We consider the problem of ranking $N$ objects starting from a set of noisy pairwise comparisons provided by a crowd of equal workers. We assume that objects are endowed with intrinsic qualities and that the probability with which an object is preferred to another depends only on the difference between the qualities of the two competitors. We propose a class of non-adaptive ranking algorithms that rely on a least-squares optimization criterion for the estimation of qualities. Such algorithms are shown to be asymptotically optimal (i.e., they require $O(\frac{N}{\epsilon ^2}\log \frac{N}{\delta })$ comparisons to be $(\epsilon, \delta)$ -PAC). Numerical results show that our schemes are very efficient also in many non-asymptotic scenarios exhibiting a performance similar to the maximum-likelihood algorithm. Moreover, we show how they can be extended to adaptive schemes and test them on real-world datasets.

中文翻译：

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对一组对象进行排名：一种基于图的最小二乘法

我们考虑排名问题 $ N $对象是从一群平等的工人提供的一组嘈杂的成对比较开始的。我们假设对象具有固有的特质，而一个对象被另一个对象偏爱的概率仅取决于两个竞争者的质量之间的差异。我们提出了一类非自适应排名算法，该算法依赖于最小二乘最优化准则来估计质量。这样的算法被证明是渐近最优的（即，它们要求$ O（\ frac {N} {\ epsilon ^ 2} \ log \ frac {N} {\ delta}）$ 比较是 $（\\ epsilon，\ delta）$ -PAC）。数值结果表明，我们的方案在许多非渐近场景中也表现出了与最大似然算法相似的性能，并且效率很高。此外，我们展示了如何将它们扩展到自适应方案并在现实世界的数据集上对其进行测试。