# Minimax semi-supervised confidence sets for multi-class classification

## TL;DR

This paper develops semi-supervised confidence set classifiers for multi-class problems, achieving faster convergence rates than supervised methods under certain assumptions, with theoretical guarantees and empirical validation.

## Contribution

It introduces a semi-supervised minimax framework for confidence set classification with controlled size, establishing convergence rates and demonstrating superiority over supervised methods.

## Key findings

- Semi-supervised estimators outperform supervised ones with enough unlabeled data.
- Achieves faster convergence rates under margin and Hölder conditions.
- Empirical results confirm theoretical convergence improvements.

## Abstract

In this work we study the semi-supervised framework of confidence set classification with controlled expected size in minimax settings. We obtain semi-supervised minimax rates of convergence under the margin assumption and a H{\"o}lder condition on the regression function. Besides, we show that if no further assumptions are made, there is no supervised method that outperforms the semi-supervised estimator proposed in this work. We establish that the best achievable rate for any supervised method is n^{--1/2} , even if the margin assumption is extremely favorable. On the contrary, semi-supervised estimators can achieve faster rates of convergence provided that sufficiently many unlabeled samples are available. We additionally perform numerical evaluation of the proposed algorithms empirically confirming our theoretical findings.