Lasso type classifiers with a reject option

We consider the problem of binary classification where one can, for a particular cost, choose not to classify an observation. We present a simple proof for the oracle inequality for the excess risk of structural risk minimizers using a lasso type penalty.

💡 Research Summary

The paper addresses binary classification problems in which a decision maker may opt to withhold a prediction for a particular observation at a predefined cost, a setting commonly referred to as a “reject option.” The authors propose a simple yet powerful framework that combines a reject option with a Lasso‑type (ℓ₁) regularization penalty, and they provide a concise proof of an oracle inequality for the excess risk of the resulting structural risk minimizer.

Problem formulation.
Let ((X_i,Y_i){i=1}^n) be i.i.d. samples with binary labels (Y_i\in{-1,+1}) and covariates (X_i\in\mathbb{R}^p). A scoring function (f\beta(x)=\beta^\top x) is used to produce a real‑valued confidence score. A threshold (\tau>0) determines whether the classifier makes a definitive prediction or rejects: if (|f_\beta(x)|\ge\tau) the sign of (f_\beta(x)) is output, otherwise the observation is rejected. The loss function incorporates both misclassification cost (c_E) and rejection cost (c_R) (typically (c_R<c_E)):

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