[论文解读] Rank Centrality: Ranking from Pair-wise Comparisons

Sahand Negahban, Sewoong Oh+1 more|arXiv (Cornell University)|Sep 8, 2012

Game Theory and Voting Systems参考文献 48被引用 18

一句话总结

该论文提出 Rank Centrality，一种计算高效的算法，通过在比较图上的随机游走来推断分数和排序，其中分数对应于平稳概率。当比较图具有正的谱间隙时，该方法在 Bradley-Terry-Luce (BTL) 模型下实现了近乎最优的样本复杂度，与最大似然估计的性能相当。

ABSTRACT

The question of aggregating pair-wise comparisons to obtain a global ranking over a collection of objects has been of interest for a very long time: be it ranking of online gamers (e.g. MSR's TrueSkill system) and chess players, aggregating social opinions, or deciding which product to sell based on transactions. In most settings, in addition to obtaining a ranking, finding `scores' for each object (e.g. player's rating) is of interest for understanding the intensity of the preferences. In this paper, we propose Rank Centrality, an iterative rank aggregation algorithm for discovering scores for objects (or items) from pair-wise comparisons. The algorithm has a natural random walk interpretation over the graph of objects with an edge present between a pair of objects if they are compared; the score, which we call Rank Centrality, of an object turns out to be its stationary probability under this random walk. To study the efficacy of the algorithm, we consider the popular Bradley-Terry-Luce (BTL) model (equivalent to the Multinomial Logit (MNL) for pair-wise comparisons) in which each object has an associated score which determines the probabilistic outcomes of pair-wise comparisons between objects. In terms of the pair-wise marginal probabilities, which is the main subject of this paper, the MNL model and the BTL model are identical. We bound the finite sample error rates between the scores assumed by the BTL model and those estimated by our algorithm. In particular, the number of samples required to learn the score well with high probability depends on the structure of the comparison graph. When the Laplacian of the comparison graph has a strictly positive spectral gap, e.g. each item is compared to a subset of randomly chosen items, this leads to dependence on the number of samples that is nearly order-optimal.

研究动机与目标

• 开发一种计算简单、数据驱动的算法，用于从成对比较数据中推断项目排序和分数。
• 确保当数据符合 Bradley-Terry-Luce (BTL) 模型时，该算法的性能与模型感知估计器相当。
• 分析在结构化比较图下，真实 BTL 分数与估计分数之间的有限样本误差率。
• 建立依赖于比较图谱性质的样本复杂度理论保证。

提出的方法

• 该算法将成对比较建模为图上的随机游走，其中节点代表项目，边代表比较。
• 项目分数被定义为该随机游走的平稳分布，由图的转移矩阵导出。
• 该方法利用图拉普拉斯矩阵的谱间隙来界定估计误差，确保在弱条件下收敛。
• 它使用集中不等式和矩阵偏差界（例如 Tropp 定理）来分析在随机图模型下比较矩阵的行为。
• 理论分析依赖于通过度数归一化将比较矩阵转换为对称形式，以应用谱理论。
• 该算法被证明对图稀疏性具有鲁棒性，在比较数据存在噪声或不完整时仍表现良好。

实验结果

研究问题

• RQ1 一个简单、迭代的算法能否在 BTL 模型下实现与最大似然估计相当的估计精度？
• RQ2 比较图的结构——特别是其谱间隙——如何影响排序估计的样本复杂度？
• RQ3 真实 BTL 分数与 Rank Centrality 估计分数之间的有限样本误差率是多少？
• RQ4 当比较稀疏或随机采样时，该算法是否仍能保持优异性能？
• RQ5 在准确性和收敛速度方面，Rank Centrality 与其它排名算法相比如何？

主要发现

• 当比较图具有严格正的谱间隙时，Rank Centrality 实现了近乎阶最优的有限样本误差率。
• 为将分数估计在给定误差界内所需的比较次数与谱间隙的倒数成比例，图越密集或连接性越好，性能越优。
• 在平均度数 $ d \geq C\log n $ 的 Erdős–Rényi 随机图模型下，该算法的误差界为 $ O\left(\sqrt{\frac{\log n}{d}}\right) $。
• 在合成 BTL 生成数据上的实证评估表明，Rank Centrality 在分数估计精度方面与最大似然估计器表现相当。
• 在收敛速度和对稀疏比较数据的鲁棒性方面，该算法优于其他流行的排名方法。
• 理论界是通过矩阵集中不等式（包括 Bernstein 不等式和 Tropp 的矩阵 Chernoff 界）推导得出，用于控制比较矩阵中的偏差。