The Implicit Regularization of Ordinary Least Squares Ensembles

Abstract

Ensemble methods that average over a collection of independent predictors that are each limited to a subsampling of both the examples and features of the training data command a significant presence in machine learning, such as the ever-popular random forest, yet the nature of the subsampling effect, particularly of the features, is not well understood. We study the case of an ensemble of linear predictors, where each individual predictor is fit using ordinary least squares on a random submatrix of the data matrix. We show that, under standard Gaussianity assumptions, when the number of features selected for each predictor is optimally tuned, the asymptotic risk of a large ensemble is equal to the asymptotic ridge regression risk, which is known to be optimal among linear predictors in this setting. In addition to eliciting this implicit regularization that results from subsampling, we also connect this ensemble to the dropout technique used in training deep (neural) networks, another strategy that has been shown to have a ridge-like regularizing effect.

arXiv comment18 pages, 4 figures. To appear in AISTATS 2020

References in corpus (5)

• Reconciling modern machine learning practice and the bias-variance trade-off
• Consistency of random forests
• Dropout Training as Adaptive Regularization
• Overfitting or perfect fitting? Risk bounds for classification and regression rules that interpolate
• Approximate Computation and Implicit Regularization for Very Large-scale Data Analysis

Cited by in corpus (7)

• Exact expressions for double descent and implicit regularization via surrogate random design
• Randomization as Regularization: A Degrees of Freedom Explanation for Random Forest Success
• Getting Better from Worse: Augmented Bagging and a Cautionary Tale of Variable Importance
• Extrapolated cross-validation for randomized ensembles
• MP-Boost: Minipatch Boosting via Adaptive Feature and Observation Sampling
• Role of Bootstrap Averaging in Generalized Approximate Message Passing
• Dropout Drops Double Descent