Fenchel 博弈视角下的欠阻尼 Langevin 动力学：加速收敛的洞见

中文摘要 AI 辅助

对于由适当阻尼的欠阻尼 Langevin 动力学控制的 $(Q_t,P_t)$，我们量化了位置边际分布关于 $\sigma$-强对数凹目标 $\pi(dq) = \frac{1}{Z}e^{-V(q)}dq$ 的 KL 散度收敛，得到 \begin{align*} \operatorname{KL}(\operatorname{Law}(Q_t) \\| \pi) \leq e^{-\sqrt{\sigma} t}\operatorname{KL}(\operatorname{Law}(Q_{0}, P_{0})\\, \\|\\, \Pi_0), \end{align*} 其中 $\Pi_0$ 表示适当选取的参考测度。当 $\pi$ 仅为对数凹时，推导出估计 \begin{align*} \operatorname{KL}(\operatorname{Law}(Q_t) \\| \pi) \leq \frac{\tau^2}{t^2}\operatorname{KL}(\operatorname{Law}(Q_{\tau}, P_{\tau})\\, \\|\\, \Pi_{\tau}) \end{align*}，其中 $\Pi_\tau$ 表示在时间 $\tau > 0$ 处相应选取的另一参考测度。两种速率均与 $\mathbb{R}^d$ 中相应加速梯度流的经典速率精确匹配。这些结果通过将欠阻尼 Langevin 动力学新颖地解释为在线采样博弈中的策略组合，并利用由虚构竞争者信息构建的成本函数估计 KL 散度而获得。

英文摘要

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.

发表机构

• King’s College London（伦敦国王学院）

机构由 AI 辅助整理，请以论文原文为准。