The Annals of Statistics

Rates of convergence in active learning

Steve Hanneke

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We study the rates of convergence in generalization error achievable by active learning under various types of label noise. Additionally, we study the general problem of model selection for active learning with a nested hierarchy of hypothesis classes and propose an algorithm whose error rate provably converges to the best achievable error among classifiers in the hierarchy at a rate adaptive to both the complexity of the optimal classifier and the noise conditions. In particular, we state sufficient conditions for these rates to be dramatically faster than those achievable by passive learning.

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Ann. Statist. Volume 39, Number 1 (2011), 333-361.

First available in Project Euclid: 3 December 2010

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Zentralblatt MATH identifier

Primary: 62L05: Sequential design 68Q32: Computational learning theory [See also 68T05] 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20] 68T05: Learning and adaptive systems [See also 68Q32, 91E40]
Secondary: 68T10: Pattern recognition, speech recognition {For cluster analysis, see 62H30} 68Q10: Modes of computation (nondeterministic, parallel, interactive, probabilistic, etc.) [See also 68Q85] 68Q25: Analysis of algorithms and problem complexity [See also 68W40] 68W40: Analysis of algorithms [See also 68Q25] 62G99: None of the above, but in this section

Active learning sequential design selective sampling statistical learning theory oracle inequalities model selection classification


Hanneke, Steve. Rates of convergence in active learning. Ann. Statist. 39 (2011), no. 1, 333--361. doi:10.1214/10-AOS843.

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Supplemental materials

  • Supplementary material: Proofs and Supplements for “Rates of Convergence in Active Learning”. The supplementary material contains three additional Appendices, namely, Appendices B, C and D. Specifically, Appendix B provides detailed proofs of Theorems 5–9, as well as several abstract lemmas from which these results are derived. Appendix C discusses the use of estimators in Algorithm 1. Finally, Appendix D includes a proof of a general minimax lower bound ∝ n^(−κ ∕ (2κ − 2)) for any nontrivial hypothesis class, generalizing a result of Castro and Nowak [12].