This paper develops a theory for group Lasso using a concept called strong group sparsity. Our result shows that group Lasso is superior to standard Lasso for strongly group-sparse signals. This provides a convincing theoretical justification for using group sparse regularization when the underlying group structure is consistent with the data. Moreover, the theory predicts some limitations of the group Lasso formulation that are confirmed by simulation studies.
Finally, we shall mention that independent of the authors, results similar to those presented in this paper have also been obtained in @cite_1 with a similar technical analysis. However, while our paper studies the general group Lasso formulation, only the special case of multi-task learning is considered in @cite_1 .
@cite_1
2739284222 2144567071 2019899889 1769664844
The Group-Lasso is a well-known tool for joint regularization in machine learning methods. While the l1,2 and the l1,∞ version have been studied in detail and efficient algorithms exist, there are still open questions regarding other l1,p variants. We characterize conditions for solutions of the l1,p Group-Lasso for all p-norms with 1 ≤ p ≤ ∞, and we present a unified active set algorithm. For all p-norms, a highly efficient projected gradient algorithm is presented. This new algorithm enables us to compare the prediction performance of many variants of the Group-Lasso in a multi-task learning setting, where the aim is to solve many learning problems in parallel which are coupled via the Group-Lasso constraint. We conduct large-scale experiments on synthetic data and on two real-world data sets. In accordance with theoretical characterizations of the different norms we observe that the weak-coupling norms with p between 1.5 and 2 consistently outperform the strong-coupling norms with p ≫ 2. We propose a novel group regularization which we call exclusive lasso. Unlike the group lasso regularizer that assumes covarying variables in groups, the proposed exclusive lasso regularizer models the scenario when variables in the same group compete with each other. Analysis is presented to illustrate the properties of the proposed regularizer. We present a framework of kernel based multi-task feature selection algorithm based on the proposed exclusive lasso regularizer. An efficient algorithm is derived to solve the related optimization problem. Experiments with document categorization show that our approach outperforms state-of-theart algorithms for multi-task feature selection. We consider a special type of multi-label learning where class assignments of training examples are incomplete. As an example, an instance whose true class assignment is (c 1 , c 2 , c 3 ) is only assigned to class c 1 when it is used as a training sample. We refer to this problem as multi-label learning with incomplete class assignment. Incompletely labeled data is frequently encountered when the number of classes is very large (hundreds as in MIR Flickr dataset) or when there is a large ambiguity between classes (e.g., jet vs plane). In both cases, it is difficult for users to provide complete class assignments for objects. We propose a ranking based multi-label learning framework that explicitly addresses the challenge of learning from incompletely labeled data by exploiting the group lasso technique to combine the ranking errors. We present a learning algorithm that is empirically shown to be efficient for solving the related optimization problem. Our empirical study shows that the proposed framework is more effective than the state-of-the-art algorithms for multi-label learning in dealing with incompletely labeled data. We consider the problem of learning a structured multi-task regression, where the output consists of multiple responses that are related by a graph and the correlated response variables are dependent on the common inputs in a sparse but synergistic manner. Previous methods such as l1 l2-regularized multi-task regression assume that all of the output variables are equally related to the inputs, although in many real-world problems, outputs are related in a complex manner. In this paper, we propose graph-guided fused lasso (GFlasso) for structured multi-task regression that exploits the graph structure over the output variables. We introduce a novel penalty function based on fusion penalty to encourage highly correlated outputs to share a common set of relevant inputs. In addition, we propose a simple yet efficient proximal-gradient method for optimizing GFlasso that can also be applied to any optimization problems with a convex smooth loss and the general class of fusion penalty defined on arbitrary graph structures. By exploiting the structure of the non-smooth ''fusion penalty'', our method achieves a faster convergence rate than the standard first-order method, sub-gradient method, and is significantly more scalable than the widely adopted second-order cone-programming and quadratic-programming formulations. In addition, we provide an analysis of the consistency property of the GFlasso model. Experimental results not only demonstrate the superiority of GFlasso over the standard lasso but also show the efficiency and scalability of our proximal-gradient method.