Efficient Euclidean projections in linear time

Jun Liu, Jieping Ye

Research output: Chapter in Book/Report/Conference proceedingConference contribution

104 Scopus citations

Abstract

We consider the problem of computing the Euclidean projection of a vector of length n onto a closed convex set including the ℓ1 and the specialized polyhedra employed in (Shalev-Shwartz & Singer, 2006). These problems have played building block roles in solving several ℓ1-norm based sparse learning problems. Existing methods have a worst-case time complexity of O(n log n). In this paper, we propose to cast both Euclidean projections as root finding problems associated with specific auxiliary functions, which can be solved in linear time via bisection. We further make use of the special structure of the auxiliary functions, and propose an improved bisection algorithm. Empirical studies demonstrate that the proposed algorithms are much more efficient than the competing ones for computing the projections.

Original languageEnglish (US)
Title of host publicationProceedings of the 26th International Conference On Machine Learning, ICML 2009
Pages657-664
Number of pages8
StatePublished - Dec 9 2009
Event26th International Conference On Machine Learning, ICML 2009 - Montreal, QC, Canada
Duration: Jun 14 2009Jun 18 2009

Publication series

NameProceedings of the 26th International Conference On Machine Learning, ICML 2009

Other

Other26th International Conference On Machine Learning, ICML 2009
CountryCanada
CityMontreal, QC
Period6/14/096/18/09

ASJC Scopus subject areas

  • Artificial Intelligence
  • Computer Networks and Communications
  • Software

Cite this

Liu, J., & Ye, J. (2009). Efficient Euclidean projections in linear time. In Proceedings of the 26th International Conference On Machine Learning, ICML 2009 (pp. 657-664). (Proceedings of the 26th International Conference On Machine Learning, ICML 2009).