We consider the problem of ranking a set of items from pairwise comparisons in the presence of features associated with the items. Recent works have established that $O(n\log(n))$ samples are needed to rank well when there is no feature information present. However, this might be sub-optimal in the presence of associated features. We introduce a new probabilistic preference model called feature-Bradley-Terry-Luce (f-BTL) model that generalizes the standard BTL model to incorporate feature information. We present a new least squares based algorithm called fBTL-LS which we show requires much lesser than $O(n\log(n))$ pairs to obtain a good ranking -- precisely our new sample complexity bound is of $O(\alpha\log \alpha)$, where $\alpha$ denotes the number of `independent items' of the set, in general $\alpha << n$. Our analysis is novel and makes use of tools from classical graph matching theory to provide tighter bounds that sheds light on the true complexity of the ranking problem, capturing the item dependencies in terms of their feature representations. This was not possible with earlier matrix completion based tools used for this problem. We also prove an information theoretic lower bound on the required sample complexity for recovering the underlying ranking, which essentially shows the tightness of our proposed algorithms. The efficacy of our proposed algorithms are validated through extensive experimental evaluations on a variety of synthetic and real world datasets.
翻译:我们考虑在与项目相关的特征存在的情况下,从对称比较对一组物品进行排序的问题。 最近的工作已经确定, 美元( n\log( n) ) 样本在缺少特征信息时, 需要以美元( n) 来进行正确排序。 但是, 在存在相关特征的情况下, 这可能是次最佳的。 我们引入了一个新的概率偏好模式, 称为特效- 布拉德利- Terry-Luce( f- BTL), 将标准 BTL 模式概括为包含特征信息。 我们的分析是新颖的, 并使用经典图表匹配理论的工具, 以提供更紧密的条框, 这比 美元( n) 夫妇要低得多。 我们的新样本复杂性被绑定为$( alpha/ alpha) 美元, 美元表示集中“ 独立项目” 的数量, 并且我们使用这个基础的排序分析工具 。 我们使用了一个基础的排序分析工具, 基础的排序分析系统, 基础的排序工具 显示, 我们的排序分析工具 基础是 。