具有均匀随机步长的Metropolis调整Langevin算法的全局谱间隙
A spectral gap for Metropolis-adjusted Langevin algorithm with a uniformly randomized step size
- School of Statistics University of Minnesota（明尼苏达大学统计学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对强凸目标分布，提出均匀随机步长的MALA算法，证明其谱间隙下界达到最优阶$(\kappa\sqrt{d})^{-1}$（含对数因子），通过新的Cheeger型不等式实现。
AI中文摘要：
设$\pi(\mathrm{d} x)\propto e^{-U(x)}\\,\mathrm{d} x$在$\mathbb R^d$上，其中$0<m\leq L<\infty$，$mI_d\preceq\nabla^2U(x)\preceq LI_d$，且$\kappa=L/m$。已知在热启动假设下，具有适当调整步长的固定步长Metropolis调整Langevin算法（MALA）的混合时间阶为$\kappa \sqrt{d}$（忽略对数因子）。相比之下，当条件数远离1时，不存在任何单一固定步长能在此目标类上一致地给出阶为$(\kappa \sqrt{d})^{-1}$的匹配谱间隙下界。我们证明具有均匀随机步长的MALA admits这种大小的谱间隙下界。在每次迭代中，这里考虑的随机步长MALA从$(0,H)$中均匀抽取$h$，并以步长$h$执行一次普通MALA转移。我们证明，当$H$的阶为$(L\sqrt{d})^{-1}$时，随机步长MALA的右谱间隙具有阶为\\[ \frac{1}{\kappa\sqrt{d}\\,[1+\log(d+1)+\log\kappa]} \\]的下界。证明中的主要新成分是一个Cheeger型不等式，用于聚合在不同步长尺度下MALA从可测集出发的单步流的估计。它允许用于控制流的尺度依赖于集合，并避免了先对流求和再应用标准Cheeger不等式所导致的额外损失。这项工作在ChatGPT的大量协助下完成，ChatGPT提出了均匀随机步长方法，发展了主要证明论证，并生成了模拟和Lean 4代码。人类作者检查并验证了数学内容，并对结果承担全部责任。
英文摘要：
Let $π(\mathrm{d} x)\propto e^{-U(x)}\, \mathrm{d} x$ on $\mathbb{R}^d$, where $U$ is continuously differentiable and $m$-strongly convex with a globally $L$-Lipschitz gradient, $0<m\leq L<\infty$, and $κ=L/m$. Fixed-step Metropolis-adjusted Langevin algorithm (MALA) has known warm-start mixing-time upper bounds of order $κ\sqrt d$, up to logarithmic factors. A spectral-gap lower bound at the corresponding scale $(κ\sqrt{d})^{-1}$ would give an upper bound of order $κ\sqrt{d}$ on the Monte Carlo asymptotic variance relative to independent sampling, uniformly over all square-integrable functions. However, when $κ$ is bounded away from~1, the best fixed-step spectral gap that can be guaranteed uniformly over this target class is at most of order $\max\{\log(κd)/(κd), e^{-cd}\}$ for some positive universal constant~$c$. We show that uniform randomization of the step size improves this worst-case guarantee. At each iteration, the algorithm draws $h\sim\operatorname{Unif}(0,H)$ and performs one ordinary MALA transition. Choosing $H$ of order $[L\sqrt{d(1+\log d+\logκ)}]^{-1}$ yields a right spectral-gap lower bound of order \[ \frac{1}{κ\sqrt{d(1+\log d+\logκ)}}, \] uniformly over the target class. Thus, given $κ> 1$, for all square-integrable functions, the ratio of the Monte Carlo asymptotic variance relative to independence sampling has an upper bound of order $\sqrt{d \log d}$. This contrasts with fixed-MALA, where the ratio can be as bad as $d/\log d$ in thew worst-case scenario. The spectral gap also gives geometric convergence of the lazy kernel from every initial density in $L^2(π)$, central limit theorems, and nonstationary mean-square error bounds. This work was developed with substantial assistance from ChatGPT.
