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Title: Robust subset selection Authors:  Ryan Thompson - Monash University (Australia) [presenting]
Abstract: The best subset selection (or `best subsets') estimator is a classic tool for sparse regression, and developments in mathematical optimisation over the past decade have made it more computationally tractable than ever. Notwithstanding its desirable statistical properties, the best subsets estimator is susceptible to outliers and can break down in the presence of a single contaminated data point. To address this issue, a robust adaption of best subsets is proposed that is highly resistant to contamination in both the response and the predictors. The adapted estimator generalises the notion of subset selection to both predictors and observations, thereby achieving robustness in addition to sparsity. This procedure, referred to as 'robust subset selection' (or `robust subsets'), is defined by a combinatorial optimisation problem for which modern discrete optimisation methods are applied. The robustness of the estimator in terms of the finite-sample breakdown point of its objective value is formally established. In support of this result, experiments on synthetic and real data are reported that demonstrate the superiority of robust subsets over best subsets in the presence of contamination. Importantly, robust subsets fare competitively across several metrics compared with popular robust adaptions of continuous shrinkage estimators.