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，该技术是线性回归模型的推广形式，其中观测协变量受到随机噪声的污染。我们首次为自适应数据收集场景下的在线（正则化）PCR提供了时间一致有限样本保证。由于固定设计场景中PCR分析的证明技术难以直接推广到在线场景，我们的结果依赖于将现代鞅集中理论工具适配到误差变量设定中。作为我们界限的应用，我们为干预措施自适应分配的面板数据实验设计提供了框架。该框架可被视为合成控制与合成干预框架的推广，其中数据通过自适应干预分配策略收集。