竞争的代价：影响尺寸异质性在高维中的重要性
The Price of Competition: Effect Size Heterogeneity Matters in High Dimensions
摘要：
在高维稀疏回归中，在固定稀疏水平的同时增加信噪比是否总是会导致更好的模型选择？对于高维稀疏回归问题，令人惊讶的是，在本文中，我们回答了这个问题的否定在政权的线性稀疏的Lasso方法，依赖于一个新的概念，我们长期效应大小异质性。粗略地说，如果一个回归系数向量的非零项具有显著不同的量值，则该向量具有高效应量异质性。从这个新的措施的观点，我们证明了假阳性率和真阳性率实现最佳的权衡一致沿Lasso路径时，该措施是最大的在一定意义上，和最坏的权衡时，它是最小的意义上，所有非零效应大小大致相等。此外，我们证明了第一次假选择发生得更早时，效应大小异质性是最小的，比当它是最大的。这两种现象的根本原因是，打个比方，在进入模型时，效应大小相同的变量之间的“竞争”。总之，我们的研究结果表明，效应量异质性应作为一个重要的补充措施，在高维回归问题的分析中的稀疏性的回归系数。我们的证明使用近似消息传递理论的技术，以及一种新的技术来估计第一个假变量的秩。
英文摘要：
In high-dimensional sparse regression, would increasing the signal-to-noise ratio while fixing the sparsity level always lead to better model selection? For high-dimensional sparse regression problems, surprisingly, in this paper we answer this question in the negative in the regime of linear sparsity for the Lasso method, relying on a new concept we term effect size heterogeneity. Roughly speaking, a regression coefficient vector has high effect size heterogeneity if its nonzero entries have significantly different magnitudes. From the viewpoint of this new measure, we prove that the false and true positive rates achieve the optimal trade-off uniformly along the Lasso path when this measure is maximal in a certain sense, and the worst trade-off is achieved when it is minimal in the sense that all nonzero effect sizes are roughly equal. Moreover, we demonstrate that the first false selection occurs much earlier when effect size heterogeneity is minimal than when it is maximal. The underlying cause of these two phenomena is, metaphorically speaking, the ``competition'' among variables with effect sizes of the same magnitude in entering the model. Taken together, our findings suggest that effect size heterogeneity shall serve as an important complementary measure to the sparsity of regression coefficients in the analysis of high-dimensional regression problems. Our proofs use techniques from approximate message passing theory as well as a novel technique for estimating the rank of the first false variable.
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