  We present a practical and powerful new framework for both unconstrained and
constrained submodular function optimization based on discrete
semidifferentials (sub- and super-differentials). The resulting algorithms,
which repeatedly compute and then efficiently optimize submodular
semigradients, offer new and generalize many old methods for submodular
optimization. Our approach, moreover, takes steps towards providing a unifying
paradigm applicable to both submodular min- imization and maximization,
problems that historically have been treated quite distinctly. The practicality
of our algorithms is important since interest in submodularity, owing to its
natural and wide applicability, has recently been in ascendance within machine
learning. We analyze theoretical properties of our algorithms for minimization
and maximization, and show that many state-of-the-art maximization algorithms
are special cases. Lastly, we complement our theoretical analyses with
supporting empirical experiments.
