Fair Best Arm Identification with Fixed Confidence

TL;DR

This work defines Fair Best Arm Identification (F-BAI), a PAC-style BAI problem augmented with per-arm fairness constraints. It derives an instance-specific lower bound on sample complexity and introduces the price of fairness, quantifying the extra samples required to satisfy fairness. The authors propose F-TaS, a Track-and-Stop-like algorithm that asymptotically matches the lower bound while guaranteeing fairness, with both model-agnostic and model-dependent fairness variants. The approach is validated through synthetic experiments and a wireless scheduling application, showing efficient sample usage and controlled fairness violations. Overall, the paper bridges fairness considerations and pure-exploration bandits, offering practical guarantees and insights for fair resource allocation problems.

Abstract

In this work, we present a novel framework for Best Arm Identification (BAI) under fairness constraints, a setting that we refer to as \textit{F-BAI} (fair BAI). Unlike traditional BAI, which solely focuses on identifying the optimal arm with minimal sample complexity, F-BAI also includes a set of fairness constraints. These constraints impose a lower limit on the selection rate of each arm and can be either model-agnostic or model-dependent. For this setting, we establish an instance-specific sample complexity lower bound and analyze the \textit{price of fairness}, quantifying how fairness impacts sample complexity. Based on the sample complexity lower bound, we propose F-TaS, an algorithm provably matching the sample complexity lower bound, while ensuring that the fairness constraints are satisfied. Numerical results, conducted using both a synthetic model and a practical wireless scheduling application, show the efficiency of F-TaS in minimizing the sample complexity while achieving low fairness violations.

Figures (18)

• Figure 1: Price of fairness for an instance with $K=30$ arms, and higher fairness rates for sub-optimal arms (see Ex. \ref{['example_case_1']}). The price of fairness $T^\star_{p}/T^\star$ scales closely with $(1-p_{\rm sum})^{-1}$
• Figure 2: Price of fairness for a MAB problem with equal gaps for different number of arms $K$. The fairness constraints are set to discourage exploration of the optimal arm by selecting $p_{a^\star}=0$ and otherwise $p_a=(w_{a}^\star+1/K)/2$ for $a\neq a^\star$. In black, on the left axis, it's depicted the ratio $T_p^\star/T^\star$. On the right axis, in dark-red, we plot the individual contributions due to $p_{\rm min}^{-1}$ and $(1-p_{\rm sum})^{-1}$.
• Figure 3: Violations for the synthetic experiments with $\delta=0.01$. Each subplot illustrates the distribution of maximum violation $\rho(t) = (\max_a p_a(\theta)-N_a(t)/t)_+$, across all rounds $t \leq \tau_\delta$ and experimental runs.
• Figure 4: Visual depiction of the scheduling environment with $K = 10$ UEs.
• Figure 5: Allocations for the scheduling experiments with $\theta$-dependent constraints. Agonistic constraints favour exploration, since $w_p^\star\approx w^\star$, while antagonistic ones discourage exploration of good arms.

Theorems & Definitions (35)

• Remark III.1
• Definition III.1
• Remark III.2
• Theorem IV.1
• Lemma IV.1
• Example IV.1
• Proposition V.1
• Corollary V.1
• Theorem V.1
• proof