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Bridging the Gap Between General and Down-Closed Convex Sets in Submodular Maximization

Loay Mualem, Murad Tukan, Moran Fledman

TL;DR

This work addresses maximizing non-monotone DR-submodular functions over general convex constraints by decomposing the constraint into a down-closed part and a general part, enabling a continuous interpolation between the guarantees known for down-closed and general convex sets. The authors develop a coordinated offline and online framework based on maintaining two vectors, one in each decomposed set, and combining them via a probabilistic-sum operator to stay within the feasible domain, guided by a carefully designed potential function. The main contributions are a polynomial-time offline algorithm with guarantees that recover $e^{-1}$ when the constraint is fully down-closed and $(1/4)(1 - m)$ in the general case, plus a guess-free variant; and an online algorithm with a regret term that scales as $\sqrt{L}$, both supported by a rigorous analysis. They demonstrate the practical value of this interpolation across offline/online revenue maximization, location summarization, and quadratic-programming-style DR-submodular objectives, with empirical results showing improvements over existing methods.

Abstract

Optimization of DR-submodular functions has experienced a notable surge in significance in recent times, marking a pivotal development within the domain of non-convex optimization. Motivated by real-world scenarios, some recent works have delved into the maximization of non-monotone DR-submodular functions over general (not necessarily down-closed) convex set constraints. Up to this point, these works have all used the minimum $\ell_\infty$ norm of any feasible solution as a parameter. Unfortunately, a recent hardness result due to Mualem \& Feldman~\cite{mualem2023resolving} shows that this approach cannot yield a smooth interpolation between down-closed and non-down-closed constraints. In this work, we suggest novel offline and online algorithms that provably provide such an interpolation based on a natural decomposition of the convex body constraint into two distinct convex bodies: a down-closed convex body and a general convex body. We also empirically demonstrate the superiority of our proposed algorithms across three offline and two online applications.

Bridging the Gap Between General and Down-Closed Convex Sets in Submodular Maximization

TL;DR

This work addresses maximizing non-monotone DR-submodular functions over general convex constraints by decomposing the constraint into a down-closed part and a general part, enabling a continuous interpolation between the guarantees known for down-closed and general convex sets. The authors develop a coordinated offline and online framework based on maintaining two vectors, one in each decomposed set, and combining them via a probabilistic-sum operator to stay within the feasible domain, guided by a carefully designed potential function. The main contributions are a polynomial-time offline algorithm with guarantees that recover when the constraint is fully down-closed and in the general case, plus a guess-free variant; and an online algorithm with a regret term that scales as , both supported by a rigorous analysis. They demonstrate the practical value of this interpolation across offline/online revenue maximization, location summarization, and quadratic-programming-style DR-submodular objectives, with empirical results showing improvements over existing methods.

Abstract

Optimization of DR-submodular functions has experienced a notable surge in significance in recent times, marking a pivotal development within the domain of non-convex optimization. Motivated by real-world scenarios, some recent works have delved into the maximization of non-monotone DR-submodular functions over general (not necessarily down-closed) convex set constraints. Up to this point, these works have all used the minimum norm of any feasible solution as a parameter. Unfortunately, a recent hardness result due to Mualem \& Feldman~\cite{mualem2023resolving} shows that this approach cannot yield a smooth interpolation between down-closed and non-down-closed constraints. In this work, we suggest novel offline and online algorithms that provably provide such an interpolation based on a natural decomposition of the convex body constraint into two distinct convex bodies: a down-closed convex body and a general convex body. We also empirically demonstrate the superiority of our proposed algorithms across three offline and two online applications.
Paper Structure (25 sections, 38 theorems, 130 equations, 3 figures, 4 algorithms)

This paper contains 25 sections, 38 theorems, 130 equations, 3 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Related work
  4. Offline DR-submodular optimization.
  5. Online (regret minimization) DR-submodular optimization.
  6. Paper organization
  7. Preliminaries
  8. Known Results and Additional Notation for DR-submodular Functions
  9. Online Optimization
  10. Our Technique
  11. Offline Maximization
  12. Algorithm Knowing F(oD(1))
  13. Guess-Free Offline Maximization
  14. Online Maximization
  15. Applications and Experimental Results
  16. ...and 10 more sections

Key Result

Lemma 2.1

Let $F\colon [0, 1]^n \to {\bR_{\geq 0}}$ be a non-negative differentiable DR-submodular function. Then,

Figures (3)

  • Figure 1: Results of the Revenue Maximization Experiments
  • Figure 2: Quadratic Programming with Uniform Distribution
  • Figure 3: Quadratic Programming with Exponential Distribution

Theorems & Definitions (69)

  • Lemma 2.1
  • Definition 2.1
  • Lemma 2.2: Lemma 4.3 of buchbinder2023constrained
  • Corollary 2.3
  • proof
  • Theorem 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.2
  • proof
  • ...and 59 more