In sparse linear bandits, a learning agent sequentially selects an action and receive reward feedback, and the reward function depends linearly on a few coordinates of the covariates of the actions. This has applications in many real-world sequential decision making problems. In this paper, we propose a simple and computationally efficient sparse linear estimation method called PopArt that enjoys a tighter $\ell_1$ recovery guarantee compared to Lasso (Tibshirani, 1996) in many problems. Our bound naturally motivates an experimental design criterion that is convex and thus computationally efficient to solve. Based on our novel estimator and design criterion, we derive sparse linear bandit algorithms that enjoy improved regret upper bounds upon the state of the art (Hao et al., 2020), especially w.r.t. the geometry of the given action set. Finally, we prove a matching lower bound for sparse linear bandits in the data-poor regime, which closes the gap between upper and lower bounds in prior work.

翻译：在稀疏线性赌博机中，学习智能体依次选择动作并接收奖励反馈，且奖励函数线性依赖于动作协变量的少数坐标。此类问题在诸多现实序贯决策问题中具有应用价值。本文提出一种名为PopArt的简单且计算高效的稀疏线性估计方法，其在诸多问题中相较于Lasso（Tibshirani, 1996）享有更紧致的$\ell_1$恢复保证。该界自然引出一个凸性的实验设计准则，因此可高效求解。基于新提出的估计量与设计准则，我们推导出稀疏线性赌博机算法，其遗憾上界较现有最优方法（Hao等, 2020）有所改进，尤其是在给定动作集的几何性质方面。最后，我们证明了数据匮乏场景下稀疏线性赌博机的匹配下界，从而弥合了先前工作中上下界之间的差距。