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Strong NP-Hardness for Sparse Optimization with Concave Penalty Functions
Yichen Chen · Dongdong Ge · Mengdi Wang · Zizhuo Wang · Yinyu Ye · Hao Yin
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.
Summary/NotesAuthor Information
Yichen Chen (Princeton University)
Dongdong Ge (Shanghai University of Finance and Economics)
Mengdi Wang (Princeton University)
Zizhuo Wang (University of Minnesota)
Yinyu Ye (Standord)
Hao Yin (Stanford University)
Related Events (a corresponding poster, oral, or spotlight)
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2017 Poster: Strong NP-Hardness for Sparse Optimization with Concave Penalty Functions »
Mon. Aug 7th 08:30 AM -- 12:00 PM Room Gallery #27
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