Regularized Unconstrained Weakly Submodular Maximization

TL;DR

We study Regularized Unconstrained Weakly-Submodular Maximization (RUWSM) and its approximate variant, RUASM, where the goal is to maximize $h(S)=f(S)-c(S)$ with $f$ monotone non-negative and $\gamma$-weakly submodular and $c$ modular. We propose a deterministic, near-linear time algorithm UP that achieves $h(\tilde{S})\ge \gamma(1-\epsilon)f(OPT)-c(OPT)-\frac{c(OPT)}{\gamma(1-\epsilon)}\log\frac{f(OPT)}{c(OPT)}$ using $\mathcal{O}(\frac{n}{\epsilon}\log \frac{n}{\gamma\epsilon})$ oracle calls, and extend the results to δ-approximate oracles with RUASM guarantees. The paper also analyzes the algorithm, provides supporting lemmas and a proof framework, and validates the approach with experiments on Profit Maximization, Directed Vertex Cover with Costs, and Bayesian A-Optimal Design, showing competitive solution quality and fast runtimes. Overall, the work delivers a fast, deterministic method for regularized unconstrained maximization under weak submodularity, with practical impact for applications involving influence, coverage, and design under costly budgets.

Abstract

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.

Figures (5)

• Figure 1: Roadmap of the proof of Theorem \ref{['th:unconstrained']}.
• Figure 2: Comparisons on p2p-Gnutella31 and NetHept networks with the application of Profit Maximization.
• Figure 3: Comparisons of average oracle evaluations over various cost penalties: p2p-Gnutella31 and NetHept networks (fixed $\lambda_1$ or $\lambda_2$) with Profit Maximization; various datasets with Vertex Cover and Bayesian A-Optimal Design applications. The results of UDG are one-run snap of ten runs.
• Figure 4: Experiments with Directed Vertex Cover (Eu-Email and Protein networks) and Bayesian A-Optimal Design (Segment and Housing data).
• Figure 5: Roadmap of the proof of Theorems \ref{['th:unconstrained-approx']} for Algorithm \ref{['alg:up']}

Theorems & Definitions (46)

• Definition 1: Submodularity
• Definition 2: $\gamma$-submodularity
• Definition 3: $\delta$-approximate
• Theorem 1
• Lemma 2
• proof
• Lemma 3
• Claim 4
• Claim 5
• proof