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Fairness in Monotone $k$-submodular Maximization: Algorithms and Applications

Yanhui Zhu, Samik Basu, A. Pavan

TL;DR

This work is the first to incorporate fairness in the context of $k$-submodular maximization, and the theoretical guarantee matches the best-known $k$-submodular maximization results without fairness constraints.

Abstract

Submodular optimization has become increasingly prominent in machine learning and fairness has drawn much attention. In this paper, we propose to study the fair $k$-submodular maximization problem and develop a $\frac{1}{3}$-approximation greedy algorithm with a running time of $\mathcal{O}(knB)$. To the best of our knowledge, our work is the first to incorporate fairness in the context of $k$-submodular maximization, and our theoretical guarantee matches the best-known $k$-submodular maximization results without fairness constraints. In addition, we have developed a faster threshold-based algorithm that achieves a $(\frac{1}{3} - ε)$ approximation with $\mathcal{O}(\frac{kn}ε \log \frac{B}ε)$ evaluations of the function $f$. Furthermore, for both algorithms, we provide approximation guarantees when the $k$-submodular function is not accessible but only can be approximately accessed. We have extensively validated our theoretical findings through empirical research and examined the practical implications of fairness. Specifically, we have addressed the question: ``What is the price of fairness?" through case studies on influence maximization with $k$ topics and sensor placement with $k$ types. The experimental results show that the fairness constraints do not significantly undermine the quality of solutions.

Fairness in Monotone $k$-submodular Maximization: Algorithms and Applications

TL;DR

This work is the first to incorporate fairness in the context of -submodular maximization, and the theoretical guarantee matches the best-known -submodular maximization results without fairness constraints.

Abstract

Submodular optimization has become increasingly prominent in machine learning and fairness has drawn much attention. In this paper, we propose to study the fair -submodular maximization problem and develop a -approximation greedy algorithm with a running time of . To the best of our knowledge, our work is the first to incorporate fairness in the context of -submodular maximization, and our theoretical guarantee matches the best-known -submodular maximization results without fairness constraints. In addition, we have developed a faster threshold-based algorithm that achieves a approximation with evaluations of the function . Furthermore, for both algorithms, we provide approximation guarantees when the -submodular function is not accessible but only can be approximately accessed. We have extensively validated our theoretical findings through empirical research and examined the practical implications of fairness. Specifically, we have addressed the question: ``What is the price of fairness?" through case studies on influence maximization with topics and sensor placement with types. The experimental results show that the fairness constraints do not significantly undermine the quality of solutions.

Paper Structure

This paper contains 22 sections, 8 theorems, 26 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Work
  4. Preliminaries
  5. Fair Greedy Algorithms
  6. Construction of $\mathbf{o}^j$
  7. Properties of $\mathbf{o}^j$ and $Q^j_i$
  8. Proof of Theorem \ref{['th:fair-greedy']}
  9. Fair Approximate $k$-submodular Maximization.
  10. A Faster Threshold Algorithm
  11. Proof of Theorem \ref{['th:fair-faster']}
  12. Fair Approximate $k$-submodular Maximization.
  13. Experiments
  14. Algorithms Compared
  15. Evaluation Metrics
  16. ...and 7 more sections

Key Result

Theorem 2

Let $\mathbf{o}$ be the optimal solution and $\mathbf{s}$ be the output of Algorithm alg:fair. The algorithm admits a $\frac{1}{3}$-approximation within $\mathcal{O}(knB)$ oracle evaluations to $f$ for Problem problem:fair .

Figures (5)

  • Figure 1: Experiments on influence maximization with $k$ topics on Digg network: budget $B$ vs. objective values (expected influence spread).
  • Figure 2: Experiments on influence maximization with $k$ topics on Digg network: budget $B$ vs. Oracle Evaluations.
  • Figure 3: Experiments on sensor placement with $k$ types on Intel Lab sensor data: budget $B$ vs. objective values (entropy).
  • Figure 4: Experiments on sensor placement with $k$ types on Intel Lab sensor data: budget $B$ vs. Oracle Evaluations.
  • Figure 5: Results of budget vs. errors for TS-greedy and IS-greedy algorithms.

Theorems & Definitions (18)

  • Definition 1: $k$-submodular functions
  • Definition 2: $\delta$-approximate $k$-submodular zheng2021maximizing
  • Definition 3
  • Remark 1
  • Definition 4
  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 8 more