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Submodular Maximization via Taylor Series Approximation

Gözde Özcan, Armin Moharrer, Stratis Ioannidis

TL;DR

This work addresses submodular maximization under matroid constraints where the objective is a composition of analytic and multilinear functions. It introduces a deterministic polynomial (Taylor-series) estimator to approximate the multilinear gradient, enabling the continuous greedy algorithm to operate with near $1-1/e$ guarantees without sampling. Under two structural assumptions, the gradient bias is shown to vanish as the Taylor degree increases, and the approach scales to problems including data summarization, influence maximization, facility location, and cache networks, with substantial speedups over sampling. The results extend continuous greedy to a broader class of objectives and offer practical efficiency gains for large-scale submodular optimization tasks, with demonstrated improvements in both accuracy and runtime.

Abstract

We study submodular maximization problems with matroid constraints, in particular, problems where the objective can be expressed via compositions of analytic and multilinear functions. We show that for functions of this form, the so-called continuous greedy algorithm attains a ratio arbitrarily close to $(1-1/e) \approx 0.63$ using a deterministic estimation via Taylor series approximation. This drastically reduces execution time over prior art that uses sampling.

Submodular Maximization via Taylor Series Approximation

TL;DR

This work addresses submodular maximization under matroid constraints where the objective is a composition of analytic and multilinear functions. It introduces a deterministic polynomial (Taylor-series) estimator to approximate the multilinear gradient, enabling the continuous greedy algorithm to operate with near guarantees without sampling. Under two structural assumptions, the gradient bias is shown to vanish as the Taylor degree increases, and the approach scales to problems including data summarization, influence maximization, facility location, and cache networks, with substantial speedups over sampling. The results extend continuous greedy to a broader class of objectives and offer practical efficiency gains for large-scale submodular optimization tasks, with demonstrated improvements in both accuracy and runtime.

Abstract

We study submodular maximization problems with matroid constraints, in particular, problems where the objective can be expressed via compositions of analytic and multilinear functions. We show that for functions of this form, the so-called continuous greedy algorithm attains a ratio arbitrarily close to using a deterministic estimation via Taylor series approximation. This drastically reduces execution time over prior art that uses sampling.

Paper Structure

This paper contains 33 sections, 12 theorems, 61 equations, 2 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction.
  2. Related Work.
  3. Technical Preliminaries.
  4. Submodularity and Matroids.
  5. Submodular Maximization Subject to Matroid Constraints.
  6. Continuous Greedy Algorithm.
  7. Multilinear Functions.
  8. Main Results
  9. Motivation and Intuition.
  10. Assumptions.
  11. A Polynomial Estimator.
  12. Time Complexity.
  13. Examples.
  14. Data Summarization (SM)lin2011classmirzasoleiman2016fast.
  15. Influence Maximization (IM) kempe2003maximizingchen2009efficient.
  16. ...and 18 more sections

Key Result

Theorem 3.1

[Calinescu et al. calinescu2011maximizing] Consider Algorithm alg:cont-greed, with $\nabla G(\mathbf{y}_k)$ replaced by $\widehat{\nabla G}(\mathbf{y}_k)$ given by (eq: sampler estimator). Set $T = \frac{10}{\delta^2} (1 + \ln{|V|})$, where $\delta = \frac{1}{40d^2|V|}$ and $d = r_{\mathcal{M}}(V)$ where $\mathbf{y}^*$ is an optimal solution to (eq: multilinProb).

Figures (2)

  • Figure 1: Trajectory of the FW algorithm. Utility of the function at the current $\mathbf{y}$ as a function of time is marked for every $10$th iteration.
  • Figure 2: Comparison of different estimators on different problems. Blue lines represent the performance of the POLY estimators and the marked points correspond to POLY1, POLY2, POLY3 respectively. Orange lines represent the performance of the SAMP estimators and the marked points correspond to SAMP1, SAMP10, SAMP100, SAMP1000 respectively.

Theorems & Definitions (12)

  • Theorem 3.1
  • Lemma 4.1
  • Lemma 5.1
  • Theorem 5.1
  • Theorem 5.2
  • Theorem 6.1
  • Theorem 6.2
  • Lemma D .1
  • Lemma G .1
  • Lemma H .1
  • ...and 2 more