  Instrumental variable (IV) methods are used to estimate causal effects in
settings with unobserved confounding, where we cannot directly experiment on
the treatment variable. Instruments are variables which only affect the outcome
indirectly via the treatment variable(s). Most IV applications focus on
low-dimensional treatments and crucially require at least as many instruments
as treatments. This assumption is restrictive: in the natural sciences we often
seek to infer causal effects of high-dimensional treatments (e.g., the effect
of gene expressions or microbiota on health and disease), but can only run few
experiments with a limited number of instruments (e.g., drugs or antibiotics).
In such underspecified problems, the full treatment effect is not identifiable
in a single experiment even in the linear case. We show that one can still
reliably recover the projection of the treatment effect onto the instrumented
subspace and develop techniques to consistently combine such partial estimates
from different sets of instruments. We then leverage our combined estimators in
an algorithm that iteratively proposes the most informative instruments at each
round of experimentation to maximize the overall information about the full
causal effect.
