Introducing the Expohedron for Efficient Pareto-optimal Fairness-Utility Amortizations in Repeated Rankings

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(n2log(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(n2log(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.