An active learning framework for multi-group mean estimation

We consider a fundamental problem where there are multiple groups whose data distributions are unknown, and an analyst would like to learn the mean of each group. We consider an active learning framework to sequentially collect $T$ samples with bandit, each period observing a sample from a chosen group. After observing a sample, the analyst may update their estimate of the mean and variance of that group and choose the next group accordingly. The objective is to dynamically collect samples to minimize the $p$-norm of the vector of variances of our mean estimators after $T$ rounds. We propose an algorithm, Variance-UCB, that selects groups according to a an upper bound on the variance estimate adjusted to the $p$-norm chosen. We show that the regret of Variance-UCB is $O(T^{-2})$ for finite $p$, and prove that no algorithm can do better. When $p$ is infinite, we recover the $O(T^{-1.5})$ obtained in \cite{activelearning, carpentier2011upper} and provide a new lower bound showing that no algorithm can do better.