  In this work, we propose a robust approach to design distributed controllers
for unknown-but-sparse linear and time-invariant systems. By leveraging modern
techniques in distributed controller synthesis and structured linear inverse
problems as applied to system identification, we show that near-optimal
distributed controllers can be learned with sub-linear sample complexity and
computed with near-linear time complexity, both measured with respect to the
dimension of the system. In particular, we provide sharp end-to-end guarantees
on the stability and the performance of the designed distributed controller and
prove that for sparse systems, the number of samples needed to guarantee robust
and near optimal performance of the designed controller can be significantly
smaller than the dimension of the system. Finally, we show that the proposed
optimization problem can be solved to global optimality with near-linear time
complexity by iteratively solving a series of small quadratic programs.
