Improved Algorithms for Nash Welfare in Linear Bandits

2026-01-30

AI Summary

This work addresses the limitations of existing linear bandit methods, which suffer from Nash regret bounds that scale with the ambient dimension \(d\) and lack a unified framework for balancing fairness and utility. The authors propose FairLinBandit, a general meta-algorithm compatible with any linear bandit strategy, designed to optimize Nash regret and extend naturally to the broader class of \(p\)-mean regret objectives. By introducing novel analytical tools, they establish the first dimension-free, optimal Nash regret bound and formally define and analyze \(p\)-mean regret in the linear bandit setting. Instantiating the framework with Phased Elimination and UCB—augmented with a new concentration inequality—the approach achieves sublinear \(p\)-mean regret across the full range of \(p\) values on real-world datasets, significantly outperforming existing baselines.

Technology Category

Machine Learning: Online Learning & BanditsGame Theory and Economic Paradigms: Fair DivisionMultiagent Systems: Mechanism Design

Application Category

Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environmentsUser Modeling, Personalization and Recommendation: Fairness-aware retrieval and rankingSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Nash regret has recently emerged as a principled fairness-aware performance metric for stochastic multi-armed bandits, motivated by the Nash Social Welfare objective. Although this notion has been extended to linear bandits, existing results suffer from suboptimality in ambient dimension $d$, stemming from proof techniques that rely on restrictive concentration inequalities. In this work, we resolve this open problem by introducing new analytical tools that yield an order-optimal Nash regret bound in linear bandits. Beyond Nash regret, we initiate the study of $p$-means regret in linear bandits, a unifying framework that interpolates between fairness and utility objectives and strictly generalizes Nash regret. We propose a generic algorithmic framework, FairLinBandit, that works as a meta-algorithm on top of any linear bandit strategy. We instantiate this framework using two bandit algorithms: Phased Elimination and Upper Confidence Bound, and prove that both achieve sublinear $p$-means regret for the entire range of $p$. Extensive experiments on linear bandit instances generated from real-world datasets demonstrate that our methods consistently outperform the existing state-of-the-art baseline.

Problem

• Nash regret
• linear bandits
• fairness
• p-means regret
• stochastic multi-armed bandits

Innovation

• Nash regret
• linear bandits
• p-means regret
• FairLinBandit
• fairness-aware learning