Post-Processing Guarantees for Classification under Linear-Fractional Performance Metrics

This paper studies binary classification under generalized performance metrics, going beyond standard misclassification error. We focus on linear-fractional measures—including the $F$-score and Jaccard index—which are widely used in class-imbalanced settings. We first provide a unified characterization of the optimal classifier, showing that it admits a thresholding structure on the regression function. This reduces the original infinite-dimensional, non-decomposable optimization problem to the estimation of a single scalar parameter. Building on this result, we analyze plug-in classifiers that combine a regression estimator with a data-driven estimate of the optimal threshold. We show that this threshold is characterized as the solution of a fixed-point equation depending only on the marginal distribution, enabling its estimation using unlabeled data and avoiding standard validation-based procedures. Our main contribution is a finite-sample analysis of this approach. We derive excess risk bounds that decompose the error into contributions from regression estimation, threshold estimation, and sampling effects, providing a modular understanding of learning under non-decomposable metrics. Under standard margin assumptions, we further establish fast convergence rates. Our results yield a unified theoretical framework for plug-in classification under linear-fractional metrics, extending prior analyses beyond specific measures such as the $F$-score. Empirical results on synthetic and real datasets support our theoretical findings.