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Toward Highly Efficient and Private Submodular Maximization via Matrix-Based Acceleration

Boyu Liu, Lianke Qin, Zhao Song, Yitan Wang, Jiale Zhao

TL;DR

This work targets efficient private submodular maximization under a cardinality constraint. It introduces a matrix-based embedding and a pair of dynamic data structures (IPE and FQFS) to convert marginal gains into fast inner-product queries, achieving a runtime of $O(\epsilon^{-2}(nd+kn+kd^2)\log(k/\delta))$ with a $(1-1/e)$-approximation up to additive terms. It further extends the framework to ($\epsilon$, $\delta$)-DP using the exponential mechanism, providing formal privacy guarantees while preserving near-optimal utility. An optional LSH extension yields further speedups, broadening applicability to large-scale, privacy-preserving submodular optimization in online and constrained settings.

Abstract

Submodular function maximization is a critical building block for diverse tasks, such as document summarization, sensor placement, and image segmentation. Yet its practical utility is often limit by the $O(knd^2)$ computational bottleneck. In this paper, we propose an integrated framework that addresses efficiency and privacy simultaneously. First, we introduce a novel matrix-based computation paradigm that accelerates function evaluations. Second, we develop approximate data structures that further streamline the optimization process, achieving a theoretical complexity of $O(ε^{-2}(nd+kn+kd^2)\log(k/δ))$. Third, we integrate ($ε, δ$)-DP guaranties to address the privacy concerns inherent in sensitive optimization tasks.

Toward Highly Efficient and Private Submodular Maximization via Matrix-Based Acceleration

TL;DR

This work targets efficient private submodular maximization under a cardinality constraint. It introduces a matrix-based embedding and a pair of dynamic data structures (IPE and FQFS) to convert marginal gains into fast inner-product queries, achieving a runtime of with a -approximation up to additive terms. It further extends the framework to (, )-DP using the exponential mechanism, providing formal privacy guarantees while preserving near-optimal utility. An optional LSH extension yields further speedups, broadening applicability to large-scale, privacy-preserving submodular optimization in online and constrained settings.

Abstract

Submodular function maximization is a critical building block for diverse tasks, such as document summarization, sensor placement, and image segmentation. Yet its practical utility is often limit by the computational bottleneck. In this paper, we propose an integrated framework that addresses efficiency and privacy simultaneously. First, we introduce a novel matrix-based computation paradigm that accelerates function evaluations. Second, we develop approximate data structures that further streamline the optimization process, achieving a theoretical complexity of . Third, we integrate ()-DP guaranties to address the privacy concerns inherent in sensitive optimization tasks.
Paper Structure (50 sections, 39 theorems, 59 equations, 12 algorithms)

This paper contains 50 sections, 39 theorems, 59 equations, 12 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Submodular functions.
  4. Differential privacy.
  5. Sketching.
  6. Preliminary
  7. Notations
  8. Goal
  9. Representation of $f(S)$
  10. Locality Sensitive Hashing.
  11. Dynamic Distance Estimation
  12. Differential Privacy
  13. Technique Overview
  14. Benchmark: Greedy Selection
  15. Analysis on Greedy Algorithms
  16. ...and 35 more sections

Key Result

Theorem 1.1

There is a submodular algorithm such that, given a submodular function maximization problem with cardinality constraint and two precision parameters $\epsilon,\delta$ as input, it runs in $O(\epsilon^{-2}(nd+kn+kd^2)\log(k/\delta))$ time and returns a solution $S$ with $f(S) \ge (1-1/e)\max_{|S|=k}\

Theorems & Definitions (75)

  • Theorem 1.1: Main result, informal version of Theorem \ref{['thm:main_formal']}
  • Definition 3.1: Submodular function
  • Definition 3.2: Quadratic embedding of $f$
  • Definition 3.3: Approximate Max-IP, ssx21
  • Theorem 3.4: Theorem 8.2, page 19, ssx21
  • Definition 3.5: Online Approximate Adaptive Distance Estimation
  • Theorem 3.6: Theorem 1.4 in cn22
  • Definition 3.7: Differential Privacy, dr14
  • Lemma 3.8: Post-Processing Lemma for DP, dr14
  • Lemma 3.9: Basic composition dmns06
  • ...and 65 more