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 Carath\'eodory 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.
翻译:我们考虑的是计算一系列排名的问题,这种排序将最大限度地提高消费方效用,同时最大限度地减少生产者方个人接触的不公平性。虽然先前的工作已经利用对饼干基质的线性或二次程序来解决这个问题,但这种方法依赖Birkhoff-von Neumann(BvN)分解,过于缓慢,无法大规模实施。在本文中,我们引入了一个几何对象,一个我们称之为Expohedron的多功能,它点代表着基于定位模型(PBM)所有可实现的商品的可实现的蛋白暴露。我们展示了它的一些属性,并展示了具有复杂性的 Carath\'odoory decomposition 算法,它展示了Carath\'emology decompositional revality $(n_2\logy),它能够以最多美元值的量的量子值表示出任何点数。 美元值的量值,它只能表示任何可能的目标暴露为最多值的分布值。此外,我们展示了我们使用这种可应用的直径直径直径的比值的值的值的值, 直径直径直径直径的值的值的值的值的值值值值值值值值值的值值值值值值值值值值值的值值值值值值值的值值值值值值值值值值值值的值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值值比值比值值值值值值值值,而不是比值是整个的比值是整个的比值, 。