  This paper introduces the \emph{confounded pure exploration transductive
linear bandit} (\texttt{CPET-LB}) problem. As a motivating example, often
online services cannot directly assign users to specific control or treatment
experiences either for business or practical reasons. In these settings,
naively comparing treatment and control groups that may result from
self-selection can lead to biased estimates of underlying treatment effects.
Instead, online services can employ a properly randomized encouragement that
incentivizes users toward a specific treatment. Our methodology provides online
services with an adaptive experimental design approach for learning the
best-performing treatment for such \textit{encouragement designs}. We consider
a more general underlying model captured by a linear structural equation and
formulate pure exploration linear bandits in this setting. Though pure
exploration has been extensively studied in standard adaptive experimental
design settings, we believe this is the first work considering a setting where
noise is confounded. Elimination-style algorithms using experimental design
methods in combination with a novel finite-time confidence interval on an
instrumental variable style estimator are presented with sample complexity
upper bounds nearly matching a minimax lower bound. Finally, experiments are
conducted that demonstrate the efficacy of our approach.
