Structured Sparsity via Alternating Direction Methods

We consider a class of sparse learning problems in high dimensional feature space regularized by a structured sparsity-inducing norm which incorporates prior knowledge of the group structure of the features. Such problems often pose a considerable challenge to optimization algorithms due to the non-smoothness and non-separability of the regularization term. In this paper, we focus on two commonly adopted sparsity-inducing regularization terms, the overlapping Group Lasso penalty l₁/l₂-norm and the l₁/l_-norm. We propose a unified framework based on the augmented Lagrangian method, under which problems with both types of regularization and their variants can be efficiently solved. As the core building-block of this framework, we develop new algorithms using an alternating partial-linearization/splitting technique, and we prove that the accelerated versions of these algorithms require O (1ε) iterations to obtain an ε-optimal solution. To demonstrate the efficiency and relevance of our algorithms, we test them on a collection of data sets and apply them to two real-world problems to compare the relative merits of the two norms.