Selective classifiers improve model reliability by abstaining on inputs the model deems uncertain. However, few practical approaches achieve the gold-standard performance of a perfect-ordering oracle that accepts examples exactly in order of correctness. Our work formalizes this shortfall as the selective-classification gap and present the first finite-sample decomposition of this gap to five distinct sources of looseness: Bayes noise, approximation error, ranking error, statistical noise, and implementation- or shift-induced slack. Crucially, our analysis reveals that monotone post-hoc calibration -- often believed to strengthen selective classifiers -- has limited impact on closing this gap, since it rarely alters the model's underlying score ranking. Bridging the gap therefore requires scoring mechanisms that can effectively reorder predictions rather than merely rescale them. We validate our decomposition on synthetic two-moons data and on real-world vision and language benchmarks, isolating each error component through controlled experiments. Our results confirm that (i) Bayes noise and limited model capacity can account for substantial gaps, (ii) only richer, feature-aware calibrators meaningfully improve score ordering, and (iii) data shift introduces a separate slack that demands distributionally robust training. Together, our decomposition yields a quantitative error budget as well as actionable design guidelines that practitioners can use to build selective classifiers which approximate ideal oracle behavior more closely.

翻译：暂无翻译