  We study the problem of estimating high-dimensional regression models
regularized by a structured sparsity-inducing penalty that encodes prior
structural information on either the input or output variables. We consider two
widely adopted types of penalties of this kind as motivating examples: (1) the
general overlapping-group-lasso penalty, generalized from the group-lasso
penalty; and (2) the graph-guided-fused-lasso penalty, generalized from the
fused-lasso penalty. For both types of penalties, due to their nonseparability
and nonsmoothness, developing an efficient optimization method remains a
challenging problem. In this paper we propose a general optimization approach,
the smoothing proximal gradient (SPG) method, which can solve structured sparse
regression problems with any smooth convex loss under a wide spectrum of
structured sparsity-inducing penalties. Our approach combines a smoothing
technique with an effective proximal gradient method. It achieves a convergence
rate significantly faster than the standard first-order methods, subgradient
methods, and is much more scalable than the most widely used interior-point
methods. The efficiency and scalability of our method are demonstrated on both
simulation experiments and real genetic data sets.
