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In this paper we analyze a family of general random block coordinate descentmethods for the minimization of $\ell_0$ regularized optimization problems,i.e. the objective function is composed of a smooth convex function and the$\ell_0$ regularization. Our family of methods covers particular cases such asrandom block coordinate gradient descent and random proximal coordinate descentmethods. We analyze necessary optimality conditions for this nonconvex $\ell_0$regularized problem and devise a separation of the set of local minima intorestricted classes based on approximation versions of the objective function.We provide a unified analysis of the almost sure convergence for this family ofblock coordinate descent algorithms and prove that, for each approximationversion, the limit points are local minima from the corresponding restrictedclass of local minimizers. Under the strong convexity assumption, we provelinear convergence in probability for our family of methods.

Iteration complexity analysis of random coordinate descent methods for $\ell_0$ regularized convex problems