On the optimal regret of collaborative personalized linear bandits

• URL: http://arxiv.org/abs/2506.15943v1
• Date: Thu, 19 Jun 2025 00:56:31 GMT
• Title: On the optimal regret of collaborative personalized linear bandits
• Authors: Bruce Huang, Ruida Zhou, Lin F. Yang, Suhas Diggavi,
• Abstract summary: This paper investigates the optimal regret achievable in collaborative personalized linear bandits.<n>We provide an information-theoretic lower bound that characterizes how the number of agents, the interaction rounds, and the degree of heterogeneity jointly affect regret.<n>Our results offer a complete characterization of when and how collaboration helps with a optimal regret bound.
• Score: 15.661920010658626
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Stochastic linear bandits are a fundamental model for sequential decision making, where an agent selects a vector-valued action and receives a noisy reward with expected value given by an unknown linear function. Although well studied in the single-agent setting, many real-world scenarios involve multiple agents solving heterogeneous bandit problems, each with a different unknown parameter. Applying single agent algorithms independently ignores cross-agent similarity and learning opportunities. This paper investigates the optimal regret achievable in collaborative personalized linear bandits. We provide an information-theoretic lower bound that characterizes how the number of agents, the interaction rounds, and the degree of heterogeneity jointly affect regret. We then propose a new two-stage collaborative algorithm that achieves the optimal regret. Our analysis models heterogeneity via a hierarchical Bayesian framework and introduces a novel information-theoretic technique for bounding regret. Our results offer a complete characterization of when and how collaboration helps with a optimal regret bound $\tilde{O}(d\sqrt{mn})$, $\tilde{O}(dm^{1-\gamma}\sqrt{n})$, $\tilde{O}(dm\sqrt{n})$ for the number of rounds $n$ in the range of $(0, \frac{d}{m \sigma^2})$, $[\frac{d}{m^{2\gamma} \sigma^2}, \frac{d}{\sigma^2}]$ and $(\frac{d}{\sigma^2}, \infty)$ respectively, where $\sigma$ measures the level of heterogeneity, $m$ is the number of agents, and $\gamma\in[0, 1/2]$ is an absolute constant. In contrast, agents without collaboration achieve a regret bound $O(dm\sqrt{n})$ at best.

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