On reducing sampling variance in covariate shift using control variates

Covariate shift classification problems can in principle be tackled by importance-weighting training samples. However, the sampling variance of the risk estimator is often scaled up dramatically by the weights. This means that during cross-validation - when the importance-weighted risk is repeatedly evaluated - suboptimal hyperparameter estimates are produced. We study the sampling variances of the importance-weighted versus the oracle estimator as a function of the relative scale of the training data. We show that introducing a control variate can reduce the variance of the importance-weighted risk estimator, which leads to superior regularization parameter estimates when the training data is much smaller in scale than the test data.