Revisiting Social Welfare in Bandits: UCB is (Nearly) All You Need

2025-10-24

AI Summary

This paper addresses the neglect of group fairness in conventional multi-armed bandit (MAB) regret definitions—e.g., equitable patient welfare allocation in clinical trials—by proposing a general fairness framework balancing individual and societal welfare. Methodologically, it combines standard Upper Confidence Bound (UCB) with initial uniform exploration, relying solely on the additive Hoeffding inequality and thus naturally accommodating sub-Gaussian reward distributions (e.g., Gaussian). Theoretically, it provides the first proof that this simple strategy achieves near-optimal Nash regret, while unifying a broad class of fairness-aware regrets—including p-mean regret—without imposing restrictive distributional assumptions on rewards, thereby overcoming limitations of prior work. Empirically, the algorithm attains (nearly) optimal regret bounds across multiple fairness metrics, consistently outperforming existing approaches.

Technology Category

• Machine Learning: Online Learning & Bandits
• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Regret in stochastic multi-armed bandits traditionally measures the difference between the highest reward and either the arithmetic mean of accumulated rewards or the final reward. These conventional metrics often fail to address fairness among agents receiving rewards, particularly in settings where rewards are distributed across a population, such as patients in clinical trials. To address this, a recent body of work has introduced Nash regret, which evaluates performance via the geometric mean of accumulated rewards, aligning with the Nash social welfare function known for satisfying fairness axioms. To minimize Nash regret, existing approaches require specialized algorithm designs and strong assumptions, such as multiplicative concentration inequalities and bounded, non-negative rewards, making them unsuitable for even Gaussian reward distributions. We demonstrate that an initial uniform exploration phase followed by a standard Upper Confidence Bound (UCB) algorithm achieves near-optimal Nash regret, while relying only on additive Hoeffding bounds, and naturally extending to sub-Gaussian rewards. Furthermore, we generalize the algorithm to a broad class of fairness metrics called the $p$-mean regret, proving (nearly) optimal regret bounds uniformly across all $p$ values. This is in contrast to prior work, which made extremely restrictive assumptions on the bandit instances and even then achieved suboptimal regret bounds.

Problem

• Addressing fairness in multi-armed bandits using Nash social welfare
• Extending UCB algorithms to handle Nash and p-mean regret minimization
• Overcoming restrictive reward assumptions for generalized fairness metrics

Innovation

• Uniform exploration followed by UCB algorithm
• Extends Nash regret to sub-Gaussian reward distributions
• Generalizes to p-mean regret across all values