Non-submodular Function Maximization subject to a Matroid Constraint, with Applications

Abstract

The standard greedy algorithm has been recently shown to enjoy approximation guarantees for constrained non-submodular nondecreasing set function maximization. While these recent results allow to better characterize the empirical success of the greedy algorithm, they are only applicable to simple cardinality constraints. In this paper, we study the problem of maximizing a non-submodular nondecreasing set function subject to a general matroid constraint. We first show that the standard greedy algorithm offers an approximation factor of $\frac{0.4{\gamma }^2}{\sqrt{\gamma r}+1}$, where $\gamma$ is the submodularity ratio of the function and $r$ is the rank of the matroid. Then, we show that the same greedy algorithm offers a constant approximation factor of $(1+1/(1-\alpha ){)}^{-1}$, where $\alpha$ is the generalized curvature of the function. In addition, we demonstrate that these approximation guarantees are applicable to several real-world applications in which the submodularity ratio and the generalized curvature can be bounded. Finally, we show that our greedy algorithm does achieve a competitive performance in practice using a variety of experiments on synthetic and real-world data.

arXiv commentAdded missing citations and changed strong submodularity ratio to generalized curvature

References in corpus (20)

• Link Prediction in Complex Networks: A Survey
• Fairness of Exposure in Rankings
• Near-optimal Nonmyopic Value of Information in Graphical Models
• Submodular meets Spectral: Greedy Algorithms for Subset Selection, Sparse Approximation and Dictionary Selection
• How Visibility and Divided Attention Constrain Social Contagion
• Guarantees for Greedy Maximization of Non-submodular Functions with Applications
• Algorithms for Approximate Minimization of the Difference Between Submodular Functions, with Applications
• Deep Reinforcement Learning of Marked Temporal Point Processes
• Maximization of Approximately Submodular Functions
• When-To-Post on Social Networks
• Learning Gaussian Tree Models: Analysis of Error Exponents and Extremal Structures
• Submodular Maximization Beyond Non-negativity: Guarantees, Fast Algorithms, and Applications
• A submodular-supermodular procedure with applications to discriminative structure learning
• Greedy Column Subset Selection: New Bounds and Distributed Algorithms
• Steering Social Activity: A Stochastic Optimal Control Point Of View
• Robust Maximization of Non-Submodular Objectives
• Smart broadcasting: Do you want to be seen?
• Weakly Submodular Functions
• VIP: Incorporating Human Cognitive Biases in a Probabilistic Model of Retweeting
• Combinatorial Penalties: Which structures are preserved by convex relaxations?

Cited by in corpus (4)

• GRAD-MATCH: Gradient Matching based Data Subset Selection for Efficient Deep Model Training
• Classification Under Human Assistance
• Training Data Subset Selection for Regression with Controlled Generalization Error
• Regression Under Human Assistance