We study learnability of linear utility functions from pairwise comparison queries. In particular, we consider two learning objectives. The first objective is to predict out-of-sample responses to pairwise comparisons, whereas the second is to approximately recover the true parameters of the utility function. We show that in the passive learning setting, linear utilities are efficiently learnable with respect to the first objective, both when query responses are uncorrupted by noise, and under Tsybakov noise when the distributions are sufficiently "nice". In contrast, we show that utility parameters are not learnable for a large set of data distributions without strong modeling assumptions, even when query responses are noise-free. Next, we proceed to analyze the learning problem in an active learning setting. In this case, we show that even the second objective is efficiently learnable, and present algorithms for both the noise-free and noisy query response settings. Our results thus exhibit a qualitative learnability gap between passive and active learning from pairwise preference queries, demonstrating the value of the ability to select pairwise queries for utility learning.

翻译：我们研究了从成对比较查询中学习线性效用函数的可学习性问题。具体而言，我们考虑两个学习目标：第一个目标是预测样本外对成对比较的响应，而第二个目标是近似恢复效用函数的真实参数。我们证明，在被动学习设置下，线性效用函数对于第一个目标是高效可学习的——无论是在查询响应未被噪声干扰的情况下，还是在噪声服从Tsybakov模型且数据分布足够"良好"的情况下。相比之下，我们表明，在没有强建模假设的情况下，即使查询响应无噪声，对于一大类数据分布而言，效用函数参数也是不可学习的。接下来，我们进一步分析了主动学习设置下的该学习问题。在这种情况下，我们证明即使是第二个目标也是高效可学习的，并针对无噪声和有噪声的查询响应设置提出了相应算法。因此，我们的结果揭示了从成对偏好查询中进行被动学习与主动学习之间存在定性的可学习性差距，这证明了为效用学习选择成对查询的能力具有重要价值。