Detangling robustness in high dimensions: composite versus model-averaged estimation

TL;DR

This paper investigates the robustness of high-dimensional regularized estimation methods, comparing model-averaged and composite approaches, and demonstrates their effectiveness in prediction tasks through simulations and real data applications.

Contribution

It introduces a framework for optimal weighting of model-averaged and composite estimators in high dimensions, accounting for regularization effects without perfect model selection.

Findings

No single method consistently outperforms others.

Model-averaged and composite estimators often outperform least-squares.

Methods are practically useful, demonstrated through audio signal reconstruction.

Abstract

Robust methods, though ubiquitous in practice, are yet to be fully understood in the context of regularized estimation and high dimensions. Even simple questions become challenging very quickly. For example, classical statistical theory identifies equivalence between model-averaged and composite quantile estimation. However, little to nothing is known about such equivalence between methods that encourage sparsity. This paper provides a toolbox to further study robustness in these settings and focuses on prediction. In particular, we study optimally weighted model-averaged as well as composite $l_1$-regularized estimation. Optimal weights are determined by minimizing the asymptotic mean squared error. This approach incorporates the effects of regularization, without the assumption of perfect selection, as is often used in practice. Such weights are then optimal for prediction quality. Through an extensive simulation study, we show that no single method systematically outperforms others. We find, however, that model-averaged and composite quantile estimators often outperform least-squares methods, even in the case of Gaussian model noise. Real data application witnesses the method's practical use through the reconstruction of compressed audio signals.