On Unconstrained Quasi-Submodular Function Optimization

TL;DR

This paper introduces algorithms for optimizing quasi-submodular functions without constraints, providing efficient lattice reduction methods that guarantee monotonic improvement and contain all optima, supported by experimental validation.

Contribution

It proposes the first algorithms for unconstrained quasi-submodular function optimization with proven efficiency and theoretical guarantees, expanding the scope of submodular optimization techniques.

Findings

Algorithms reduce lattices in O(n) iterations

Objective function values strictly increase or decrease monotonically

All local and global optima are within the reduced lattices

Abstract

With the extensive application of submodularity, its generalizations are constantly being proposed. However, most of them are tailored for special problems. In this paper, we focus on quasi-submodularity, a universal generalization, which satisfies weaker properties than submodularity but still enjoys favorable performance in optimization. Similar to the diminishing return property of submodularity, we first define a corresponding property called the {\em single sub-crossing}, then we propose two algorithms for unconstrained quasi-submodular function minimization and maximization, respectively. The proposed algorithms return the reduced lattices in $\mathcal{O}(n)$ iterations, and guarantee the objective function values are strictly monotonically increased or decreased after each iteration. Moreover, any local and global optima are definitely contained in the reduced lattices. Experimental results verify the effectiveness and efficiency of the proposed algorithms on lattice reduction.