Learning Curves for Heterogeneous Feature-Subsampled Ridge Ensembles

Feature bagging is a well-established ensembling method which aims to reduce prediction variance by training estimators in an ensemble on random subsamples or projections of features. Typically, ensembles are chosen to be homogeneous, in the sense the the number of feature dimensions available to an estimator is uniform across the ensemble. Here, we introduce heterogeneous feature ensembling, with estimators built on varying number of feature dimensions, and consider its performance in a linear regression setting. We study an ensemble of linear predictors, each fit using ridge regression on a subset of the available features. We allow the number of features included in these subsets to vary. Using the replica trick from statistical physics, we derive learning curves for ridge ensembles with deterministic linear masks. We obtain explicit expressions for the learning curves in the case of equicorrelated data with an isotropic feature noise. Using the derived expressions, we investigate the effect of subsampling and ensembling, finding sharp transitions in the optimal ensembling strategy in the parameter space of noise level, data correlations, and data-task alignment. Finally, we suggest variable-dimension feature bagging as a strategy to mitigate double descent for robust machine learning in practice.