We study the problem of dynamic batch learning in high-dimensional sparse linear contextual bandits, where a decision maker, under a given maximum-number-of-batch constraint and only able to observe rewards at the end of each batch, can dynamically decide how many individuals to include in the next batch (at the end of the current batch) and what personalized action-selection scheme to adopt within each batch. Such batch constraints are ubiquitous in a variety of practical contexts, including personalized product offerings in marketing and medical treatment selection in clinical trials. We characterize the fundamental learning limit in this problem via a regret lower bound and provide a matching upper bound (up to log factors), thus prescribing an optimal scheme for this problem. To the best of our knowledge, our work provides the first inroad into a theoretical understanding of dynamic batch learning in high-dimensional sparse linear contextual bandits. Notably, even a special case of our result -- when no batch constraint is present -- yields that the simple exploration-free algorithm using the LASSO estimator already achieves the minimax optimal regret bound for standard online learning in high-dimensional linear contextual bandits (for the no-margin case), a result that appears unknown in the emerging literature of high-dimensional contextual bandits.
翻译:我们研究在高度分散的线性背景土匪中动态批量学习的问题,在这个批量中,决策者在给定的最大批量限制下,只能观察每批末端的奖赏,能够动态地决定下批(在本批末端)要包括多少人,以及每批中要采用什么个性化的行动选择计划。这种批量限制在各种实际环境中普遍存在,包括在临床试验中销售和医疗治疗选择中提供个性化产品。我们通过低感应圈来描述这一问题的基本学习限度,并提供一个匹配的上限(最多为日志因素),从而为这一问题规定一个最佳办法。根据我们的最佳知识,我们的工作首次在理论上理解了在高度分散的线性线性背景土匪中进行动态批次学习的情况。值得注意的是,即使我们的结果的一个特殊案例 -- -- 当没有批量限制时 -- -- 也表明,使用LASSO估计器的简单探索性算法已经实现了在高度直线性直线性直线型背景土匪中标准在线学习的微成型最佳遗憾(在不甚为背景的图像中出现)。