  It is of increasing importance to develop learning methods for ranking. In
contrast to many learning objectives, however, the ranking problem presents
difficulties due to the fact that the space of permutations is not smooth. In
this paper, we examine the class of rank-linear objective functions, which
includes popular metrics such as precision and discounted cumulative gain. In
particular, we observe that expectations of these gains are completely
characterized by the marginals of the corresponding distribution over
permutation matrices. Thus, the expectations of rank-linear objectives can
always be described through locations in the Birkhoff polytope, i.e.,
doubly-stochastic matrices (DSMs). We propose a technique for learning
DSM-based ranking functions using an iterative projection operator known as
Sinkhorn normalization. Gradients of this operator can be computed via
backpropagation, resulting in an algorithm we call Sinkhorn propagation, or
SinkProp. This approach can be combined with a wide range of gradient-based
approaches to rank learning. We demonstrate the utility of SinkProp on several
information retrieval data sets.
