This technical report studies the problem of ranking from pairwise comparisons in the classical Bradley-Terry-Luce (BTL) model, with a focus on score estimation. For general graphs, we show that, with sufficiently many samples, maximum likelihood estimation (MLE) achieves an entrywise estimation error matching the Cram\'er-Rao lower bound, which can be stated in terms of effective resistances; the key to our analysis is a connection between statistical estimation and iterative optimization by preconditioned gradient descent. We are also particularly interested in graphs with locality, where only nearby items can be connected by edges; our analysis identifies conditions under which locality does not hurt, i.e. comparing the scores between a pair of items that are far apart in the graph is nearly as easy as comparing a pair of nearby items. We further explore divide-and-conquer algorithms that can provably achieve similar guarantees even in the regime with the sparsest samples, while enjoying certain computational advantages. Numerical results validate our theory and confirm the efficacy of the proposed algorithms.

翻译：本技术报告研究经典Bradley-Terry-Luce（BTL）模型中基于成对比较的排序问题，重点关注分数估计。对于一般图，我们证明在样本量充足时，最大似然估计（MLE）能实现与Cramér-Rao下界相匹配的逐项估计误差，该下界可用有效电阻表述；分析的关键在于建立统计估计与预处理梯度下降迭代优化之间的联系。我们尤其关注具有局部性的图结构（仅邻近项目可通过边连接），分析指明了局部性不损害比较性能的条件——即图中相距较远的项目对之间的分数比较几乎与邻近项目对同样容易。我们进一步探索了分治算法，该算法即便在样本最稀疏的机制下也能证明地实现类似保障，同时具备计算优势。数值结果验证了理论分析并确认了所提算法的有效性。