Variable Selection is Hard
Proceedings of The 28th Conference on Learning Theory, PMLR 40:696-709, 2015.
Variable selection for sparse linear regression is the problem of finding, given an m\times p matrix B and a target vector \bfy, a sparse vector \bfx such that B\bfx approximately equals \bfy. Assuming a standard complexity hypothesis, we show that no polynomial-time algorithm can find a k’-sparse \bfx with \|B\bfx-\bfy\|^2\le h(m,p), where k’=k⋅2^\log ^1-δ p and h(m,p)= p^C_1 m^1-C_2, where δ>0,C_1>0,C_2>0 are arbitrary. This is true even under the promise that there is an unknown k-sparse vector \bfx^* satisfying B\bfx^*=\bfy. We prove a similar result for a statistical version of the problem in which the data are corrupted by noise. To the authors’ knowledge, these are the first hardness results for sparse regression that apply when the algorithm simultaneously has k’>k and h(m,p)>0.