  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.
