{"title":"Minimax semi-supervised set-valued approach to multi-class classification","authors":"Evgenii Chzhen, Christophe Denis, Mohamed Hebiri","doi":"10.3150/20-BEJ1313","DOIUrl":null,"url":null,"abstract":"We study supervised and semi-supervised algorithms in the set-valued classification framework with controlled expected size. While the former methods can use only n labeled samples, the latter are able to make use of N additional unlabeled data. We obtain semi-supervised minimax rates of convergence under the α-margin assumption and a β-Hölder condition on the conditional distribution of labels. Our analysis implies that if no further assumption is made, there is no supervised method that outperforms the semi-supervised estimator proposed in this work – the best achievable rate for any supervised method is O(n−1/2), even if the margin assumption is extremely favorable; on the contrary, the developed semi-supervised estimator can achieve faster O((n/ logn)−(1+α)β/(2β+d)) rate of convergence provided that sufficiently many unlabeled samples are available. We also show that under additional smoothness assumption, supervised methods are able to achieve faster rates and the unlabeled sample cannot improve the rate of convergence. Finally, a numerical study supports our theory and emphasizes the relevance of the assumptions we required from an empirical perspective.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bernoulli","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3150/20-BEJ1313","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 5
Abstract
We study supervised and semi-supervised algorithms in the set-valued classification framework with controlled expected size. While the former methods can use only n labeled samples, the latter are able to make use of N additional unlabeled data. We obtain semi-supervised minimax rates of convergence under the α-margin assumption and a β-Hölder condition on the conditional distribution of labels. Our analysis implies that if no further assumption is made, there is no supervised method that outperforms the semi-supervised estimator proposed in this work – the best achievable rate for any supervised method is O(n−1/2), even if the margin assumption is extremely favorable; on the contrary, the developed semi-supervised estimator can achieve faster O((n/ logn)−(1+α)β/(2β+d)) rate of convergence provided that sufficiently many unlabeled samples are available. We also show that under additional smoothness assumption, supervised methods are able to achieve faster rates and the unlabeled sample cannot improve the rate of convergence. Finally, a numerical study supports our theory and emphasizes the relevance of the assumptions we required from an empirical perspective.
期刊介绍:
BERNOULLI is the journal of the Bernoulli Society for Mathematical Statistics and Probability, issued four times per year. The journal provides a comprehensive account of important developments in the fields of statistics and probability, offering an international forum for both theoretical and applied work.
BERNOULLI will publish:
Papers containing original and significant research contributions: with background, mathematical derivation and discussion of the results in suitable detail and, where appropriate, with discussion of interesting applications in relation to the methodology proposed.
Papers of the following two types will also be considered for publication, provided they are judged to enhance the dissemination of research:
Review papers which provide an integrated critical survey of some area of probability and statistics and discuss important recent developments.
Scholarly written papers on some historical significant aspect of statistics and probability.