Strong NPHardness for Sparse Optimization with Concave Penalty Functions
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Proceedings of the 34th International Conference on Machine Learning, PMLR 70:740747, 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 NPhard 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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