Active Heteroscedastic Regression

An active learner is given a model class $\Theta$, a large sample of unlabeled data drawn from an underlying distribution and access to a labeling oracle which can provide a label for any of the unlabeled instances. The goal of the learner is to find a model $\theta \in \Theta$ that fits the data to a given accuracy while making as few label queries to the oracle as possible. In this work, we consider a theoretical analysis of the label requirement of active learning for regression under a heteroscedastic noise model. Previous work has looked at active regression either with no model mismatch~\cite{chaudhuri2015convergence} or with arbitrary model mismatch~\cite{sabato2014active}. In the first case, active learning provided no improvement even in the simple case where the unlabeled examples were drawn from Gaussians. In the second case, under arbitrary model mismatch, the algorithm either required a very high running time or a large number of labels. We provide bounds on the convergence rates of active and passive learning for heteroscedastic regression, where the noise depends on the instance. Our results illustrate that just like in binary classification, some partial knowledge of the nature of the noise can lead to significant gains in the label requirement of active learning.