  Submodular functions and variants, through their ability to characterize
diversity and coverage, have emerged as a key tool for data selection and
summarization. Many recent approaches to learn submodular functions suffer from
limited expressiveness. In this work, we propose FLEXSUBNET, a family of
flexible neural models for both monotone and non-monotone submodular functions.
To fit a latent submodular function from (set, value) observations, FLEXSUBNET
applies a concave function on modular functions in a recursive manner. We do
not draw the concave function from a restricted family, but rather learn from
data using a highly expressive neural network that implements a differentiable
quadrature procedure. Such an expressive neural model for concave functions may
be of independent interest. Next, we extend this setup to provide a novel
characterization of monotone \alpha-submodular functions, a recently introduced
notion of approximate submodular functions. We then use this characterization
to design a novel neural model for such functions. Finally, we consider
learning submodular set functions under distant supervision in the form of
(perimeter-set, high-value-subset) pairs. This yields a novel subset selection
method based on an order-invariant, yet greedy sampler built around the above
neural set functions. Our experiments on synthetic and real data show that
FLEXSUBNET outperforms several baselines.
