Year: 2026
Type: preprint
Abstract: For $(Q_t,P_t)$ governed by suitably damped underdamped Langevin dynamics, we quantify the convergence in KL divergence of the positional marginal to a $σ$-strongly log-concave target $π(dq) = \frac{1}{Z}e^{-V(q)}dq$ as \begin{align*} \operatorname{KL}(\operatorname{Law}(Q_t) \| π) \leq e^{-\sqrtσ t}\operatorname{KL}(\operatorname{Law}(Q_{0}, P_{0})\, \|\, Π_0), \end{align*} where $Π_0$ denotes an appropriately selected reference measure. When $π$ is merely log-concave, the estimate \begin{align*} \operatorname{KL}(\operatorname{Law}(Q_t) \| π) \leq \frac{τ^2}{t^2}\operatorname{KL}(\operatorname{Law}(Q_τ, P_τ)\, \|\, Π_τ) \end{align*} is derived, where $Π_τ$ denotes another correspondingly chosen reference measure at time $τ> 0$. Both rates match precisely the canonical rates of the corresponding accelerated gradient flows in $\mathbb{R}^d$. They are achieved by the novel interpretation of the underdamped Langevin dynamics as a combination of strategies in an online sampling game and by estimating the KL divergence using a cost function informed by fictitious competitors.
Source: arXiv (Cornell University)
Authors: Borkowski, Alexandra, Nüsken, Nikolas
Institution: King's College London
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212017251