Regularized Unconstrained Weakly Submodular Maximization
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 ${O}(\frac{n}{\epsilon}\log \frac{n}{\gamma \epsilon})$ oracle calls to function $h$, and outputs a set ${S}$ such that $h({S}) \geq \gamma(1-\epsilon)f(OPT)-c(OPT)-\frac{c(OPT)}{\gamma(1-\epsilon)}\log\frac{f(OPT)}{c(OPT)}$, where $\gamma$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2024
- DOI:
- 10.48550/arXiv.2408.04620
- arXiv:
- arXiv:2408.04620
- Bibcode:
- 2024arXiv240804620Z
- Keywords:
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- Computer Science - Data Structures and Algorithms
- E-Print:
- To appear in CIKM'24. Full paper including omitted proofs