Strong NP-Hardness for Sparse Optimization with Concave Penalty Functions

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Yichen Chen, Dongdong Ge, Mengdi Wang, Zizhuo Wang, Yinyu Ye, Hao Yin ;
Proceedings of the 34th International Conference on Machine Learning, PMLR 70:740-747, 2017.

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 $\mathcal{O}(n^{c_1}d^{c_2})$-optimal solution to the regularized sparse optimization problem is strongly NP-hard for any $c_1, c_2\in [0,1)$ such that $c_1+c_2<1$. 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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