  We consider a non-convex constrained Lagrangian formulation of a fundamental
bi-criteria optimization problem for variable selection in statistical
learning; the two criteria are a smooth (possibly) nonconvex loss function,
measuring the fitness of the model to data, and the latter function is a
difference-of-convex (DC) regularization, employed to promote some extra
structure on the solution, like sparsity. This general class of nonconvex
problems arises in many big-data applications, from statistical machine
learning to physical sciences and engineering. We develop the first unified
distributed algorithmic framework for these problems and establish its
asymptotic convergence to d-stationary solutions. Two key features of the
method are: i) it can be implemented on arbitrary networks (digraphs) with
(possibly) time-varying connectivity; and ii) it does not require the
restrictive assumption that the (sub)gradient of the objective function is
bounded, which enlarges significantly the class of statistical learning
problems that can be solved with convergence guarantees.
