发表机构
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 中文总结
本文通过谱间隙比较随机扫描与确定性扫描在分量式马尔可夫链中的计算成本，证明随机扫描成本至多为确定性的两倍（吉布斯情形），并推广到一般可逆更新，给出反向成本比的阶数界限。
AI 中文摘要
吉布斯采样器，以及更一般的分量式马尔可夫链蒙特卡洛算法（如Metropolis-within-Gibbs），可以通过随机扫描或确定性扫描更新来实现。收敛速度在多大程度上依赖于这种扫描规则的选择一直是一个长期存在的问题。我们通过$L^2$谱间隙来研究这个问题，以分量更新的次数来衡量计算成本。对于$d$块的吉布斯采样器，随机扫描的成本至多是任何固定确定性扫描成本的两倍，而反向成本比至多为$d^2$量级。我们将这些比较扩展到全局分块压缩条件下的通用可逆分量式更新。若$K_j$表示块$j$的更新，$P_j$表示其对应的吉布斯更新，且对于$j=1,\ldots,d$，有$\\|K_j-P_j\\|\leq \lambda_0<1$，则随机扫描的成本至多是确定性扫描成本的$2/(1-\lambda_0)$倍，而反向成本比至多为$d^2/(1-\lambda_0)$量级。例子表明，随机扫描相对于确定性扫描的成本界限是渐近精确的，且反向比较中$d$与$(1-\lambda_0)^{-1}$的联合依赖无法被统一改进。这项工作得到了生成式人工智能的辅助，包括在Lean中对数学结果进行形式化验证。人类作者审查并验证了数学内容，并对结果负全部责任。
英文摘要
Gibbs samplers, and more generally component-wise Markov chain Monte Carlo algorithms such as Metropolis-within-Gibbs, can be implemented using either random-scan or deterministic-scan updates. How much convergence can depend on this choice of scanning rule has been a longstanding question. We study this problem through $L^2$ spectral gaps, measuring computational cost in units of component updates. For a $d$-block Gibbs sampler, the cost of random scan is at most twice that of any fixed deterministic scan, while the reverse cost ratio is at most of order $d^2$. We extend these comparisons to general reversible component-wise updates under the global block-wise contraction condition. If $K_j$ denotes the update of block $j$ and $P_j$ its Gibbs counterpart, and $\|K_j-P_j\|\leq λ_0<1$ for $j=1,\ldots,d$, then the cost of random scan is at most $2/(1-λ_0)$ times that of deterministic scan, while the reverse cost ratio is of order at most $d^2/(1-λ_0)$. Examples show that the cost bounds for random scan relative to deterministic scan are asymptotically sharp and that the joint dependence on $d$ and $(1-λ_0)^{-1}$ in the reverse comparison cannot be improved uniformly. This work was assisted by generative AI, including for formal verification of mathematical results in Lean. The human author reviewed and verified the mathematical content and takes full responsibility for the results.
