Improved Algorithms for Nash Welfare in Linear Bandits

TL;DR

This paper introduces new analytical tools to achieve order-optimal Nash regret bounds in linear bandits and proposes a flexible framework for fairness-aware regret minimization, extending to p-means regret with strong empirical results.

Contribution

It provides the first order-optimal Nash regret bounds for linear bandits and introduces a general framework for p-means regret, unifying fairness and utility objectives.

Findings

Achieves order-optimal Nash regret bounds in linear bandits.

Proposes FairLinBandit, a meta-algorithm for fairness-aware regret minimization.

Demonstrates superior empirical performance over existing baselines.

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.