Improved Algorithms for Nash Welfare in Linear Bandits

TL;DR

This work tackles fairness in stochastic linear bandits by introducing Nash regret and $p$-means regret, and then presenting FairLinBandit, a two-phase meta-algorithm. Phase I uses a data-adaptive exploration that combines D-optimal design and John ellipsoid initialization, producing a stopping time $\tau$ that enables Phase II to employ standard optimistic linear-bandit algorithms (e.g., LinUCB, LinPE) with near-optimal confidence control. The authors establish order-optimal Nash regret $\widetilde{O}(d/\sqrt{T})$ and derive $p$-means regret bounds that interpolate between fairness and utility across all $p$, with a unified reduction framework to plug Phase II into any optimistic method. Empirical results on real-world linear-bandit instances demonstrate faster convergence and greater stability than the prior LinNash baseline, highlighting practical impact in fairness-aware sequential decision making.

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.

Figures (3)

• Figure 1: Numerical results comparing Nash regret for different algorithm runs. (a) and (b) showcase the better performance of FairLinPE and FairLinUCB over LinNash on MSLR-WEB10K and Yahoo! LTRC dataset, respectively for $T=10^8$ rounds. (c) compares runs over $T=10^7$ rounds and demonstrates the instability of LinNash for shorter time horizons. (d), (e), (f) compare the $p-$mean regret for FairLinPE and FairLinUCB at $p=0.5$ , -0.5, and -1.5. The LinUCB procedure performs better than LinPE across all values of $p$.
• Figure 2: Numerical results showing the effect of variation of $p$ and $d$ on the regret. (a) and (c) show the effect of variation of $p$ on the $p-$ mean regret for FairLinPE and FairLinUCB, respectively, on the MSLR-WEB10K dataset. (b) and (d) show the effect of variation of $d$ on the Nash regret for FairLinPE and FairLinUCB, respectively, on the Yahoo! Learning To Rank Challenge dataset.
• Figure 3: Numerical results showing the effect of variation of the number of arms on the Nash regret. (a) and (b) show the effect on FairLinPE and FairLinUCB, respectively.

Theorems & Definitions (53)

• remark 1
• remark 2
• remark 3
• theorem 1
• remark 4: Lower bound
• theorem 2
• lemma 1: Informal; Number of Rounds in Phase I
• lemma 2: Informal; Near optimality of Phase II arms
• lemma 3: Chernoff Bound
• lemma 4