  We propose and analyze batch greedy heuristics for cardinality constrained
maximization of non-submodular non-decreasing set functions. We consider the
standard greedy paradigm, along with its distributed greedy and stochastic
greedy variants. Our theoretical guarantees are characterized by the
combination of submodularity and supermodularity ratios. We argue how these
parameters define tight modular bounds based on incremental gains, and provide
a novel reinterpretation of the classical greedy algorithm using the
minorize-maximize (MM) principle. Based on that analogy, we propose a new class
of methods exploiting any plausible modular bound. In the context of optimal
experimental design for linear Bayesian inverse problems, we bound the
submodularity and supermodularity ratios when the underlying objective is based
on mutual information. We also develop novel modular bounds for the mutual
information in this setting, and describe certain connections to polyhedral
combinatorics. We discuss how algorithms using these modular bounds relate to
established statistical notions such as leverage scores and to more recent
efforts such as volume sampling. We demonstrate our theoretical findings on
synthetic problems and on a real-world climate monitoring example.

Optimization and Control;Discrete Mathematics;Data Structures and Algorithms;Machine Learning