  Learning neural set functions becomes increasingly more important in many
applications like product recommendation and compound selection in AI-aided
drug discovery. The majority of existing works study methodologies of set
function learning under the function value oracle, which, however, requires
expensive supervision signals. This renders it impractical for applications
with only weak supervisions under the Optimal Subset (OS) oracle, the study of
which is surprisingly overlooked. In this work, we present a principled yet
practical maximum likelihood learning framework, termed as EquiVSet, that
simultaneously meets the following desiderata of learning set functions under
the OS oracle: i) permutation invariance of the set mass function being
modeled; ii) permission of varying ground set; iii) minimum prior; and iv)
scalability. The main components of our framework involve: an energy-based
treatment of the set mass function, DeepSet-style architectures to handle
permutation invariance, mean-field variational inference, and its amortized
variants. Thanks to the elegant combination of these advanced architectures,
empirical studies on three real-world applications (including Amazon product
recommendation, set anomaly detection, and compound selection for virtual
screening) demonstrate that EquiVSet outperforms the baselines by a large
margin.
