M is an indicator matrix of size L × V , where L is the sum of the total sizes of all groups, and its ones show the link between the actual weights θ and their copies v. Following Yogatama and Smith (2014b) a constrained optimization problem is formed, that can be transformed to an augmented Lagrangian problem: Essentially, the problem becomes the iterative update of θ, v and u: Convergence Yogatama and Smith (2014b) proved that ADMM for sparse overlapping group lasso converges.
the overall expected risk (Vapnik, 1991) is the weighted sum of two components: the empirical risk and a regularization penalty term, expression referred to as "Loss+Penalty" by Hastie et al.