Optimizing Costly Functions with Simple Constraints: A Limited-Memory Projected Quasi-Newton Algorithm

Mark Schmidt, Ewout Berg, Michael Friedlander, Kevin Murphy
; Proceedings of the Twelth International Conference on Artificial Intelligence and Statistics, PMLR 5:456-463, 2009.

Abstract

An optimization algorithm for minimizing a smooth function over a convex set is described. Each iteration of the method computes a descent direction by minimizing, over the original constraints, a diagonal plus low-rank quadratic approximation to the function. The quadratic approximation is constructed using a limited-memory quasi-Newton update. The method is suitable for large-scale problems where evaluation of the function is substantially more expensive than projection onto the constraint set. Numerical experiments on one-norm regularized test problems indicate that the proposed method is competitive with state-of-the-art methods such as bound-constrained L-BFGS and orthant-wise descent. We further show that the method generalizes to a wide class of problems, and substantially improves on state-of-the-art methods for problems such as learning the structure of Gaussian graphical models and Markov random fields.

Cite this Paper

BibTeX
@InProceedings{pmlr-v5-schmidt09a, title = {Optimizing Costly Functions with Simple Constraints: A Limited-Memory Projected Quasi-Newton Algorithm}, author = {Mark Schmidt and Ewout Berg and Michael Friedlander and Kevin Murphy}, booktitle = {Proceedings of the Twelth International Conference on Artificial Intelligence and Statistics}, pages = {456--463}, year = {2009}, editor = {David van Dyk and Max Welling}, volume = {5}, series = {Proceedings of Machine Learning Research}, address = {Hilton Clearwater Beach Resort, Clearwater Beach, Florida USA}, month = {16--18 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v5/schmidt09a/schmidt09a.pdf}, url = {http://proceedings.mlr.press/v5/schmidt09a.html}, abstract = {An optimization algorithm for minimizing a smooth function over a convex set is described. Each iteration of the method computes a descent direction by minimizing, over the original constraints, a diagonal plus low-rank quadratic approximation to the function. The quadratic approximation is constructed using a limited-memory quasi-Newton update. The method is suitable for large-scale problems where evaluation of the function is substantially more expensive than projection onto the constraint set. Numerical experiments on one-norm regularized test problems indicate that the proposed method is competitive with state-of-the-art methods such as bound-constrained L-BFGS and orthant-wise descent. We further show that the method generalizes to a wide class of problems, and substantially improves on state-of-the-art methods for problems such as learning the structure of Gaussian graphical models and Markov random fields.} }
Endnote
%0 Conference Paper %T Optimizing Costly Functions with Simple Constraints: A Limited-Memory Projected Quasi-Newton Algorithm %A Mark Schmidt %A Ewout Berg %A Michael Friedlander %A Kevin Murphy %B Proceedings of the Twelth International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2009 %E David van Dyk %E Max Welling %F pmlr-v5-schmidt09a %I PMLR %J Proceedings of Machine Learning Research %P 456--463 %U http://proceedings.mlr.press %V 5 %W PMLR %X An optimization algorithm for minimizing a smooth function over a convex set is described. Each iteration of the method computes a descent direction by minimizing, over the original constraints, a diagonal plus low-rank quadratic approximation to the function. The quadratic approximation is constructed using a limited-memory quasi-Newton update. The method is suitable for large-scale problems where evaluation of the function is substantially more expensive than projection onto the constraint set. Numerical experiments on one-norm regularized test problems indicate that the proposed method is competitive with state-of-the-art methods such as bound-constrained L-BFGS and orthant-wise descent. We further show that the method generalizes to a wide class of problems, and substantially improves on state-of-the-art methods for problems such as learning the structure of Gaussian graphical models and Markov random fields.
RIS
TY - CPAPER TI - Optimizing Costly Functions with Simple Constraints: A Limited-Memory Projected Quasi-Newton Algorithm AU - Mark Schmidt AU - Ewout Berg AU - Michael Friedlander AU - Kevin Murphy BT - Proceedings of the Twelth International Conference on Artificial Intelligence and Statistics PY - 2009/04/15 DA - 2009/04/15 ED - David van Dyk ED - Max Welling ID - pmlr-v5-schmidt09a PB - PMLR SP - 456 DP - PMLR EP - 463 L1 - http://proceedings.mlr.press/v5/schmidt09a/schmidt09a.pdf UR - http://proceedings.mlr.press/v5/schmidt09a.html AB - An optimization algorithm for minimizing a smooth function over a convex set is described. Each iteration of the method computes a descent direction by minimizing, over the original constraints, a diagonal plus low-rank quadratic approximation to the function. The quadratic approximation is constructed using a limited-memory quasi-Newton update. The method is suitable for large-scale problems where evaluation of the function is substantially more expensive than projection onto the constraint set. Numerical experiments on one-norm regularized test problems indicate that the proposed method is competitive with state-of-the-art methods such as bound-constrained L-BFGS and orthant-wise descent. We further show that the method generalizes to a wide class of problems, and substantially improves on state-of-the-art methods for problems such as learning the structure of Gaussian graphical models and Markov random fields. ER -
APA
Schmidt, M., Berg, E., Friedlander, M. & Murphy, K.. (2009). Optimizing Costly Functions with Simple Constraints: A Limited-Memory Projected Quasi-Newton Algorithm. Proceedings of the Twelth International Conference on Artificial Intelligence and Statistics, in PMLR 5:456-463

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