发表机构
The Hong Kong Polytechnic University; University of Kentucky（香港理工大学; 肯塔基大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究布拉德利 - 特里模型中策梅洛型迭代的收敛性，通过推导同步和异步更新下局部收敛因子的闭式表达式及谱分析，发现异步更新时α = 0最优，其加速不仅因参数选择，更因异步更新，数值实验证实理论。
AI 中文摘要
策梅洛算法是计算布拉德利 - 特里（BT）模型中最大似然估计器的经典方法，但实际收敛可能较慢。为加速计算，纽曼引入了一族由α参数化的策梅洛型不动点迭代，α = 1时恢复为策梅洛算法。经验证据表明α = 0通常收敛快得多，但加速机制不明。本文通过系统的局部收敛分析提供理论见解。推导同步和异步更新下局部收敛因子的闭式表达式，通过相关雅可比矩阵的谱分析分析其对α的依赖性。对于同步更新，α < 1时算法可能不收敛，在总体BT模型下局部收敛因子在α中是拟凸的。相比之下，异步更新总是局部收敛的，在一致有序二分比较图的总体BT模型下局部收敛因子在α中单调递增，证明α = 0在此设置下的最优性。还建立了BT模型下总体收敛因子的渐近近似结果，证明其实际相关性。数值实验证实了理论。我们的分析补充了现有收敛结果，表明α = 0的加速不仅源于参数选择，更重要的是源于异步更新的使用。
英文摘要
Zermelo's algorithm is a classical method for computing the maximum likelihood estimator in the Bradley--Terry (BT) model, but its convergence can be slow in practice. To accelerate computation, Newman introduced a family of Zermelo-type fixed-point iterations parameterized by $α$, with Zermelo's algorithm recovered at $α=1$. Empirical evidence suggests that the choice $α=0$ often converges substantially faster, making it a promising alternative, yet the mechanism underlying this acceleration remains elusive. This paper provides theoretical insight into this phenomenon through a systematic local convergence analysis. We derive closed-form expressions for local convergence factors under synchronous and asynchronous updates and analyze their dependence on $α$ via spectral analysis of the associated Jacobian matrices. For synchronous updates, we show that the algorithm may fail to converge when $α<1$, and its local convergence factor is quasi-convex in $α$ under the population BT model. In contrast, asynchronous updates are always locally convergent, and their local convergence factor is provably monotonically increasing in $α$ under the population BT model of consistently ordered bipartite comparison graphs, establishing the optimality of $α=0$ in this setting. We further establish asymptotic approximation results for the population convergence factors under the BT model, justifying their practical relevance. Numerical experiments on synthetic and real-world datasets confirm the theory. Our analysis complements existing convergence results and shows that the acceleration of $α=0$ arises not only from the parameter choice but, more importantly, from the use of asynchronous updates.
Comments46 pages
