  We study sparse group Lasso for high-dimensional double sparse linear
regression, where the parameter of interest is simultaneously element-wise and
group-wise sparse. This problem is an important instance of the simultaneously
structured model -- an actively studied topic in statistics and machine
learning. In the noiseless case, matching upper and lower bounds on sample
complexity are established for the exact recovery of sparse vectors and for
stable estimation of approximately sparse vectors, respectively. In the noisy
case, upper and matching minimax lower bounds for estimation error are
obtained. We also consider the debiased sparse group Lasso and investigate its
asymptotic property for the purpose of statistical inference. Finally,
numerical studies are provided to support the theoretical results.
