A Framework for Sample Efficient Interval Estimation with Control Variates

We consider the problem of estimating confidence intervals for the mean of a random variable, where the goal is to produce the smallest possible interval for a given number of samples. While minimax optimal algorithms are known for this problem in the general case, improved performance is possible under additional assumptions. In particular, we design an estimation algorithm to take advantage of side information in the form of a control variate, leveraging order statistics. Under certain conditions on the quality of the control variates, we show improved asymptotic efficiency compared to existing estimation algorithms. Empirically, we demonstrate superior performance on several real world surveying and estimation tasks where we use the output of regression models as the control variates.