  The problem of causal inference with panel data is a central econometric
question. The following is a fundamental version of this problem: Let $M^*$ be
a low rank matrix and $E$ be a zero-mean noise matrix. For a `treatment' matrix
$Z$ with entries in $\{0,1\}$ we observe the matrix $O$ with entries $O_{ij} :=
M^*_{ij} + E_{ij} + \mathcal{T}_{ij} Z_{ij}$ where $\mathcal{T}_{ij} $ are
unknown, heterogenous treatment effects. The problem requires we estimate the
average treatment effect $\tau^* := \sum_{ij} \mathcal{T}_{ij} Z_{ij} /
\sum_{ij} Z_{ij}$. The synthetic control paradigm provides an approach to
estimating $\tau^*$ when $Z$ places support on a single row. This paper extends
that framework to allow rate-optimal recovery of $\tau^*$ for general $Z$, thus
broadly expanding its applicability. Our guarantees are the first of their type
in this general setting. Computational experiments on synthetic and real-world
data show a substantial advantage over competing estimators.
