A Parameterized Family of Meta-Submodular Functions
This work addresses a bottleneck in optimization for machine learning applications, offering a unified framework for handling both submodular and supermodular functions, though it appears incremental in extending existing methods.
The paper tackles the intractability of supermodular maximization problems by introducing a parameterized family of functions called γ-meta-submodular, which includes submodular and supermodular functions, and develops local search algorithms with approximation factors dependent on γ.
Submodular function maximization has found a wealth of new applications in machine learning models during the past years. The related supermodular maximization models (submodular minimization) also offer an abundance of applications, but they appeared to be highly intractable even under simple cardinality constraints. Hence, while there are well-developed tools for maximizing a submodular function subject to a matroid constraint, there is much less work on the corresponding supermodular maximization problems. We give a broad parameterized family of monotone functions which includes submodular functions and a class of supermodular functions containing diversity functions. Functions in this parameterized family are called \emph{$γ$-meta-submodular}. We develop local search algorithms with approximation factors that depend only on the parameter $γ$. We show that the $γ$-meta-submodular families include well-known classes of functions such as meta-submodular functions ($γ=0$), metric diversity functions and proportionally submodular functions (both with $γ=1$), diversity functions based on negative-type distances or Jensen-Shannon divergence (both with $γ=2$), and $σ$-semi metric diversity functions ($γ= σ$).