We study the sequential batch learning problem in linear contextual bandits with finite action sets, where the decision maker is constrained to split incoming individuals into (at most) a fixed number of batches and can only observe outcomes for the individuals within a batch at the batch's end. Compared to both standard online contextual bandits learning or offline policy learning in contexutal bandits, this sequential batch learning problem provides a finer-grained formulation of many personalized sequential decision making problems in practical applications, including medical treatment in clinical trials, product recommendation in e-commerce and adaptive experiment design in crowdsourcing. We study two settings of the problem: one where the contexts are arbitrarily generated and the other where the contexts are \textit{iid} drawn from some distribution. In each setting, we establish a regret lower bound and provide an algorithm, whose regret upper bound nearly matches the lower bound. As an important insight revealed therefrom, in the former setting, we show that the number of batches required to achieve the fully online performance is polynomial in the time horizon, while for the latter setting, a pure-exploitation algorithm with a judicious batch partition scheme achieves the fully online performance even when the number of batches is less than logarithmic in the time horizon. Together, our results provide a near-complete characterization of sequential decision making in linear contextual bandits when batch constraints are present.

我们研究了线性环境中上下文臂中的顺序批处理学习问题,其中决策者被限制将个体分成（至多）固定数量的批处理,并且只能在批处理结束时观察批处理内的个体的结果.我们研究了问题的两种设置:一种是上下文是任意生成的,另一种是上下文是从某个分布中iid抽取的。在每个环境下，我们确定了遗憾下界，并提供了一个算法，其遗憾上界几乎与下界相匹配。

有限动作线性背景下的顺序批次学习