An Augmented Lagrangian Method For Exclusive Lasso Problems

Group lasso have been extensively studied in both the statistics and machine learning fields,it can guarantee inter-group level sparsity,but not within intra-group level sparsity.Exclusive lasso(also known as elitist lasso)was born to achieve inter-group and intra-group sparsity.The exclusive lasso regularization has become pop-ular because of its superior performance in structural sparsity.Its complex nature poses di culties for the computation of high-dimensional machine learning models involving such a regularizer.In this paper,we propose an e(?)ective augmented La-grangian(ALM)method for exclusive lasso problems,and use superlinear convergent inexact semi-smooth Newton method to solve the subproblems.We systematically study the proximal mapping of conjugate functions of exclusive lasso regularizer and the corresponding generalized Jacobian.It is shown that the PPA iteration of the original problem is equivalent to the ALM iteration of the dual problem,and then it is proved that the calmness of the KKT canonically perturbed stationary point set-valued map is equivalent to the calmness of the set-valued mapS_{P P A}.The sequence x^kconverges Q-superlinearly,and the Q-superlinear convergence of the ALM algorithm.