  Many real-world multi-label prediction problems involve set-valued
predictions that must satisfy specific requirements dictated by downstream
usage. We focus on a typical scenario where such requirements, separately
encoding $\textit{value}$ and $\textit{cost}$, compete with each other. For
instance, a hospital might expect a smart diagnosis system to capture as many
severe, often co-morbid, diseases as possible (the value), while maintaining
strict control over incorrect predictions (the cost). We present a general
pipeline, dubbed as FavMac, to maximize the value while controlling the cost in
such scenarios. FavMac can be combined with almost any multi-label classifier,
affording distribution-free theoretical guarantees on cost control. Moreover,
unlike prior works, it can handle real-world large-scale applications via a
carefully designed online update mechanism, which is of independent interest.
Our methodological and theoretical contributions are supported by experiments
on several healthcare tasks and synthetic datasets - FavMac furnishes higher
value compared with several variants and baselines while maintaining strict
cost control. Our code is available at https://github.com/zlin7/FavMac
