Adaptive Linear Estimating Equations

TL;DR

This paper introduces a debiased estimator for adaptive linear regression that ensures asymptotic normality, facilitating accurate inference in sequential data collection scenarios like multi-armed bandits.

Contribution

It proposes a novel adaptive linear estimating equation method that achieves asymptotic normality while maintaining non-asymptotic performance, bridging two paradigms of adaptive inference.

Findings

Estimator achieves asymptotic normality under adaptive data collection.

Retains non-asymptotic performance comparable to least squares.

Connects non-asymptotic and asymptotic inference paradigms.

Abstract

Sequential data collection has emerged as a widely adopted technique for enhancing the efficiency of data gathering processes. Despite its advantages, such data collection mechanism often introduces complexities to the statistical inference procedure. For instance, the ordinary least squares (OLS) estimator in an adaptive linear regression model can exhibit non-normal asymptotic behavior, posing challenges for accurate inference and interpretation. In this paper, we propose a general method for constructing debiased estimator which remedies this issue. It makes use of the idea of adaptive linear estimating equations, and we establish theoretical guarantees of asymptotic normality, supplemented by discussions on achieving near-optimal asymptotic variance. A salient feature of our estimator is that in the context of multi-armed bandits, our estimator retains the non-asymptotic performance of the least square estimator while obtaining asymptotic normality property. Consequently, this work helps connect two fruitful paradigms of adaptive inference: a) non-asymptotic inference using concentration inequalities and b) asymptotic inference via asymptotic normality.