Best subset selection is robust against design dependence
最优子集选择对设计相关性的稳健性AI 解读 ↓
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最优子集选择对设计相关性的稳健性
- 研究问题
- 当目标稀疏度不小于真实稀疏度、解释变量高度相关时，最优子集及可计算的近优子集能否恢复或筛入真实模型？
- 方法与思路
- 作者以不依赖受限特征值的可识别性间隔条件分析变量选择，并将优化误差纳入理论，结合两阶段迭代硬阈值算法。
- 材料陈述的贡献
- 摘要称该条件足以且近乎必要地保证模型恢复，相关性质可传递至近优子集，并给出对数步数内可靠筛选的算法。
- 阅读关注点
- 应核对可识别性间隔如何随样本与信号变化、近优残差平方和的容许误差，以及筛选后精确搜索的计算成本。
- 阅读边界
- 这是 arXiv 预印本，同行评审及期刊接收未核验；摘要未列明设计与噪声条件，模拟比较不能证明普遍优越。
原文摘要
Best subset selection (BSS) is widely known as the holy grail for high-dimensional variable selection. Nevertheless, the notorious NP-hardness of BSS substantially restricts its practical application and also discourages its theoretical development to some extent, particularly in the current era of big data. In this paper, we investigate the variable selection properties of BSS when its target sparsity is greater than or equal to the true sparsity. Our main message is that BSS is robust against design dependence in terms of achieving model consistency and sure screening, and more importantly, that such robustness can be propagated to the near best subsets that are computationally tangible. Specifically, we introduce an identifiability margin condition that is free of restricted eigenvalues and show that it is sufficient and nearly necessary for BSS to exactly recover the true model. A relaxed version of this condition is also sufficient for BSS to achieve the sure screening property. Moreover, taking optimization error into account, we find that all the established statistical properties for the exact best subset carry over to any near best subset whose residual sum of squares is close enough to that of the best one. In particular, a two-stage fully corrective iterative hard thresholding (IHT) algorithm can provably find a sparse sure screening subset within logarithmic steps; another round of exact BSS within this set can recover the true model. The simulation studies and real data examples show that IHT yields lower false discovery rates and higher true positive rates than the competing approaches including LASSO, SCAD and Sure Independence Screening (SIS), especially under highly correlated design.