  We study two mixed robust/average-case submodular partitioning problems that
we collectively call Submodular Partitioning. These problems generalize both
purely robust instances of the problem (namely max-min submodular fair
allocation (SFA) and min-max submodular load balancing (SLB) and also
generalize average-case instances (that is the submodular welfare problem (SWP)
and submodular multiway partition (SMP). While the robust versions have been
studied in the theory community, existing work has focused on tight
approximation guarantees, and the resultant algorithms are not, in general,
scalable to very large real-world applications. This is in contrast to the
average case, where most of the algorithms are scalable. In the present paper,
we bridge this gap, by proposing several new algorithms (including those based
on greedy, majorization-minimization, minorization-maximization, and relaxation
algorithms) that not only scale to large sizes but that also achieve
theoretical approximation guarantees close to the state-of-the-art, and in some
cases achieve new tight bounds. We also provide new scalable algorithms that
apply to additive combinations of the robust and average-case extreme
objectives. We show that these problems have many applications in machine
learning (ML). This includes: 1) data partitioning and load balancing for
distributed machine algorithms on parallel machines; 2) data clustering; and 3)
multi-label image segmentation with (only) Boolean submodular functions via
pixel partitioning. We empirically demonstrate the efficacy of our algorithms
on real-world problems involving data partitioning for distributed optimization
of standard machine learning objectives (including both convex and deep neural
network objectives), and also on purely unsupervised (i.e., no supervised or
semi-supervised learning, and no interactive segmentation) image segmentation.
