  The estimation of causal treatment effects from observational data is a
fundamental problem in causal inference. To avoid bias, the effect estimator
must control for all confounders. Hence practitioners often collect data for as
many covariates as possible to raise the chances of including the relevant
confounders. While this addresses the bias, this has the side effect of
significantly increasing the number of data samples required to accurately
estimate the effect due to the increased dimensionality. In this work, we
consider the setting where out of a large number of covariates $X$ that satisfy
strong ignorability, an unknown sparse subset $S$ is sufficient to include to
achieve zero bias, i.e. $c$-equivalent to $X$. We propose a common objective
function involving outcomes across treatment cohorts with nonconvex joint
sparsity regularization that is guaranteed to recover $S$ with high probability
under a linear outcome model for $Y$ and subgaussian covariates for each of the
treatment cohort. This improves the effect estimation sample complexity so that
it scales with the cardinality of the sparse subset $S$ and $\log |X|$, as
opposed to the cardinality of the full set $X$. We validate our approach with
experiments on treatment effect estimation.
