Regularized Unconstrained Weakly Submodular Maximization

Submodular optimization finds applications in machine learning and data mining. In this paper, we study the problem of maximizing functions of the form $h = f-c$, where $f$ is a monotone, non-negative, weakly submodular set function and $c$ is a modular function. We design a deterministic approximation algorithm that runs with ${O}(\frac{n}ε\log \frac{n}{γε})$ oracle calls to function $h$, and outputs a set ${S}$ such that $h({S}) \geq γ(1-ε)f(OPT)-c(OPT)-\frac{c(OPT)}{γ(1-ε)}\log\frac{f(OPT)}{c(OPT)}$, where $γ$ is the submodularity ratio of $f$. Existing algorithms for this problem either admit a worse approximation ratio or have quadratic runtime. We also present an approximation ratio of our algorithm for this problem with an approximate oracle of $f$. We validate our theoretical results through extensive empirical evaluations on real-world applications, including vertex cover and influence diffusion problems for submodular utility function $f$, and Bayesian A-Optimal design for weakly submodular $f$. Our experimental results demonstrate that our algorithms efficiently achieve high-quality solutions.