  Traditional models of active learning assume a learner can directly
manipulate or query a covariate $X$ in order to study its relationship with a
response $Y$. However, if $X$ is a feature of a complex system, it may be
possible only to indirectly influence $X$ by manipulating a control variable
$Z$, a scenario we refer to as Indirect Active Learning. Under a nonparametric
model of Indirect Active Learning with a fixed budget, we study minimax
convergence rates for estimating the relationship between $X$ and $Y$ locally
at a point, obtaining different rates depending on the complexities and noise
levels of the relationships between $Z$ and $X$ and between $X$ and $Y$. We
also identify minimax rates for passive learning under comparable assumptions.
In many cases, our results show that, while there is an asymptotic benefit to
active learning, this benefit is fully realized by a simple two-stage learner
that runs two passive experiments in sequence.
