We investigate the problem of probably approximately correct and fair (PACF) ranking of items by adaptively evoking pairwise comparisons. Given a set of $n$ items that belong to disjoint groups, our goal is to find an $(\epsilon, \delta)$-PACF-Ranking according to a fair objective function that we propose. We assume access to an oracle, wherein, for each query, the learner can choose a pair of items and receive stochastic winner feedback from the oracle. Our proposed objective function asks to minimize the $\ell_q$ norm of the error of the groups, where the error of a group is the $\ell_p$ norm of the error of all the items within that group, for $p, q \geq 1$. This generalizes the objective function of $\epsilon$-Best-Ranking, proposed by Saha & Gopalan (2019). By adopting our objective function, we gain the flexibility to explore fundamental fairness concepts like equal or proportionate errors within a unified framework. Adjusting parameters $p$ and $q$ allows tailoring to specific fairness preferences. We present both group-blind and group-aware algorithms and analyze their sample complexity. We provide matching lower bounds up to certain logarithmic factors for group-blind algorithms. For a restricted class of group-aware algorithms, we show that we can get reasonable lower bounds. We conduct comprehensive experiments on both real-world and synthetic datasets to complement our theoretical findings.

翻译：我们研究通过自适应地引发成对比较来实现（近似正确且公平）项目排序的问题。给定一组属于不相交群体的 $n$ 个项目，我们的目标是根据我们提出的公平目标函数找到一个 $(\epsilon, \delta)$-PACF-排序。假设可以访问一个预言机，其中学习者可以为每次查询选择一对项目，并从预言机接收随机胜者反馈。我们提出的目标函数要求最小化群体误差的 $\ell_q$ 范数，其中群体的误差是该群体内所有项目误差的 $\ell_p$ 范数，$p, q \geq 1$。这推广了 Saha & Gopalan (2019) 提出的 $\epsilon$-最优排序目标函数。通过采用我们的目标函数，我们能够在一个统一框架内灵活地探索基本公平概念，如相等或成比例的误差。调整参数 $p$ 和 $q$ 可以根据特定的公平偏好进行定制。我们提出了群体盲和群体感知算法，并分析了它们的样本复杂度。我们为群体盲算法提供了匹配的下界（达到某些对数因子）。对于群体感知算法的一个受限类别，我们表明可以获得合理的下界。我们在真实世界和合成数据集上进行了全面的实验，以补充我们的理论发现。