On the Time Derivative of the KL Divergence for a Generalized Langevin Annealing Scheme
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908661596553216 |
|---|---|
| author | Habring, Andreas |
| author_facet | Habring, Andreas |
| contents | 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 $\frac{\mathrm{d}}{\mathrm{d} t}\mathrm{KL}(q_t|p_t)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_11956 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Time Derivative of the KL Divergence for a Generalized Langevin Annealing Scheme Habring, Andreas Optimization and Control Probability 65C40, 65C05, 68U10, 65C60 G.3; I.4.5 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 $\frac{\mathrm{d}}{\mathrm{d} t}\mathrm{KL}(q_t|p_t)$. |
| title | On the Time Derivative of the KL Divergence for a Generalized Langevin Annealing Scheme |
| topic | Optimization and Control Probability 65C40, 65C05, 68U10, 65C60 G.3; I.4.5 |
| url | https://arxiv.org/abs/2511.11956 |