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Online Multi-Class Selection with Group Fairness Guarantee

Faraz Zargari, Hossein Nekouyan, Lyndon Hallett, Bo Sun, Xiaoqi Tan

TL;DR

This work studies Online Multi-class Selection (OMcS) with group fairness guarantees under online, irrevocable allocations with offline resources. It introduces a lossless relax-and-round framework that preserves fractional performance in the integral setting, addressing the challenges of multi-labeled agents and overlapping class constraints. The paper develops optimal deterministic and randomized algorithms for Group Fairness by Quantity (GFQ) and a PF-based approach for beta-Proportional Fairness, supported by a novel fractional relaxation and a lossless rounding scheme. A learning-augmented variant (LiLA) leverages untrusted advice to improve practical fairness-efficiency trade-offs while maintaining worst-case guarantees. Overall, the results provide tight fairness and efficiency guarantees and a practical pathway to improved online fair allocation in multi-labeled, real-world settings.

Abstract

We study the online multi-class selection problem with group fairness guarantees, where limited resources must be allocated to sequentially arriving agents. Our work addresses two key limitations in the existing literature. First, we introduce a novel lossless rounding scheme that ensures the integral algorithm achieves the same expected performance as any fractional solution. Second, we explicitly address the challenges introduced by agents who belong to multiple classes. To this end, we develop a randomized algorithm based on a relax-and-round framework. The algorithm first computes a fractional solution using a resource reservation approach -- referred to as the set-aside mechanism -- to enforce fairness across classes. The subsequent rounding step preserves these fairness guarantees without degrading performance. Additionally, we propose a learning-augmented variant that incorporates untrusted machine-learned predictions to better balance fairness and efficiency in practical settings.

Online Multi-Class Selection with Group Fairness Guarantee

TL;DR

This work studies Online Multi-class Selection (OMcS) with group fairness guarantees under online, irrevocable allocations with offline resources. It introduces a lossless relax-and-round framework that preserves fractional performance in the integral setting, addressing the challenges of multi-labeled agents and overlapping class constraints. The paper develops optimal deterministic and randomized algorithms for Group Fairness by Quantity (GFQ) and a PF-based approach for beta-Proportional Fairness, supported by a novel fractional relaxation and a lossless rounding scheme. A learning-augmented variant (LiLA) leverages untrusted advice to improve practical fairness-efficiency trade-offs while maintaining worst-case guarantees. Overall, the results provide tight fairness and efficiency guarantees and a practical pathway to improved online fair allocation in multi-labeled, real-world settings.

Abstract

We study the online multi-class selection problem with group fairness guarantees, where limited resources must be allocated to sequentially arriving agents. Our work addresses two key limitations in the existing literature. First, we introduce a novel lossless rounding scheme that ensures the integral algorithm achieves the same expected performance as any fractional solution. Second, we explicitly address the challenges introduced by agents who belong to multiple classes. To this end, we develop a randomized algorithm based on a relax-and-round framework. The algorithm first computes a fractional solution using a resource reservation approach -- referred to as the set-aside mechanism -- to enforce fairness across classes. The subsequent rounding step preserves these fairness guarantees without degrading performance. Additionally, we propose a learning-augmented variant that incorporates untrusted machine-learned predictions to better balance fairness and efficiency in practical settings.
Paper Structure (30 sections, 8 theorems, 58 equations, 6 figures, 5 algorithms)

This paper contains 30 sections, 8 theorems, 58 equations, 6 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Work
  4. Problem Formulation and Preliminaries
  5. $\mathsf{OMcS}$: Problem Statement and Assumptions
  6. Efficiency and Fairness Metrics
  7. $\mathsf{OMcS}$ with Group Fairness by Quantity
  8. Warm Up: An Optimal Deterministic Set-Aside Algorithm
  9. Optimal Randomized Algorithm: $\textsc{r-SetAside-gfq}$ for $\mathsf{OMcS}$ with $\mathsf{GFQ}$
  10. $\mathsf{OMcS}$ with $\beta$-Proportional Fairness
  11. Relaxation Phase (lines \ref{['relaxation_begin']}-\ref{['relaxation_end']}): A Novel Fractional Set-Aside Algorithm
  12. Rounding Phase (lines \ref{['rounding_begin']}-\ref{['rounding_end']}): A Lossless Online Rounding Scheme
  13. Improving Group Fairness via Learning-Augmented Algorithms
  14. Conclusion and Future Work
  15. Numerical Experiments
  16. ...and 15 more sections

Key Result

Theorem 3.1

$\textsc{d-SetAside-gfq}$ achieves the optimal competitive ratio of among all deterministic algorithms, denoted by $\alpha^*$, if and only if the threshold $\boldsymbol{\lambda}^* = \{\lambda_0^*, \lambda_1^*, \ldots, \lambda_{\tau}^*, \ldots, \lambda_{B-M}^*\}$ is designed as follows: (i) If $\frac (ii) If $\frac{B}{\alpha^*}< M$: In this case, $\lambda^*_{B-M} = \theta_K$ and $\{\alpha^*, \lambd

Figures (6)

  • Figure 1: Comparison of $\textsc{d-SetAside-gfq}$ and $\textsc{r-SetAside-gfq}$.
  • Figure 2: Utilities and resource allocations of each class under different algorithms; $\theta_1=5$, $\theta_2 = 10$ and $\theta_3 = 15$.
  • Figure 3: CDF of empirical competitive ratios of different algorithms.
  • Figure 4: Variation of the empirical fairness guarantee w.r.t. the adversarial probability $\xi$.
  • Figure 5: The red and yellow curves ($\textsc{d-SetAside-gfq}$) show the impact of $\mathsf{GFQ}$ constraint compare to the blue curve which shows the thresholds without any fairness consideration ($\mathsf{d}$-Dynamic). This figure shows that higher total $\mathsf{GFQ}$ requirements leads to having lower admission thresholds. Here we set $B=20$, $\theta_1 = 5$ and $\theta_2=10$.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Definition 1: Group Fairness by Quantity ($\mathsf{GFQ}$)
  • Definition 2: $\beta$-Proportional Fairness ($\beta$-$\mathsf{PF}$)
  • Theorem 3.1: $\mathsf{OMcS}$ with $\mathsf{GFQ}$: Optimal Deterministic Algorithm
  • Theorem 3.2: $\mathsf{OMcS}$ with $\mathsf{GFQ}$: Optimal Randomized Algorithm
  • Theorem 4.1
  • Corollary 4.2: Pareto-optimality
  • Theorem 4.3: $\mathsf{OMcS}$ with $\beta$-$\mathsf{PF}$
  • Definition 3: Advice Model of $\mathsf{OMcS}$ with $\beta$-$\mathsf{PF}$
  • Theorem 5.1
  • Corollary 5.2
  • ...and 5 more