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.

翻译：主成分回归（PCR）是一种用于固定设计误差变量回归的流行技术，这是线性回归设置的一种推广，其中观测到的协变量受到随机噪声的污染。我们首次为在线（正则化）PCR提供了时间均匀有限样本保证，适用于数据自适应收集的场景。由于固定设计设置中用于分析PCR的证明技术不易直接扩展到在线设置，我们的结果依赖于将现代鞅集中工具适配到误差变量设置中。作为我们界限的一个应用，我们提供了一个在干预措施自适应分配的面板数据设置中进行实验设计的框架。我们的框架可以被视为合成控制和合成干预框架的推广，其中数据是通过自适应干预分配策略收集的。