On Robustness of Principal Component Regression
On Robustness of Principal Component Regression
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DOI：
10.1080/01621459.2021.1928513
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发表时间：
2021-07-18
影响因子：
3.7
通讯作者：
Song, Dogyoon
中科院分区：
文献类型：
--
作者：
Agarwal, Anish;Shah, Devavrat;Song, Dogyoon
主成分回归（PCR）是一种简单但功能强大且应用广泛的方法。当协变量表现为低秩结构时，其有效性得到了很好的证实。然而，它处理有噪声、缺失和混合值（即离散和连续）协变量设置的能力尚不清楚，仍然是一个重要的公开挑战。作为这项工作的主要贡献，我们在这方面建立了PCR的稳健性，没有任何变化，并提供了有意义的有限样本分析。为此，我们建立了PCR相当于通过硬奇异值阈值（HSVT）预处理协变量矩阵后进行线性回归。因此，在使用观测数据进行反事实分析的背景下，我们表明PCR等同于最近提出的合成控制方法的鲁棒变体，即鲁棒合成控制（RSC）。作为一个直接的结果，我们获得了以前缺失的RSC估计量的有限样本分析。作为对综合控制文献的重要贡献，我们建立了（近似）线性综合控制存在于广义因子模型或潜在变量模型的设置中；在传统的文献中，综合控制的存在需要作为一个公理来假定。我们进一步讨论了PCR相对于噪声的鲁棒性的一个令人惊讶的含义，即，PCR可以学习一个很好的预测模型，即使协变量被巧妙地转换以保持差分隐私。最后，这项工作通过建立关于l2的更强的保证来推进HSVT的最先进的分析，无穷范数而不是Frobenius范数，这是在矩阵估计文献中通常做的，这可能是它自己的兴趣。
Principal component regression (PCR) is a simple, but powerful and ubiquitously utilized method. Its effectiveness is well established when the covariates exhibit low-rank structure. However, its ability to handle settings with noisy, missing, and mixed-valued, that is, discrete and continuous, covariates is not understood and remains an important open challenge. As the main contribution of this work, we establish the robustness of PCR, without any change, in this respect and provide meaningful finite-sample analysis. To do so, we establish that PCR is equivalent to performing linear regression after preprocessing the covariate matrix via hard singular value thresholding (HSVT). As a result, in the context of counterfactual analysis using observational data, we show PCR is equivalent to the recently proposed robust variant of the synthetic control method, known as robust synthetic control (RSC). As an immediate consequence, we obtain finite-sample analysis of the RSC estimator that was previously absent. As an important contribution to the synthetic controls literature, we establish that an (approximate) linear synthetic control exists in the setting of a generalized factor model, or latent variable model; traditionally in the literature, the existence of a synthetic control needs to be assumed to exist as an axiom. We further discuss a surprising implication of the robustness property of PCR with respect to noise, that is, PCR can learn a good predictive model even if the covariates are tactfully transformed to preserve differential privacy. Finally, this work advances the state-of-the-art analysis for HSVT by establishing stronger guarantees with respect to the l2,infinity -norm rather than the Frobenius norm as is commonly done in the matrix estimation literature, which may be of interest in its own right.