  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 (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. We demonstrate the usefulness of our bounds by
applying them to the domain of panel data, a ubiquitous setting in econometrics
and statistics. As our first application, 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. Our second application is a procedure for
learning such an intervention assignment policy in a setting where units arrive
sequentially to be treated. In addition to providing theoretical performance
guarantees (as measured by regret), we show that our method empirically
outperforms a baseline which does not leverage error-in-variables regression.
