Minimax semi-supervised set-valued approach to multi-class classification

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 Hölder 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, semisupervised 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.

Optimal rates for F-score binary classification

“…This result is the main reason why our conclusions on the semi-supervised estimation is different from the ones in [5,18]. For instance, in [5] the authors also obtain a final decision rule by thresholding on some estimated level.…”

“…Our analysis implies that in the setting of binary classification with F-score the semi-supervised techniques are not superior to the supervised ones. In contrast, in [5] the authors showed that in the context of confidence set classification semi-supervised classifiers might outperform it supervised counterparts. -From the theoretical point of view, the most relevant reference is a recent work of [23],…”

“…Q2. : In view of recent results of [5], we wonder if the introduction of unlabeled dataset can improve classification algorithms in the context of F-score.…”

“…For instance, in [5] the authors also obtain a final decision rule by thresholding on some estimated level. However, in the present work the difference between θ * and θ is controlled via ℓ 1 -norm of difference of CDF's, whereas in [5] they control a similar quantity through Wassertstein infinity distance.…”

Set-valued classification -- overview via a unified framework

“…That is to say that the optimal set-valued classifier Γ * (F P ,C P ) exists, is unique, and is deterministic. We refer to (Chzhen, Denis and Hebiri, 2021;Sadinle, Lei and Wasserman, 2019a) for a broad discussion on the role of Assumption 1. In particular, (Sadinle, Lei and Wasserman, 2019a) describe a way to bypass this assumption by considering randomized set-valued classifiers.…”