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Structured Sparsity via Alternating Direction Methods

Zhiwei (Tony) Qin (zq2107***at***columbia.edu)
Donald Goldfarb (goldfarb***at***columbia.edu)

Abstract: 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 $l_1/l_2$-norm and the $l_1/l_\infty$-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 $O(\frac{1}{\sqrt{\epsilon}})$ iterations to obtain an $\epsilon$-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.

Keywords: structured sparsity, overlapping Group Lasso, alternating directions methods, variable splitting, augmented Lagrangian

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Citation:

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Entry Submitted: 05/03/2011
Entry Accepted: 05/03/2011
Entry Last Modified: 12/14/2011

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