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

Learning optimal policies from historical data enables personalization in a
wide variety of applications including healthcare, digital recommendations, and
online education. The growing policy learning literature focuses on settings
where the data collection rule stays fixed throughout the experiment. However,
adaptive data collection is becoming more common in practice, from two primary
sources: 1) data collected from adaptive experiments that are designed to
improve inferential efficiency; 2) data collected from production systems that
progressively evolve an operational policy to improve performance over time
(e.g. contextual bandits). Yet adaptivity complicates the optimal policy
identification ex post, since samples are dependent, and each treatment may not
receive enough observations for each type of individual. In this paper, we make
initial research inquiries into addressing the challenges of learning the
optimal policy with adaptively collected data. We propose an algorithm based on
generalized augmented inverse propensity weighted (AIPW) estimators, which
non-uniformly reweight the elements of a standard AIPW estimator to control
worst-case estimation variance. We establish a finite-sample regret upper bound
for our algorithm and complement it with a regret lower bound that quantifies
the fundamental difficulty of policy learning with adaptive data. When equipped
with the best weighting scheme, our algorithm achieves minimax rate optimal
regret guarantees even with diminishing exploration. Finally, we demonstrate
our algorithm's effectiveness using both synthetic data and public benchmark
datasets.

本文探讨在自适应数据收集环境下如何使用基于加权的估计算法来学习最优策略，提出了基于广义增强的倾向性加权（AIPW）估计器的算法，并建立了有限样本遗憾上限，证明最优权重方案下，算法即使在减少探索数据的情况下也能实现最小化的遗憾保证。