Near-optimal inference in adaptive linear regression

TL;DR

This paper introduces online debiasing estimators for adaptive linear regression that correct distributional anomalies, achieve asymptotic normality, and provide accurate confidence intervals, with applications in bandits, time series, and active learning.

Contribution

It proposes a novel family of online debiasing estimators that leverage covariance structure to improve inference in adaptive linear regression, establishing asymptotic normality and minimax optimality.

Findings

Estimators achieve asymptotic normality under mild conditions.

Provided asymptotically exact confidence intervals.

Demonstrated effectiveness in bandit, time series, and active learning applications.

Abstract

When data is collected in an adaptive manner, even simple methods like ordinary least squares can exhibit non-normal asymptotic behavior. As an undesirable consequence, hypothesis tests and confidence intervals based on asymptotic normality can lead to erroneous results. We propose a family of online debiasing estimators to correct these distributional anomalies in least squares estimation. Our proposed methods take advantage of the covariance structure present in the dataset and provide sharper estimates in directions for which more information has accrued. We establish an asymptotic normality property for our proposed online debiasing estimators under mild conditions on the data collection process and provide asymptotically exact confidence intervals. We additionally prove a minimax lower bound for the adaptive linear regression problem, thereby providing a baseline by which to compare estimators. There are various conditions under which our proposed estimators achieve the minimax lower bound. We demonstrate the usefulness of our theory via applications to multi-armed bandit, autoregressive time series estimation, and active learning with exploration.