The distribution of Ridgeless least squares interpolators
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无岭最小二乘插值估计量的分布
- 研究问题
- 研究无岭最小ℓ2范数插值估计量在过参数化线性回归中的精确随机行为，即其作为统计估计量的分布特征。
- 方法与思路
- 作者提出在高维设定下，将该插值估计量的分布刻画为关联高斯序列模型中带正正则化岭估计量的分布，适用于一般非高斯随机设计，并统一推广到正正则化岭估计量。
- 材料陈述的贡献
- 摘要称给出了插值估计量分布的一般性刻画，量化了隐含正则化；由此得到一类加权ℓq风险的完整刻画，此前仅q=2已知；并揭示广义与k折交叉验证调优ℓ2预测风险可同时达到最优的样本内、预测与估计风险及去偏置信区间长度。
- 阅读关注点
- 应核对分布刻画所需的高维渐近框架与识别条件、非高斯设计的适用范围、加权ℓq风险结论的推导细节，以及交叉验证最优性结论所依赖的假设与风险定义。
- 阅读边界
- 仅凭摘要无法判断定理的严格假设、证明正确性与数值表现；摘要未说明具体收敛速率、有限样本性质及与既有结果的差异程度。该文为arXiv预印本，未经同行评审验证。
原文摘要
The Ridgeless minimum $\ell_2$-norm interpolator in overparametrized linear regression has attracted considerable attention in recent years in both machine learning and statistics communities. While it seems to defy conventional wisdom that overfitting leads to poor prediction, recent theoretical research on its $\ell_2$-type risks reveals that its norm minimizing property induces an `implicit regularization' that helps prediction in spite of interpolation. This paper takes a further step that aims at understanding its precise stochastic behavior as a statistical estimator. Specifically, we characterize the distribution of the Ridgeless interpolator in high dimensions, in terms of a Ridge estimator in an associated Gaussian sequence model with positive regularization, which provides a precise quantification of the prescribed implicit regularization in the most general distributional sense. Our distributional characterizations hold for general non-Gaussian random designs and extend uniformly to positively regularized Ridge estimators. As a direct application, we obtain a complete characterization for a general class of weighted $\ell_q$ risks of the Ridge(less) estimators that are previously only known for $q=2$ by random matrix methods. These weighted $\ell_q$ risks not only include the standard prediction and estimation errors, but also include the non-standard covariate shift settings. Our uniform characterizations further reveal a surprising feature of the commonly used generalized and $k$-fold cross-validation schemes: tuning the estimated $\ell_2$ prediction risk by these methods alone lead to simultaneous optimal $\ell_2$ in-sample, prediction and estimation risks, as well as the optimal length of debiased confidence intervals.