x_i}\hat{\mu}_k|\Pi_{x_i}\pi)$ for $i=1,\dots, 4$ where for $\nu\in\mathcal{P}(\mathbb{R}^d)$, $\Pi_{x_i}\nu$ denotes the $x_t$ marginal of $\nu$. Rows show results for different annealing speeds controlled by the parameter $T$. Columns show the $\mathop{\mathrm{KL}}\limits$ divergence of the marginals $x_i$ for $i=1,\dots, 4$.

Theorems & Definitions (61)

• Definition 1: Log-Sobolev inequality
• Theorem 4.1: Existence of strong solutions of the time-inhomogeneous Langevin diffusion
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 4.2: Forward-KL convergence of the time-inhomogeneous Langevin diffusion
• proof
• Lemma 3