Ask for More Than Bayes Optimal: A Theory of Indecisions for Classification

Abstract

Selective classification is a powerful tool for automated decision-making in high-risk scenarios, allowing classifiers to make only highly confident decisions while abstaining when uncertainty is too high. Given a target classification accuracy, our goal is to minimize the number of indecisions, which are observations that we do not automate. For problems that are hard, the target accuracy may not be achievable without using indecisions. In contrast, by using indecisions, we are able to control the misclassification rate to any user-specified level, even below the Bayes optimal error rate, while minimizing the frequency of identifying an indecision. We provide a full characterization of the minimax risk in selective classification, proving key continuity and monotonicity properties that enable optimal indecision selection. Our results extend to hypothesis testing, where we control type II error given a fixed type I error, introducing a novel perspective in selective inference. We analyze the impact of estimating the regression function , showing that plug-in classifiers remain consistent and that accuracy-based calibration effectively controls indecision levels. Additionally, we develop finite-sample calibration methods and identify cases where no training data is needed under the Monotone Likelihood Ratio (MLR) property. In the binary Gaussian mixture model, we establish sharp phase transition results, demonstrating that minimal indecisions can yield near-optimal accuracy even with suboptimal class separation. These findings highlight the potential of selective classification to significantly reduce misclassification rates with a relatively small cost in terms of indecisions.

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Reproductions