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A Unified Framework for Analyzing and Optimizing a Class of Convex Fairness Measures

Man Yiu Tsang, Karmel S. Shehadeh

TL;DR

A new framework that unifies different fairness measures into a general, parameterized class of convex fairness measures suitable for optimization contexts, and derives an equivalent dual representation of these measures as a robustified order-based fairness measure over their dual sets.

Abstract

We propose a new framework that unifies different fairness measures into a general, parameterized class of convex fairness measures suitable for optimization contexts. First, we propose a new class of order-based fairness measures, discuss their properties, and derive an axiomatic characterization for such measures. Then, we introduce the class of convex fairness measures, discuss their properties, and derive an equivalent dual representation of these measures as a robustified order-based fairness measure over their dual sets. Importantly, this dual representation renders a unified mathematical expression and an alternative geometric characterization for convex fairness measures through their dual sets. Moreover, it allows us to develop a unified framework for optimization problems with a convex fairness measure objective or constraint, including unified reformulations and solution methods. In addition, we provide stability results that quantify the impact of employing different convex fairness measures on the optimal value and solution of the resulting fairness-promoting optimization problem. Finally, we present numerical results demonstrating the computational efficiency of our unified framework over traditional ones and illustrating our stability results.

A Unified Framework for Analyzing and Optimizing a Class of Convex Fairness Measures

TL;DR

A new framework that unifies different fairness measures into a general, parameterized class of convex fairness measures suitable for optimization contexts, and derives an equivalent dual representation of these measures as a robustified order-based fairness measure over their dual sets.

Abstract

We propose a new framework that unifies different fairness measures into a general, parameterized class of convex fairness measures suitable for optimization contexts. First, we propose a new class of order-based fairness measures, discuss their properties, and derive an axiomatic characterization for such measures. Then, we introduce the class of convex fairness measures, discuss their properties, and derive an equivalent dual representation of these measures as a robustified order-based fairness measure over their dual sets. Importantly, this dual representation renders a unified mathematical expression and an alternative geometric characterization for convex fairness measures through their dual sets. Moreover, it allows us to develop a unified framework for optimization problems with a convex fairness measure objective or constraint, including unified reformulations and solution methods. In addition, we provide stability results that quantify the impact of employing different convex fairness measures on the optimal value and solution of the resulting fairness-promoting optimization problem. Finally, we present numerical results demonstrating the computational efficiency of our unified framework over traditional ones and illustrating our stability results.
Paper Structure (46 sections, 22 theorems, 68 equations, 7 figures, 8 tables, 2 algorithms)

This paper contains 46 sections, 22 theorems, 68 equations, 7 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Notation
  4. Relevant Literature
  5. Axioms for Fairness Measures
  6. Order-Based Fairness Measures
  7. Convex Fairness Measures
  8. Optimization with Convex Fairness Measures
  9. Minimizing Order-based Fairness Measures
  10. Minimizing Convex Fairness Measures
  11. Stability Analysis
  12. Numerical Experiments
  13. Fair Facility Location
  14. Fair Resource Allocation
  15. Conclusion
  16. ...and 31 more sections

Key Result

Proposition 1

For any $\bm{w}\in\mathcal{W}$, the order-based fairness measure $\nu_{\bm{w}}:\mathbb{R}^N\rightarrow\mathbb{R}$ defined as $\nu_{\bm{w}}(\bm{u})=\sum_{i=1}^N w_i u_{(i)}$ satisfies Axioms axiom:continuity, axiom:normalization, axiom:symmetry, axiom:Schur_convex, axiom:trans_invariance, and axiom:p

Figures (7)

  • Figure 1: Illustration of the set $\mathcal{S}^N=\{\bm{w}\in\mathbb{R}_{\uparrow}^N\mid \bm{1}^\top\bm{w}=0\}$ when (a) $N=2$ and (b) $N=3$
  • Figure 2: Dual sets $\mathcal{W}_\nu$ (in red) of fairness measures (iv)--(vi) as subsets of $\mathcal{S}^3$ (in brown)
  • Figure 3: Hausdorff distance $d_H$ between the dual sets of $\nu^\text{MaxMAD}$ and $\nu\in\{\nu^\text{MAD},\nu^\text{GD},\nu^\text{SMaxPD}\}$
  • Figure 4: Average difference in optimal value $\texttt{val\_diff}_k(N,\gamma)$ for different values of $N$ and $\gamma$
  • Figure 5: Average difference in optimal solution $\texttt{sol\_diff}_k(N,\gamma)$ for different values of $N$ and $\gamma$
  • ...and 2 more figures

Theorems & Definitions (42)

  • Remark 1
  • Definition 4.1: Order-Based Fairness Measure
  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Definition 5.1
  • Remark 2
  • Theorem 3
  • Remark 3
  • Remark 4
  • ...and 32 more