高效质量矩阵估计：基于高斯冷却的方法
Efficient Mass Matrix Estimation with Gaussian Cooling
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- UC Berkeley（加州大学伯克利分校）
- Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
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中文总结 AI 辅助
提出基于高斯冷却的多阶段预处理算法，用于高效估计对数凹分布的质量矩阵，复杂度随条件数对数增长，并在一阶拒绝采样下给出查询复杂度上界及格点φ^4模型数值验证。
中文摘要 AI 辅助
我们提出了一种高效预处理对数凹和对数光滑分布的算法，其复杂度随底层分布的条件数呈对数增长。基于高斯冷却，我们的多阶段方法近似地从一系列良态分布中采样，以构建冷却调度中每个后续阶段的预处理器。该方法受马尔可夫链蒙特卡洛（MCMC）中现有质量矩阵估计实践方法的启发，即在预热期间从产生的样本中估计目标分布的有效预处理器。然而，我们的方法在概念上不同于现有方法，并且允许进行理论分析。我们的算法框架灵活，允许几乎任何对数凹采样算法作为每个冷却阶段内的子程序。例如，使用一阶拒绝采样（FORS）作为采样器，对于一大类条件数为κ的对数凹和对数光滑分布，我们方法的一阶查询复杂度为O~(d^{4/3} log κ)，对于平移不变分布则为O~(d^{1/3} log κ)。在平移不变设置中，我们为格点φ^4模型（一种简单的格点场论）提供了数值实验。
英文摘要
We introduce an algorithm for efficiently preconditioning log-concave and log-smooth distributions that scales logarithmically with the condition number of the underlying distribution. Based on Gaussian cooling, our multistage method approximately samples from a sequence of well-conditioned distributions to construct a preconditioner for each subsequent stage of the cooling schedule. This method is motivated by existing practical approaches for mass matrix estimation in Markov chain Monte Carlo (MCMC), in which an effective preconditioner for the target distribution is estimated from samples produced during a warm-up period. However, our approach is conceptually distinct from existing approaches and moreover permits theoretical analysis. Our algorithmic framework is flexible, allowing essentially any log-concave sampling algorithm to act as the subroutine within each cooling stage. For instance, using first-order rejection sampling (FORS) as the sampler, the first-order query complexity of our method is $\tilde{O}(d^{4/3} \log κ)$ for a large class of log-concave and log-smooth distributions with condition number $κ$ and $\tilde{O}(d^{1/3} \log κ)$ for translation-invariant distributions. In the translation-invariant setting, we provide numerical experiments for the lattice $ϕ^4$ model, a simple lattice field theory.
