Approximate Submodular Functions and Performance Guarantees
This work addresses optimization challenges in practical applications where objectives are not submodular, offering theoretical tools for performance analysis, though it is incremental in extending submodular methods.
The paper tackles the problem of maximizing non-submodular set functions by approximating them with submodular functions, introducing δ-approximation and greedy curvature to quantify errors and derive performance guarantees. It provides necessary conditions for such approximations and validates tightened bounds with real data, achieving linear complexity.
We consider the problem of maximizing non-negative non-decreasing set functions. Although most of the recent work focus on exploiting submodularity, it turns out that several objectives we encounter in practice are not submodular. Nonetheless, often we leverage the greedy algorithms used in submodular functions to determine a solution to the non-submodular functions. Hereafter, we propose to address the original problem by \emph{approximating} the non-submodular function and analyze the incurred error, as well as the performance trade-offs. To quantify the approximation error, we introduce a novel concept of $δ$-approximation of a function, which we used to define the space of submodular functions that lie within an approximation error. We provide necessary conditions on the existence of such $δ$-approximation functions, which might not be unique. Consequently, we characterize this subspace which we refer to as \emph{region of submodularity}. Furthermore, submodular functions are known to lead to different sub-optimality guarantees, so we generalize those dependencies upon a $δ$-approximation into the notion of \emph{greedy curvature}. Finally, we used this latter notion to simplify some of the existing results and efficiently (i.e., linear complexity) determine tightened bounds on the sub-optimality guarantees using objective functions commonly used in practical setups and validate them using real data.