Distributed Nonconvex Optimization for Sparse Representation

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.