  Many re-ranking strategies in search systems rely on stochastic ranking
policies, encoded as Doubly-Stochastic (DS) matrices, that satisfy desired
ranking constraints in expectation, e.g., Fairness of Exposure (FOE). These
strategies are generally two-stage pipelines: \emph{i)} an offline re-ranking
policy construction step and \emph{ii)} an online sampling of rankings step.
Building a re-ranking policy requires repeatedly solving a constrained
optimization problem, one for each issued query. Thus, it is necessary to
recompute the optimization procedure for any new/unseen query. Regarding
sampling, the Birkhoff-von-Neumann decomposition (BvND) is the favored approach
to draw rankings from any DS-based policy. However, the BvND is too costly to
compute online. Hence, the BvND as a sampling solution is memory-consuming as
it can grow as $\gO(N\, n^2)$ for $N$ queries and $n$ documents.
  This paper offers a novel, fast, lightweight way to predict fair stochastic
re-ranking policies: Constrained Meta-Optimal Transport (CoMOT). This method
fits a neural network shared across queries like a learning-to-rank system. We
also introduce Gumbel-Matching Sampling (GumMS), an online sampling approach
from DS-based policies. Our proposed pipeline, CoMOT + GumMS, only needs to
store the parameters of a single model, and it generalizes to unseen queries.
We empirically evaluated our pipeline on the TREC 2019 and 2020 datasets under
FOE constraints. Our experiments show that CoMOT rapidly predicts fair
re-ranking policies on held-out data, with a speed-up proportional to the
average number of documents per query. It also displays fairness and ranking
performance similar to the original optimization-based policy. Furthermore, we
empirically validate the effectiveness of GumMS to approximate DS-based
policies in expectation.
