Probabilistic Permutation Graph Search: Black-Box Optimization for Fairness in Ranking

There are several measures for fairness in ranking, based on different underlying assumptions and perspectives. \acPL optimization with the REINFORCE algorithm can be used for optimizing black-box objective functions over permutations. In particular, it can be used for optimizing fairness measures. However, though effective for queries with a moderate number of repeating sessions, \acPL optimization has room for improvement for queries with a small number of repeating sessions. In this paper, we present a novel way of representing permutation distributions, based on the notion of permutation graphs. Similar to~\acPL, our distribution representation, called~\acPPG, can be used for black-box optimization of fairness. Different from~\acPL, where pointwise logits are used as the distribution parameters, in~\acPPG pairwise inversion probabilities together with a reference permutation construct the distribution. As such, the reference permutation can be set to the best sampled permutation regarding the objective function, making~\acPPG suitable for both deterministic and stochastic rankings. Our experiments show that~\acPPG, while comparable to~\acPL for larger session repetitions (i.e., stochastic ranking), improves over~\acPL for optimizing fairness metrics for queries with one session (i.e., deterministic ranking). Additionally, when accurate utility estimations are available, e.g., in tabular models, the performance of \acPPG in fairness optimization is significantly boosted compared to lower quality utility estimations from a learning to rank model, leading to a large performance gap with PL. Finally, the pairwise probabilities make it possible to impose pairwise constraints such as "item $d_1$ should always be ranked higher than item $d_2$.'' Such constraints can be used to simultaneously optimize the fairness metric and control another objective such as ranking performance.