Multilabel reductions: what is my loss optimising?

Multilabel classification is a challenging problem arising in applications ranging from information retrieval to image tagging. A popular approach to this problem is to employ a reduction to a suitable series of binary or multiclass problems (e.g., computing a softmax based cross-entropy over the relevant labels). While such methods have seen empirical success, less is understood about how well they approximate two fundamental performance measures: precision@$k$ and recall@$k$. In this paper, we study five commonly used reductions, including the one-versus-all reduction, a reduction to multiclass classification, and normalised versions of the same, wherein the contribution of each instance is normalised by the number of relevant labels. Our main result is a formal justification of each reduction: we explicate their underlying risks, and show they are each consistent with respect to either precision or recall. Further, we show that in general no reduction can be optimal for both measures. We empirically validate our results, demonstrating scenarios where normalised reductions yield recall gains over unnormalised counterparts.