Improved Algorithms for Nash Welfare in Linear Bandits

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.

DP-NCB: Privacy Preserving Fair Bandits

Multi-armed bandit algorithms are fundamental tools for sequential decision-making under uncertainty, with widespread applications across domains such as clinical trials and personalized decision-making. As bandit algorithms are increasingly deployed in these socially sensitive settings, it becomes critical to protect user data privacy and ensure fair treatment across decision rounds. While prior work has independently addressed privacy and fairness in bandit settings, the question of whether both objectives can be achieved simultaneously has remained largely open. Existing privacy-preserving bandit algorithms typically optimize average regret, a utilitarian measure, whereas fairness-aware approaches focus on minimizing Nash regret, which penalizes inequitable reward distributions, but often disregard privacy concerns. To bridge this gap, we introduce Differentially Private Nash Confidence Bound (DP-NCB)-a novel and unified algorithmic framework that simultaneously ensures $\epsilon$-differential privacy and achieves order-optimal Nash regret, matching known lower bounds up to logarithmic factors. The framework is sufficiently general to operate under both global and local differential privacy models, and is anytime, requiring no prior knowledge of the time horizon. We support our theoretical guarantees with simulations on synthetic bandit instances, showing that DP-NCB incurs substantially lower Nash regret than state-of-the-art baselines. Our results offer a principled foundation for designing bandit algorithms that are both privacy-preserving and fair, making them suitable for high-stakes, socially impactful applications.