Abstract One natural way to express preferences over items is to represent them in the form of pairwise comparisons, from which a model is learned in order to predict further preferences. In this setting, if an item a is preferred to the item b, then it is natural to consider that the preference still holds after multiplying both vectors by a positive scalar (e.g., 2 a ≻ 2 b ). Such invariance to scaling is satisfied in maximum margin learning approaches for pairs of test vectors, but not for the preference input pairs, i.e., scaling the inputs in a different way could result in a different preference relation being learned. In addition to the scaling of preference inputs, maximum margin methods are also sensitive to the way used for normalizing (scaling) the features, which is an essential pre-processing phase for these methods. In this paper, we define and analyse more cautious preference relations that are invariant to the scaling of features, or preference inputs, or both simultaneously; this leads to computational methods for testing dominance with respect to the induced relations, and for generating optimal solutions (i.e., best items) among a set of alternatives. In our experiments, we compare the relations and their associated optimality sets based on their decisiveness, computation time and cardinality of the optimal set.

中文翻译：

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缩放不变的最大边际偏好学习

摘要 表达对项目偏好的一种自然方法是以成​​对比较的形式表示它们，从中学习模型以预测进一步的偏好。在这个设置中，如果一个项目 a 比项目 b 更受欢迎，那么很自然地认为，在将两个向量乘以一个正标量（例如，2 a ≻ 2 b ）后，偏好仍然成立。这种缩放不变性在测试向量对的最大裕度学习方法中得到满足，但对于偏好输入对则不满足，即以不同的方式缩放输入可能导致学习到不同的偏好关系。除了偏好输入的缩放之外，最大边际方法也对用于规范化（缩放）特征的方式敏感，这是这些方法必不可少的预处理阶段。在本文中，我们定义和分析更谨慎的偏好关系，这些关系对特征的缩放或偏好输入或两者同时不变；这导致了用于测试相对于诱导关系的优势的计算方法，以及用于在一组备选方案中生成最佳解决方案（即，最佳项目）的计算方法。在我们的实验中，我们根据关系的决定性、计算时间和最优集的基数来比较关系及其相关的最优集。