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Practical Parallel Algorithms for Non-Monotone Submodular Maximization

Shuang Cui, Kai Han, Jing Tang, Xueying Li, Aakas Zhiyuli, Hanxiao Li

TL;DR

This paper studies the problem of non-monotone submodular maximization subject to a knapsack constraint, and proposes the first combinatorial algorithm achieving an $(8+\epsilon)$-approximation under $\mathcal{O}(\log n)$ adaptive complexity, which is \textit{optimal} up to a factor of $log n$.

Abstract

Submodular maximization has found extensive applications in various domains within the field of artificial intelligence, including but not limited to machine learning, computer vision, and natural language processing. With the increasing size of datasets in these domains, there is a pressing need to develop efficient and parallelizable algorithms for submodular maximization. One measure of the parallelizability of a submodular maximization algorithm is its adaptive complexity, which indicates the number of sequential rounds where a polynomial number of queries to the objective function can be executed in parallel. In this paper, we study the problem of non-monotone submodular maximization subject to a knapsack constraint, and propose the first combinatorial algorithm achieving an $(8+ε)$-approximation under $\mathcal{O}(\log n)$ adaptive complexity, which is \textit{optimal} up to a factor of $\mathcal{O}(\log\log n)$. Moreover, we also propose the first algorithm with both provable approximation ratio and sublinear adaptive complexity for the problem of non-monotone submodular maximization subject to a $k$-system constraint. As a by-product, we show that our two algorithms can also be applied to the special case of submodular maximization subject to a cardinality constraint, and achieve performance bounds comparable with those of state-of-the-art algorithms. Finally, the effectiveness of our approach is demonstrated by extensive experiments on real-world applications.

Practical Parallel Algorithms for Non-Monotone Submodular Maximization

TL;DR

This paper studies the problem of non-monotone submodular maximization subject to a knapsack constraint, and proposes the first combinatorial algorithm achieving an -approximation under adaptive complexity, which is \textit{optimal} up to a factor of .

Abstract

Submodular maximization has found extensive applications in various domains within the field of artificial intelligence, including but not limited to machine learning, computer vision, and natural language processing. With the increasing size of datasets in these domains, there is a pressing need to develop efficient and parallelizable algorithms for submodular maximization. One measure of the parallelizability of a submodular maximization algorithm is its adaptive complexity, which indicates the number of sequential rounds where a polynomial number of queries to the objective function can be executed in parallel. In this paper, we study the problem of non-monotone submodular maximization subject to a knapsack constraint, and propose the first combinatorial algorithm achieving an -approximation under adaptive complexity, which is \textit{optimal} up to a factor of . Moreover, we also propose the first algorithm with both provable approximation ratio and sublinear adaptive complexity for the problem of non-monotone submodular maximization subject to a -system constraint. As a by-product, we show that our two algorithms can also be applied to the special case of submodular maximization subject to a cardinality constraint, and achieve performance bounds comparable with those of state-of-the-art algorithms. Finally, the effectiveness of our approach is demonstrated by extensive experiments on real-world applications.
Paper Structure (20 sections, 16 theorems, 50 equations, 4 figures, 1 table, 5 algorithms)

This paper contains 20 sections, 16 theorems, 50 equations, 4 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Algorithms for the SKP
  4. Algorithms for the SSP
  5. Algorithms for the SMC
  6. Preliminaries
  7. Approximation Algorithms
  8. The RandBatch Procedure
  9. The Algorithm for the SKP
  10. The Algorithm for the SSP
  11. Extensions for Cardinality Constraint
  12. Performance Evaluation
  13. Applications
  14. Experiments for the SKP
  15. Additional Experiments for the SKP
  16. ...and 5 more sections

Key Result

Theorem 1

For every constant $\epsilon>0$, there is an algorithm without using multi-linear extension that achieves a $(1/2-\epsilon)$-approximation for the unconstrained submodular maximization problem using $\mathcal{O}(\frac{1}{\epsilon}\log\frac{1}{\epsilon})$ adaptive rounds with $\mathcal{O}(\frac{n}{\e

Figures (4)

  • Figure 1: The figure compares the implemented algorithms on utility and adaptivity, where the plotted utilities are normalized by the largest utility achieved by all algorithms.
  • Figure 2: Comparison of our algorithms with the ENE algorithm ( ? ) on a small instance of movie recommendation. Similar to Figure \ref{['fig:exp-knapsack']}, the plotted utilities are normalized by the best utility achieved by the implemented algorithms.
  • Figure 3: The figure compares the implemented algorithms on utility and adaptivity, where the plotted utilities are normalized by the largest utility achieved by all algorithms
  • Figure 4: The figure compares the implemented algorithms on utility and adaptivity, where the plotted utilities are normalized by the largest utility achieved by all algorithms

Theorems & Definitions (34)

  • Definition 1: Submodular Function, defined by ( ? )
  • Definition 2: Independence System
  • Definition 3: $k$-system
  • Definition 4: Knapsack Constraint
  • Theorem 1: Theorem A.1 in the full version of ( ? )
  • Lemma 1
  • Lemma 2
  • proof : Proof of Lemma \ref{['lm:density']}
  • proof : Proof of Lemma \ref{['lm:properties']}
  • Lemma 3
  • ...and 24 more