引入外面体实现重复排序下有效Pareto最优公平效用摊销
Introducing the Expohedron for Efficient Pareto-optimal Fairness-Utility Amortizations in Repeated Rankings
摘要：
我们考虑的问题，计算一系列的排名，最大限度地提高消费者的效用，同时最大限度地减少生产者的个人不公平的曝光。虽然以前的工作已经解决了这个问题，使用线性或二次规划的双随机矩阵，这种方法，依赖于伯克霍夫-冯诺依曼（BvN）分解，太慢，在大规模实施。在本文中，我们介绍了一个几何对象，一个多面体，我们称之为多面体，其点表示所有可实现的曝光项目的位置为基础的模型（PBM）。我们展示了它的一些属性，并制定了一个Carathéodory分解算法的复杂性$O(n^2\log(n))$能够表示任何点内的多面体作为一个凸和最多$n$顶点，其中$n$是项目的数量排名。这样的分解使得可以将任何可行的目标暴露表示为至多$n$等级上的分布。此外，我们表明，我们可以使用这个多面体恢复多目标公平效用优化问题的整个帕累托前沿，使用一个简单的几何过程与复杂性$O(n^2\log(n))$。我们的方法比较有利的线性或二次规划基线的算法复杂性和经验运行时间，适用于任何优点，是一个非递减函数的项目相关性。此外，我们的解决方案可以表示为一个分布，只有$n$排列，而不是$(n-1)^2 + 1$实现BvN分解。我们在合成和真实世界的数据集上进行实验，证实了我们的理论结果。
英文摘要：
We consider the problem of computing a sequence of rankings that maximizes consumer-side utility while minimizing producer-side individual unfairness of exposure. While prior work has addressed this problem using linear or quadratic programs on bistochastic matrices, such approaches, relying on Birkhoff-von Neumann (BvN) decompositions, are too slow to be implemented at large scale. In this paper we introduce a geometrical object, a polytope that we call expohedron, whose points represent all achievable exposures of items for a Position Based Model (PBM). We exhibit some of its properties and lay out a Carathéodory decomposition algorithm with complexity $O(n^2\log(n))$ able to express any point inside the expohedron as a convex sum of at most $n$ vertices, where $n$ is the number of items to rank. Such a decomposition makes it possible to express any feasible target exposure as a distribution over at most $n$ rankings. Furthermore we show that we can use this polytope to recover the whole Pareto frontier of the multi-objective fairness-utility optimization problem, using a simple geometrical procedure with complexity $O(n^2\log(n))$. Our approach compares favorably to linear or quadratic programming baselines in terms of algorithmic complexity and empirical runtime and is applicable to any merit that is a non-decreasing function of item relevance. Furthermore our solution can be expressed as a distribution over only $n$ permutations, instead of the $(n-1)^2 + 1$ achieved with BvN decompositions. We perform experiments on synthetic and real-world datasets, confirming our theoretical results.
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