Near-Exponential Savings for Mean Estimation with Active Learning

TL;DR

This paper introduces PartiBandits, an active learning algorithm that achieves near-exponential savings in label complexity for mean estimation of a multi-class variable by adaptively partitioning data and using UCB strategies.

Contribution

The paper proposes a novel two-stage active learning algorithm that combines UCB and disagreement-based methods, achieving minimax optimal convergence rates for mean estimation.

Findings

Achieves near-exponential label savings with respect to N.

Convergence rates are minimax optimal in classical settings.

Demonstrates effectiveness through simulations with electronic health records.

Abstract

We study the problem of efficiently estimating the mean of a $k$-class random variable, $Y$, using a limited number of labels, $N$, in settings where the analyst has access to auxiliary information (i.e.: covariates) $X$ that may be informative about $Y$. We propose an active learning algorithm ("PartiBandits") to estimate $\mathbb{E}[Y]$. The algorithm yields an estimate, $\widehat{\mu}_{\text{PB}}$, such that $\left( \widehat{\mu}_{\text{PB}} - \mathbb{E}[Y]\right)^2$ is $\tilde{\mathcal{O}}\left( \frac{\nu + \exp(c \cdot (-N/\log(N))) }{N} \right)$, where $c > 0$ is a constant and $\nu$ is the risk of the Bayes-optimal classifier. PartiBandits is essentially a two-stage algorithm. In the first stage, it learns a partition of the unlabeled data that shrinks the average conditional variance of $Y$. In the second stage it uses a UCB-style subroutine ("WarmStart-UCB") to request labels from each stratum round-by-round. Both the main algorithm's and the subroutine's convergence rates are minimax optimal in classical settings. PartiBandits bridges the UCB and disagreement-based approaches to active learning despite these two approaches being designed to tackle very different tasks. We illustrate our methods through simulation using nationwide electronic health records. Our methods can be implemented using the PartiBandits package in R.