Indirect Active Learning

TL;DR

This paper explores the problem of learning the relationship between a covariate and response when the covariate can only be influenced indirectly through a control variable, analyzing minimax rates for estimation under this scenario.

Contribution

It introduces a nonparametric framework for Indirect Active Learning, deriving minimax convergence rates and showing simple passive strategies can match active learning benefits.

Findings

Active learning offers asymptotic advantages in indirect influence scenarios.

Two-stage passive experiments can achieve the same rates as active learning.

Different convergence rates depend on the complexities and noise levels of the relationships.

Abstract

Traditional models of active learning assume a learner can directly manipulate or query a covariate $X$ in order to study its relationship with a response $Y$. However, if $X$ is a feature of a complex system, it may be possible only to indirectly influence $X$ by manipulating a control variable $Z$, a scenario we refer to as Indirect Active Learning. Under a nonparametric model of Indirect Active Learning with a fixed budget, we study minimax convergence rates for estimating the relationship between $X$ and $Y$ locally at a point, obtaining different rates depending on the complexities and noise levels of the relationships between $Z$ and $X$ and between $X$ and $Y$. We also identify minimax rates for passive learning under comparable assumptions. In many cases, our results show that, while there is an asymptotic benefit to active learning, this benefit is fully realized by a simple two-stage learner that runs two passive experiments in sequence.