Robust Multi-Model Subset Selection

Outlying observations can be challenging to handle and adversely affect subsequent analyses, especially in data with increasing dimensional complexity. Although outliers are not always undesired anomalies in the data and may possess valuable insights, only methods that are robust to outliers are able to accurately identify them and resist their influence. In this paper, we propose Robust Multi-Model Subset Selection (RMSS), a method that generates an ensemble of sparse and diverse predictive models that are resistant to outliers. We show that the ensembles generally outperform single-model sparse and robust methods. Cross-validation is used to tune parameters to control levels of sparsity, diversity and robustness. We establish the finite-sample breakdown point of the models generated by RMSS, including that of the Robust Best Subset Selection (RBSS) estimator as a special case. In addition, we develop a tailored computing algorithm to learn the ensemble by leveraging recent developments in $ \ell_0 $ optimization. Our extensive numerical experiments on synthetic and artificially contaminated real datasets from bioinformatics and cheminformatics demonstrate the competitive advantage of our method over state-of-the-art single-model methods. The appendix contains all theoretical proofs, additional algorithmic and computational details, and the code and data to reproduce our numerical results.