退火欠阻尼朗之万动力学
Annealed Underdamped Langevin Dynamics
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中文总结 AI 辅助
本文提出退火欠阻尼朗之万采样方法，针对对数凹分布，通过辅助过程克服布朗运动退化问题，获得总变差距离下 $\mathcal{O}(\epsilon^{-2})$ 的迭代复杂度，优于退火过阻尼方法的 $\mathcal{O}(\epsilon^{-6})$，并有望推广至非对数凹目标。
中文摘要 AI 辅助
在采样中，不同种类的退火、温度调节或其他逐次逼近方法被广泛使用和研究。在本工作中，我们研究了针对形式为 $\pi(x) \propto e^{-\Psi(x)}$ 的对数凹分布（其中势函数 $\Psi:\mathbb{R}^d \rightarrow \mathbb{R}$）的退火欠阻尼朗之万采样。即，我们假设可以访问一个一般逼近分布族 $(\pi_{\tau})_{\tau\in [0,1]}$ 的得分，其中 $\pi_0=\pi$。该分布族被用于一个时间非齐次的欠阻尼朗之万过程，其目标分布随时间变化为 $\pi_{\tau(t)}$，并采用适当的退火调度 $t\mapsto \tau(t)$，使得当 $t\to T$ 时 $\tau(t)\downarrow 0$，其中 $T>0$。虽然布朗运动的退化性禁止了传统的 Girsanov 论证，但我们通过设计一个适当的辅助过程来规避这一问题。据我们所知，这是使用时间非齐次欠阻尼朗之万动力学进行采样的首个（定量）结果。此外，我们获得了总变差距离下的迭代复杂度为 $\mathcal{O}(\epsilon^{-2})$，而退火过阻尼朗之万采样的复杂度为 $\mathcal{O}(\epsilon^{-6})$。虽然我们目前假设凸性，但我们的证明策略适用于非对数凹目标的推广，并为若干未来研究方向提供了可能。
英文摘要
In sampling, different flavours of annealing, tempering, or other successive approximation approaches are widely used and studied. In this work we investigate annealed underdamped Langevin sampling for logconcave distributions of the form $π(x) \propto e^{-Ψ(x)}$ for a potential $Ψ:\mathbb{R}^d \rightarrow \mathbb{R}$. That is, we assume access to the score of a general approximating family of distributions $(π_τ)_{τ\in [0,1]}$ with $π_0=π$. This family is used in a time-inhomogeneous underdamped Langevin process with moving target $π_{τ(t)}$ and an appropriate annealing schedule $t\mapsto τ(t)$ such that $τ(t)\downarrow 0$ as $t\to T$ for some $T>0$. While the degeneracy of the Brownian motion prohibits conventional Girsanov arguments, we are able to circumvent this issue by relying on an appropriately designed auxiliary process. To the best of our knowledge these are the first (quantitative) results for sampling using time-inhomogeneous underdamped Langevin dynamics. Moreover, we obtain an iteration complexity of $\mathcal{O}(ε^{-2})$ in total variation distance in comparison to $\mathcal{O}(ε^{-6})$ for annealed overdamped Langevin sampling. While we currently assume convexity, our proof strategies are amenable to generalization for non-logconcave targets and enable for several future research directions.
