  In many statistical learning problems, it is desired that the optimal
solution conforms to an a priori known sparsity structure represented by a
directed acyclic graph. Inducing such structures by means of convex
regularizers requires nonsmooth penalty functions that exploit group
overlapping. Our study focuses on evaluating the proximal operator of the
Latent Overlapping Group lasso developed by Jacob et al. (2009). We implemented
an Alternating Direction Method of Multiplier with a sharing scheme to solve
large-scale instances of the underlying optimization problem efficiently. In
the absence of strong convexity, global linear convergence of the algorithm is
established using the error bound theory. More specifically, the paper
contributes to establishing primal and dual error bounds when the nonsmooth
component in the objective function does not have a polyhedral epigraph. We
also investigate the effect of the graph structure on the speed of convergence
of the algorithm. Detailed numerical simulation studies over different graph
structures supporting the proposed algorithm and two applications in learning
are provided.

Optimization and Control;Computation