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k-Support and Ordered Weighted Sparsity for Overlapping Groups: Hardness and Algorithms
Cong Han Lim · Stephen Wright

Mon Dec 04 06:30 PM -- 10:30 PM (PST) @ Pacific Ballroom #56

The k-support and OWL norms generalize the l1 norm, providing better prediction accuracy and better handling of correlated variables. We study the norms obtained from extending the k-support norm and OWL norms to the setting in which there are overlapping groups. The resulting norms are in general NP-hard to compute, but they are tractable for certain collections of groups. To demonstrate this fact, we develop a dynamic program for the problem of projecting onto the set of vectors supported by a fixed number of groups. Our dynamic program utilizes tree decompositions and its complexity scales with the treewidth. This program can be converted to an extended formulation which, for the associated group structure, models the k-group support norms and an overlapping group variant of the ordered weighted l1 norm. Numerical results demonstrate the efficacy of the new penalties.

Author Information

Cong Han Lim (Georgia Tech)
Stephen Wright (UW-Madison)

Steve Wright is a Professor of Computer Sciences at the University of Wisconsin-Madison. His research interests lie in computational optimization and its applications to science and engineering. Prior to joining UW-Madison in 2001, Wright was a Senior Computer Scientist (1997-2001) and Computer Scientist (1990-1997) at Argonne National Laboratory, and Professor of Computer Science at the University of Chicago (2000-2001). He is the past Chair of the Mathematical Optimization Society (formerly the Mathematical Programming Society), the leading professional society in optimization, and a member of the Board of the Society for Industrial and Applied Mathematics (SIAM). Wright is the author or co-author of four widely used books in numerical optimization, including "Primal Dual Interior-Point Methods" (SIAM, 1997) and "Numerical Optimization" (with J. Nocedal, Second Edition, Springer, 2006). He has also authored over 85 refereed journal papers on optimization theory, algorithms, software, and applications. He is coauthor of widely used interior-point software for linear and quadratic optimization. His recent research includes algorithms, applications, and theory for sparse optimization (including applications in compressed sensing and machine learning).

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