Least Ambiguous Set-Valued Classifiers with Bounded Error Levels

TL;DR

This paper proposes a framework for multiclass set-valued classifiers that guarantees user-defined confidence levels while minimizing ambiguity, with theoretical derivations and practical estimators that improve handling ambiguous instances.

Contribution

It introduces a novel framework for bounded-error set-valued classification, deriving oracle classifiers and practical estimators with theoretical guarantees.

Findings

Oracle classifiers based on level sets of conditional probabilities.

Estimators with strong asymptotic and finite-sample properties.

Solutions for handling empty set outputs in practical applications.

Abstract

In most classification tasks there are observations that are ambiguous and therefore difficult to correctly label. Set-valued classifiers output sets of plausible labels rather than a single label, thereby giving a more appropriate and informative treatment to the labeling of ambiguous instances. We introduce a framework for multiclass set-valued classification, where the classifiers guarantee user-defined levels of coverage or confidence (the probability that the true label is contained in the set) while minimizing the ambiguity (the expected size of the output). We first derive oracle classifiers assuming the true distribution to be known. We show that the oracle classifiers are obtained from level sets of the functions that define the conditional probability of each class. Then we develop estimators with good asymptotic and finite sample properties. The proposed estimators build on existing single-label classifiers. The optimal classifier can sometimes output the empty set, but we provide two solutions to fix this issue that are suitable for various practical needs.