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Practical $0.385$-Approximation for Submodular Maximization Subject to a Cardinality Constraint

Murad Tukan, Loay Mualem, Moran Feldman

TL;DR

This work tackles non-monotone submodular maximization under a cardinality constraint, a problem with strong NP-hardness and gaps between theory and practice. It introduces a novel algorithmic framework that combines a fast local search component with a guided stochastic greedy refinement to achieve a robust $0.385$-approximation while constraining the query complexity to $O(n+k^2)$. Theoretical guarantees are complemented by empirical evaluation on Movie Recommendation, Image Summarization, and Revenue Maximization, where the method consistently outperforms prior practical algorithms and shows reduced variance. Overall, the approach offers a scalable, practically efficient alternative to existing methods for non-monotone submodular maximization in real-world ML applications, with potential for future linear-time realizations.

Abstract

Non-monotone constrained submodular maximization plays a crucial role in various machine learning applications. However, existing algorithms often struggle with a trade-off between approximation guarantees and practical efficiency. The current state-of-the-art is a recent $0.401$-approximation algorithm, but its computational complexity makes it highly impractical. The best practical algorithms for the problem only guarantee $1/e$-approximation. In this work, we present a novel algorithm for submodular maximization subject to a cardinality constraint that combines a guarantee of $0.385$-approximation with a low and practical query complexity of $O(n+k^2)$. Furthermore, we evaluate the empirical performance of our algorithm in experiments based on various machine learning applications, including Movie Recommendation, Image Summarization, and more. These experiments demonstrate the efficacy of our approach.

Practical $0.385$-Approximation for Submodular Maximization Subject to a Cardinality Constraint

TL;DR

This work tackles non-monotone submodular maximization under a cardinality constraint, a problem with strong NP-hardness and gaps between theory and practice. It introduces a novel algorithmic framework that combines a fast local search component with a guided stochastic greedy refinement to achieve a robust -approximation while constraining the query complexity to . Theoretical guarantees are complemented by empirical evaluation on Movie Recommendation, Image Summarization, and Revenue Maximization, where the method consistently outperforms prior practical algorithms and shows reduced variance. Overall, the approach offers a scalable, practically efficient alternative to existing methods for non-monotone submodular maximization in real-world ML applications, with potential for future linear-time realizations.

Abstract

Non-monotone constrained submodular maximization plays a crucial role in various machine learning applications. However, existing algorithms often struggle with a trade-off between approximation guarantees and practical efficiency. The current state-of-the-art is a recent -approximation algorithm, but its computational complexity makes it highly impractical. The best practical algorithms for the problem only guarantee -approximation. In this work, we present a novel algorithm for submodular maximization subject to a cardinality constraint that combines a guarantee of -approximation with a low and practical query complexity of . Furthermore, we evaluate the empirical performance of our algorithm in experiments based on various machine learning applications, including Movie Recommendation, Image Summarization, and more. These experiments demonstrate the efficacy of our approach.
Paper Structure (29 sections, 15 theorems, 68 equations, 4 figures, 6 algorithms)

This paper contains 29 sections, 15 theorems, 68 equations, 4 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Preliminaries
  4. Method
  5. Fast local search
  6. Guided stochastic greedy
  7. $0.385$-Approximation guarantee
  8. Experiments
  9. Personalized movie recommendation
  10. Personalized image summarization
  11. Revenue maximization
  12. Conclusion
  13. A warmup version of our algorithm
  14. Local search
  15. Handling the case where $E_{+}$ is not holding.
  16. ...and 14 more sections

Key Result

Theorem 3.1

There exists an algorithm that given a positive integer $k$, a value $\varepsilon \in (0, 1)$, and a non-negative submodular function $f\colon 2^{\mathcal{N}} \to \mathbb{R}$, outputs a set $S \subseteq \mathcal{N}$ of size at most $k$ that, with probability at least $1-\varepsilon$, obeys Furthermore, the query complexity of the above algorithm is $O_{\varepsilon}(n + k^2)$.

Figures (4)

  • Figure 1: Experimental results for Personalized Movie Recommendation. Each plot includes the output of our algorithm in comparison to previous state-of-the-art practical algorithms as described in the beginning of this section for various amounts of movies $k$.
  • Figure 2: Results for a varying number of images $k$ concerning the personalized image summarization problem involving CIFAR10 and CIFAR100 datasets.
  • Figure 3: Results for a varying number of users concerning the revenue maximization problem on the Advogato and Facebook network datasets.
  • Figure 4: Results for a varying number of images $k$ with respect to the Image summarization problem on the Tiny ImageNet dataset.

Theorems & Definitions (26)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3: Approximation guarantee
  • Lemma 3.4
  • proof
  • Lemma A.0
  • proof
  • Theorem A.1
  • proof
  • Lemma A.2
  • ...and 16 more