  A key challenge with machine learning approaches for ranking is the gap
between the performance metrics of interest and the surrogate loss functions
that can be optimized with gradient-based methods. This gap arises because
ranking metrics typically involve a sorting operation which is not
differentiable w.r.t. the model parameters. Prior works have proposed
surrogates that are loosely related to ranking metrics or simple smoothed
versions thereof. We propose PiRank, a new class of differentiable surrogates
for ranking, which employ a continuous, temperature-controlled relaxation to
the sorting operator. We show that PiRank exactly recovers the desired metrics
in the limit of zero temperature and scales favorably with the problem size,
both in theory and practice. Empirically, we demonstrate that PiRank
significantly improves over existing approaches on publicly available
internet-scale learning-to-rank benchmarks.
