On the Time Derivative of the KL Divergence for a Generalized Langevin Annealing Scheme

TL;DR

This paper rigorously derives the time derivative of the KL divergence between the evolving distribution of a Langevin diffusion process and a time-dependent target distribution, clarifying conditions for its existence.

Contribution

It provides a rigorous derivation of the time derivative of the KL divergence for a generalized Langevin annealing scheme, addressing gaps in previous analyses.

Findings

Derived the explicit formula for the time derivative of KL divergence.

Clarified conditions under which the derivative exists.

Enhanced understanding of convergence analysis in Langevin-based methods.

Abstract

Consider the Langevin diffusion process $\mathrm{d} X_t = \nabla \log p_t(X_t) + \sqrt{2}\mathrm{d} W_t$ guided by the time-dependent probability density $p_t(x)$. Let $q_t$ be the density of $X_t$. Recently, in order to analyze convergence in the Kullback-Leibler divergence, the time derivative of $t\mapsto \mathrm{KL}(q_t|p_t)$ has been used in several works without investigating in detail when such a derivative exists. In this short manuscript we provide a rigorous derivation of the quantity $\frac{\mathrm{d}}{\mathrm{d} t}\mathrm{KL}(q_t|p_t)$.