Principal component regression (PCR) is a popular technique for fixed-design
error-in-variables regression, a generalization of the linear regression
setting in which the observed covariates are corrupted with random noise. We
provide the first time-uniform finite sample guarantees for online
(regularized) PCR whenever data is collected adaptively. Since the proof
techniques for analyzing PCR in the fixed design setting do not readily extend
to the online setting, our results rely on adapting tools from modern
martingale concentration to the error-in-variables setting. As an application
of our bounds, we provide a framework for experiment design in panel data
settings when interventions are assigned adaptively. Our framework may be
thought of as a generalization of the synthetic control and synthetic
interventions frameworks, where data is collected via an adaptive intervention
assignment policy.

本文提供了一种基于主成分回归 (PCR) 的在线方法以及关于在线之下的样本保证方法和应用于面板数据设置中的实验设计框架，是一种思想的扩展。

面板数据自适应主成分回归

Adaptive Principal Component Regression with Applications to Panel Data

This work obtains novel finite sample guarantees for Principal Component
Analysis (PCA). These hold even when the corrupting noise is non-isotropic, and
a part (or all of it) is data-dependent. Because of the latter, in general, the
noise and the true data are correlated. The results in this work are a
significant improvement over those given in our earlier work where this
"correlated-PCA" problem was first studied. In fact, in certain regimes, our
results imply that the sample complexity required to achieve subspace recovery
error that is a constant fraction of the noise level is near-optimal. Useful
corollaries of our result include guarantees for PCA in sparse data-dependent
noise and for PCA with missing data. An important application of the former is
in proving correctness of the subspace update step of a popular online
algorithm for dynamic robust PCA.

本文研究了主成分分析的新颖有限样本保证，包括在非等向噪声和数据依赖性噪声的情况下，这是前人未曾解决的相关问题，并给出了相应的保证， 同时在稀疏数据相关噪声和缺失数据的情况下进行了讨论。