  We study sparse linear regression over a network of agents, modeled as an
undirected graph (with no centralized node). The estimation problem is
formulated as the minimization of the sum of the local LASSO loss functions
plus a quadratic penalty of the consensus constraint -- the latter being
instrumental to obtain distributed solution methods. While penalty-based
consensus methods have been extensively studied in the optimization literature,
their statistical and computational guarantees in the high dimensional setting
remain unclear. This work provides an answer to this open problem. Our
contribution is two-fold. First, we establish statistical consistency of the
estimator: under a suitable choice of the penalty parameter, the optimal
solution of the penalized problem achieves near optimal minimax rate
$\mathcal{O}(s \log d/N)$ in $\ell_2$-loss, where $s$ is the sparsity value,
$d$ is the ambient dimension, and $N$ is the total sample size in the network
-- this matches centralized sample rates. Second, we show that the
proximal-gradient algorithm applied to the penalized problem, which naturally
leads to distributed implementations, converges linearly up to a tolerance of
the order of the centralized statistical error -- the rate scales as
$\mathcal{O}(d)$, revealing an unavoidable speed-accuracy dilemma.Numerical
results demonstrate the tightness of the derived sample rate and convergence
rate scalings.
