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Biometrika 2009 96(3):513-527; doi:10.1093/biomet/asp038
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© 2009 Biometrika Trust

Article

Adaptive regularization using the entire solution surface

S. Wu, X. Shen and C. J. Geyer

School of Statistics, University of Minnesota, 313 Ford Hall, 224 Church Street S. E., Minneapolis, Minnesota 55455, U.S.A. swu{at}stat.umn.edu xshen{at}stat.umn.edu charlie{at}stat.umn.edu

Received for publication 1 November 2007. Revision received 1 January 2009.
  Abstract

Several sparseness penalties have been suggested for delivery of good predictive performance in automatic variable selection within the framework of regularization. All assume that the true model is sparse. We propose a penalty, a convex combination of the L1- and L{infty}-norms, that adapts to a variety of situations including sparseness and nonsparseness, grouping and nongrouping. The proposed penalty performs grouping and adaptive regularization. In addition, we introduce a novel homotopy algorithm utilizing subgradients for developing regularization solution surfaces involving multiple regularizers. This permits efficient computation and adaptive tuning. Numerical experiments are conducted using simulation. In simulated and real examples, the proposed penalty compares well against popular alternatives.

Key Words: Homotopy • Lasso • L1-norm • L{infty}-norm • Subgradient • Support vector machine • Variable grouping and selection


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