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Conference Paper: Efficient Euclidean projections onto the intersection of norm balls
Title | Efficient Euclidean projections onto the intersection of norm balls |
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Authors | |
Keywords | Auxiliary functions Bisection algorithms Building blockes Classical methods Empirical studies |
Issue Date | 2012 |
Publisher | International Machine Learning Society. |
Citation | The 29th International Conference on Machine Learning (ICML 2012). Edinburgh, Scotland, UK., 27 June-3 July 2012 In Proceedings of the 29th International Conference on Machine Learning, ICML-12, 2012, v. 1, p. 433-440 How to Cite? |
Abstract | Using sparse-inducing norms to learn robust models has received increasing attention from many fields for its attractive properties. Projection-based methods have been widely applied to learning tasks constrained by such norms. As a key building block of these methods, an efficient operator for Euclidean projection onto the intersection of ℓ 1 and ℓ 1,q norm balls (q = 2 or ∞) is proposed in this paper. We prove that the projection can be reduced to finding the root of an auxiliary function which is piecewise smooth and monotonic. Hence, a bisection algorithm is sufficient to solve the problem. We show that the time complexity of our solution is O(n + g log g) for q = 2 and O(n log n) for q = ∞), where n is the dimensionality of the vector to be projected and g is the number of disjoint groups; we confirm this complexity by experimentation. Empirical study reveals that our method achieves significantly better performance than classical methods in terms of running time and memory usage. We further show that embedded with our efficient projection operator, projection-based algorithms can solve regression problems with composite norm constraints more efficiently than other methods and give superior accuracy. Copyright 2012 by the author(s)/owner(s). |
Persistent Identifier | http://hdl.handle.net/10722/164926 |
ISBN |
DC Field | Value | Language |
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dc.contributor.author | Su, H | en_US |
dc.contributor.author | Yu, W | en_US |
dc.contributor.author | Li, FF | en_US |
dc.date.accessioned | 2012-09-20T08:12:25Z | - |
dc.date.available | 2012-09-20T08:12:25Z | - |
dc.date.issued | 2012 | en_US |
dc.identifier.citation | The 29th International Conference on Machine Learning (ICML 2012). Edinburgh, Scotland, UK., 27 June-3 July 2012 In Proceedings of the 29th International Conference on Machine Learning, ICML-12, 2012, v. 1, p. 433-440 | en_US |
dc.identifier.isbn | 9781450312851 | - |
dc.identifier.uri | http://hdl.handle.net/10722/164926 | - |
dc.description.abstract | Using sparse-inducing norms to learn robust models has received increasing attention from many fields for its attractive properties. Projection-based methods have been widely applied to learning tasks constrained by such norms. As a key building block of these methods, an efficient operator for Euclidean projection onto the intersection of ℓ 1 and ℓ 1,q norm balls (q = 2 or ∞) is proposed in this paper. We prove that the projection can be reduced to finding the root of an auxiliary function which is piecewise smooth and monotonic. Hence, a bisection algorithm is sufficient to solve the problem. We show that the time complexity of our solution is O(n + g log g) for q = 2 and O(n log n) for q = ∞), where n is the dimensionality of the vector to be projected and g is the number of disjoint groups; we confirm this complexity by experimentation. Empirical study reveals that our method achieves significantly better performance than classical methods in terms of running time and memory usage. We further show that embedded with our efficient projection operator, projection-based algorithms can solve regression problems with composite norm constraints more efficiently than other methods and give superior accuracy. Copyright 2012 by the author(s)/owner(s). | - |
dc.language | eng | en_US |
dc.publisher | International Machine Learning Society. | en_US |
dc.relation.ispartof | Proceedings of the 29th International Conference on Machine Learning, ICML-12 | en_US |
dc.subject | Auxiliary functions | - |
dc.subject | Bisection algorithms | - |
dc.subject | Building blockes | - |
dc.subject | Classical methods | - |
dc.subject | Empirical studies | - |
dc.title | Efficient Euclidean projections onto the intersection of norm balls | en_US |
dc.type | Conference_Paper | en_US |
dc.identifier.email | Su, H: haosu@cs.stanford.edu | - |
dc.identifier.email | Yu, W: wyu@cs.hku.hk | - |
dc.identifier.email | Li, FF: feifeili@cs.stanford.edu | - |
dc.description.nature | postprint | - |
dc.identifier.scopus | eid_2-s2.0-84867132579 | - |
dc.identifier.hkuros | 209420 | en_US |
dc.identifier.volume | 1 | - |
dc.identifier.spage | 433 | - |
dc.identifier.epage | 440 | - |
dc.publisher.place | Germany | - |
dc.description.other | The 29th International Conference on Machine Learning (ICML 2012). Edinburgh, Scotland, UK., 27 June-3 July 2012 In Proceedings of the 29th International Conference on Machine Learning, ICML-12, 2012, v. 1, p. 433-440 | - |