On Ensembling vs Merging: Least Squares and Random Forests under Covariate Shift

It has been postulated and observed in practice that for prediction problems in which covariate data can be naturally partitioned into clusters, ensembling algorithms based on suitably aggregating models trained on individual clusters often perform substantially better than methods that ignore the clustering structure in the data. In this paper, we provide theoretical support to these empirical observations by asymptotically analyzing linear least squares and random forest regressions under a linear model. Our main results demonstrate that the benefit of ensembling compared to training a single model on the entire data, often termed 'merging', might depend on the underlying bias and variance interplay of the individual predictors to be aggregated. In particular, under both fixed and high dimensional linear models, we show that merging is asymptotically superior to optimal ensembling techniques for linear least squares regression due to the unbiased nature of least squares prediction. In contrast, for random forest regression under fixed dimensional linear models, our bounds imply a strict benefit of ensembling over merging. Finally, we also present numerical experiments to verify the validity of our asymptotic results across different situations.