Generalizing Fair Top-$k$ Selection: An Integrative Approach

TL;DR

This work studies the problem of finding a fair (linear) scoring function with multiple protected groups while also minimizing the disparity from a reference scoring function, and introduces an alternative disparity measure that may yield a more stable scoring function under small weight perturbations.

Abstract

Fair top-$k$ selection, which ensures appropriate proportional representation of members from minority or historically disadvantaged groups among the top-$k$ selected candidates, has drawn significant attention. We study the problem of finding a fair (linear) scoring function with multiple protected groups while also minimizing the disparity from a reference scoring function. This generalizes the prior setup, which was restricted to the single-group setting without disparity minimization. Previous studies imply that the number of protected groups may have a limited impact on the runtime efficiency. However, driven by the need for experimental exploration, we find that this implication overlooks a critical issue that may affect the fairness of the outcome. Once this issue is properly considered, our hardness analysis shows that the problem may become computationally intractable even for a two-dimensional dataset and small values of $k$. However, our analysis also reveals a gap in the hardness barrier, enabling us to recover the efficiency for the case of small $k$ when the number of protected groups is sufficiently small. Furthermore, beyond measuring disparity as the "distance" between the fair and the reference scoring functions, we introduce an alternative disparity measure$\unicode{x2014}$utility loss$\unicode{x2014}$that may yield a more stable scoring function under small weight perturbations. Through careful engineering trade-offs that balance implementation complexity, robustness, and performance, our augmented two-pronged solution demonstrates strong empirical performance on real-world datasets, with experimental observations also informing algorithm design and implementation decisions.

Figures (5)

• Figure 1: The structure and interplay of key components in this work. Solid arrows denote the primary workflow and direct influences, while dashed arrows indicate feedback and motivation.
• Figure 2: $(k-1)$-level (triangular mesh) and $V$ (dashed region on the $x$-$y$ plane) in 3-D. The dotted region (on the $x$-$y$ plane) represents a projected fair cell $\mathcal{F}$ and the point inside $V$ represents the input reference weight vector $w^o$. When minimizing the $w$ difference, the resulting fair weight vector will lie on the boundary of $\mathcal{F}$. In contrast, minimizing the utility loss enables a stable weight vector by selecting a weight vector inside $\mathcal{F}$ and away from its boundary.
• Figure 3: (a)$w$ difference minimization in 2-D, where the interval represents $V$ and the vertical dashed line $w^o$ represents the reference weight vector. The bidirectional sweep-line algorithm works by sweeping from $w^o$ to $ub$ and to $lb$. (b) For the weight vectors $\hat{w}$ and $\widetilde{w}$ with $k = 3$, $l_p$ (resp. $l_q$) is in the top-$k$ subset of $\hat{w}$ (resp. $\widetilde{w}$) but not in that of $\widetilde{w}$ (resp. $\hat{w}$). Intuitively, $l_p$ lies above $l_q$ at $x = w_x^o$ since the two lines intersect at a point between $x = \hat{w}_x$ and $x = \widetilde{w}_x$, where their orders change.
• Figure 4: Runtime experimental results for 2-D datasets.
• Figure 5: Runtime experimental results for multi-dimensional datasets ($3 \leq d \leq 6$).

Theorems & Definitions (19)

• Example 1
• Example 2
• Definition 1
• Example 3
• Theorem 1
• proof
• Remark 1
• Theorem 2
• proof
• Corollary 1