Strong NP-hardness for sparse optimization with concave penalty functions

Yichen Chen; Dongdong Ge; Mengdi Wang; Zizhuo Wang; Yinyu Ye; Hao Yin

Strong NP-hardness for sparse optimization with concave penalty functions

Yichen Chen, Dongdong Ge, Mengdi Wang, Zizhuo Wang, Yinyu Ye, Hao Yin

Industrial and Systems Engineering

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

4 Scopus citations

Abstract

Consider the regularized sparse minimization problem, which involves empirical sums of loss functions for n data points (each of dimension d) and a nonconvex sparsity penalty. We prove that finding an O(n^C1d^C2)-optimal solution to the regularized sparse optimization problem is strongly NP-hard for any C₁, C₂ ∈ [0, 1) such that C₁ + C₂ < 1. The result applies to a broad class of loss functions and sparse penalty functions. It suggests that one cannot even approximately solve the sparse optimization problem in polynomial time, unless P = NP.

Original language	English (US)
Title of host publication	34th International Conference on Machine Learning, ICML 2017
Publisher	International Machine Learning Society (IMLS)
Pages	1230-1251
Number of pages	22
ISBN (Electronic)	9781510855144
State	Published - 2017
Event	34th International Conference on Machine Learning, ICML 2017 - Sydney, Australia Duration: Aug 6 2017 → Aug 11 2017

Publication series

Name	34th International Conference on Machine Learning, ICML 2017
Volume	2

Other

Other	34th International Conference on Machine Learning, ICML 2017
Country/Territory	Australia
City	Sydney
Period	8/6/17 → 8/11/17

Bibliographical note

Publisher Copyright:
© 2017 by the author(s).

Keywords

Computational complexity
Concave penalty
NP-hardness
Nonconvex optimization
Sparsity

Cite this

Strong NP-hardness for sparse optimization with concave penalty functions. / Chen, Yichen; Ge, Dongdong; Wang, Mengdi; Wang, Zizhuo; Ye, Yinyu; Yin, Hao.

34th International Conference on Machine Learning, ICML 2017. International Machine Learning Society (IMLS), 2017. p. 1230-1251 (34th International Conference on Machine Learning, ICML 2017; Vol. 2).

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

Chen, Y, Ge, D, Wang, M, Wang, Z, Ye, Y & Yin, H 2017, Strong NP-hardness for sparse optimization with concave penalty functions. in 34th International Conference on Machine Learning, ICML 2017. 34th International Conference on Machine Learning, ICML 2017, vol. 2, International Machine Learning Society (IMLS), pp. 1230-1251, 34th International Conference on Machine Learning, ICML 2017, Sydney, Australia, 8/6/17.

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abstract = "Consider the regularized sparse minimization problem, which involves empirical sums of loss functions for n data points (each of dimension d) and a nonconvex sparsity penalty. We prove that finding an O(nC1dC2)-optimal solution to the regularized sparse optimization problem is strongly NP-hard for any C1, C2 ∈ [0, 1) such that C1 + C2 < 1. The result applies to a broad class of loss functions and sparse penalty functions. It suggests that one cannot even approximately solve the sparse optimization problem in polynomial time, unless P = NP.",

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AU - Wang, Zizhuo

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AU - Yin, Hao

PY - 2017

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N2 - Consider the regularized sparse minimization problem, which involves empirical sums of loss functions for n data points (each of dimension d) and a nonconvex sparsity penalty. We prove that finding an O(nC1dC2)-optimal solution to the regularized sparse optimization problem is strongly NP-hard for any C1, C2 ∈ [0, 1) such that C1 + C2 < 1. The result applies to a broad class of loss functions and sparse penalty functions. It suggests that one cannot even approximately solve the sparse optimization problem in polynomial time, unless P = NP.

AB - Consider the regularized sparse minimization problem, which involves empirical sums of loss functions for n data points (each of dimension d) and a nonconvex sparsity penalty. We prove that finding an O(nC1dC2)-optimal solution to the regularized sparse optimization problem is strongly NP-hard for any C1, C2 ∈ [0, 1) such that C1 + C2 < 1. The result applies to a broad class of loss functions and sparse penalty functions. It suggests that one cannot even approximately solve the sparse optimization problem in polynomial time, unless P = NP.

KW - Computational complexity

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KW - Nonconvex optimization

KW - Sparsity

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ER -

Strong NP-hardness for sparse optimization with concave penalty functions

Abstract

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Keywords

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