Nearly unbiased variable selection under minimax concave penalty
极小极大凹惩罚下的近无偏变量选择AI 解读 ↓
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极小极大凹惩罚下的近无偏变量选择
- 研究问题
- 在变量数远大于样本量的线性回归中，如何兼顾变量选择的偏差、计算可行性、解路径连续性与选择一致性？
- 方法与思路
- 作者提出由极小极大凹惩罚MCP与PLUS算法组成的MC+方法，并用SURE构造自由度和风险估计，辅助惩罚水平选择。
- 材料陈述的贡献
- 摘要称，MC+在相应条件下具有符号恢复与极小极大收敛性质，并给出风险估计无偏性及部分惩罚估计的连续性条件。
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- 应核对设计矩阵与信号条件、主分支解的定义，以及非凸局部极小值如何影响选择一致性和噪声水平估计。
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原文摘要
We propose MC+, a fast, continuous, nearly unbiased and accurate method of penalized variable selection in high-dimensional linear regression. The LASSO is fast and continuous, but biased. The bias of the LASSO may prevent consistent variable selection. Subset selection is unbiased but computationally costly. The MC+ has two elements: a minimax concave penalty (MCP) and a penalized linear unbiased selection (PLUS) algorithm. The MCP provides the convexity of the penalized loss in sparse regions to the greatest extent given certain thresholds for variable selection and unbiasedness. The PLUS computes multiple exact local minimizers of a possibly nonconvex penalized loss function in a certain main branch of the graph of critical points of the penalized loss. Its output is a continuous piecewise linear path encompassing from the origin for infinite penalty to a least squares solution for zero penalty. We prove that at a universal penalty level, the MC+ has high probability of matching the signs of the unknowns, and thus correct selection, without assuming the strong irrepresentable condition required by the LASSO. This selection consistency applies to the case of p≫n, and is proved to hold for exactly the MC+ solution among possibly many local minimizers. We prove that the MC+ attains certain minimax convergence rates in probability for the estimation of regression coefficients in ℓr balls. We use the SURE method to derive degrees of freedom and Cp-type risk estimates for general penalized LSE, including the LASSO and MC+ estimators, and prove their unbiasedness. Based on the estimated degrees of freedom, we propose an estimator of the noise level for proper choice of the penalty level. For full rank designs and general sub-quadratic penalties, we provide necessary and sufficient conditions for the continuity of the penalized LSE. Simulation results overwhelmingly support our claim of superior variable selection properties and demonstrate the computational efficiency of the proposed method.