An accelerated Newton's method for projections onto the ℓ1-ball

Paul Rodriguez

doi:10.1109/MLSP.2017.8168161

An accelerated Newton's method for projections onto the ℓ₁-ball

Paul Rodriguez

Sección de Electricidad y Electrónica

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

7 Scopus citations

Abstract

We present a simple and computationally efficient algorithm, based on the accelerated Newton's method, to solve the root finding problem associated with the projection onto the ℓ₁-ball problem. Considering an interpretation of the Michelot's algorithm as Newton method, our algorithm can be understood as an accelerated version of the Michelot's algorithm, that needs significantly less major iterations to converge to the solution. Although the worst-case performance of the propose algorithm is O(n²), it exhibits in practice an O(n) performance and it is empirically demonstrated that it is competitive or faster than existing methods.

Original language	English
Title of host publication	2017 IEEE International Workshop on Machine Learning for Signal Processing, MLSP 2017 - Proceedings
Editors	Naonori Ueda, Jen-Tzung Chien, Tomoko Matsui, Jan Larsen, Shinji Watanabe
Publisher	IEEE Computer Society
Pages	1-4
Number of pages	4
ISBN (Electronic)	9781509063413
DOIs	https://doi.org/10.1109/MLSP.2017.8168161
State	Published - 5 Dec 2017
Event	2017 IEEE International Workshop on Machine Learning for Signal Processing, MLSP 2017 - Tokyo, Japan Duration: 25 Sep 2017 → 28 Sep 2017

Publication series

Name	IEEE International Workshop on Machine Learning for Signal Processing, MLSP
Volume	2017-September
ISSN (Print)	2161-0363
ISSN (Electronic)	2161-0371

Conference

Conference	2017 IEEE International Workshop on Machine Learning for Signal Processing, MLSP 2017
Country/Territory	Japan
City	Tokyo
Period	25/09/17 → 28/09/17

Keywords

Accelerated Newton's method
Simplex
ℓ-Norm ball

Access to Document

10.1109/MLSP.2017.8168161

Cite this

Rodriguez, P. (2017). An accelerated Newton's method for projections onto the ℓ₁-ball. In N. Ueda, J.-T. Chien, T. Matsui, J. Larsen, & S. Watanabe (Eds.), 2017 IEEE International Workshop on Machine Learning for Signal Processing, MLSP 2017 - Proceedings (pp. 1-4). (IEEE International Workshop on Machine Learning for Signal Processing, MLSP; Vol. 2017-September). IEEE Computer Society. https://doi.org/10.1109/MLSP.2017.8168161

Rodriguez, Paul. / An accelerated Newton's method for projections onto the ℓ₁-ball. 2017 IEEE International Workshop on Machine Learning for Signal Processing, MLSP 2017 - Proceedings. editor / Naonori Ueda ; Jen-Tzung Chien ; Tomoko Matsui ; Jan Larsen ; Shinji Watanabe. IEEE Computer Society, 2017. pp. 1-4 (IEEE International Workshop on Machine Learning for Signal Processing, MLSP).

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abstract = "We present a simple and computationally efficient algorithm, based on the accelerated Newton's method, to solve the root finding problem associated with the projection onto the ℓ1-ball problem. Considering an interpretation of the Michelot's algorithm as Newton method, our algorithm can be understood as an accelerated version of the Michelot's algorithm, that needs significantly less major iterations to converge to the solution. Although the worst-case performance of the propose algorithm is O(n2), it exhibits in practice an O(n) performance and it is empirically demonstrated that it is competitive or faster than existing methods.",

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Rodriguez, P 2017, An accelerated Newton's method for projections onto the ℓ₁-ball. in N Ueda, J-T Chien, T Matsui, J Larsen & S Watanabe (eds), 2017 IEEE International Workshop on Machine Learning for Signal Processing, MLSP 2017 - Proceedings. IEEE International Workshop on Machine Learning for Signal Processing, MLSP, vol. 2017-September, IEEE Computer Society, pp. 1-4, 2017 IEEE International Workshop on Machine Learning for Signal Processing, MLSP 2017, Tokyo, Japan, 25/09/17. https://doi.org/10.1109/MLSP.2017.8168161

An accelerated Newton's method for projections onto the ℓ₁-ball. / Rodriguez, Paul.
2017 IEEE International Workshop on Machine Learning for Signal Processing, MLSP 2017 - Proceedings. ed. / Naonori Ueda; Jen-Tzung Chien; Tomoko Matsui; Jan Larsen; Shinji Watanabe. IEEE Computer Society, 2017. p. 1-4 (IEEE International Workshop on Machine Learning for Signal Processing, MLSP; Vol. 2017-September).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Rodriguez P. An accelerated Newton's method for projections onto the ℓ₁-ball. In Ueda N, Chien JT, Matsui T, Larsen J, Watanabe S, editors, 2017 IEEE International Workshop on Machine Learning for Signal Processing, MLSP 2017 - Proceedings. IEEE Computer Society. 2017. p. 1-4. (IEEE International Workshop on Machine Learning for Signal Processing, MLSP). doi: 10.1109/MLSP.2017.8168161