Multiclass Learnability and the ERM Principle

Amit Daniely, Sivan Sabato, Shai Ben-David, Shai Shalev-Shwartz

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


Multiclass learning is an area of growing practical relevance, for which the currently available theory is still far from providing satisfactory understanding. We study the learnability of multiclass prediction, and derive upper and lower bounds on the sample complexity of multiclass hypothesis classes in different learning models: batch/online, realizable/unrealizable, full information/bandit feedback. Our analysis reveals a surprising phenomenon: In the multiclass setting, in sharp contrast to binary classification, not all Empirical Risk Minimization (ERM) algorithms are equally successful. We show that there exist hypotheses classes for which some ERM learners have lower sample complexity than others. Furthermore, there are classes that are learnable by some ERM learners, while other ERM learner will fail to learn them. We propose a principle for designing good ERM learners, and use this principle to prove tight bounds on the sample complexity of learning symmetric multiclass hypothesis classes (that is, classes that are invariant under any permutation of label names). We demonstrate the relevance of the theory by analyzing the sample complexity of two widely used hypothesis classes: generalized linear multiclass models and reduction trees. We also obtain some practically relevant conclusions.

Original languageEnglish GB
Title of host publicationProceedings of the 24nd Annual Conference on Learning Theory (COLT), JMLR Workshop and Conference Proceedings
EditorsSham M. Kakade, Ulrike von Luxburg
Number of pages26
StatePublished - 2011
Externally publishedYes
Event24th International Conference on Learning Theory, COLT 2011 - Budapest, Hungary
Duration: 9 Jul 201111 Jul 2011

Publication series

NameProceedings of Machine Learning Research
ISSN (Electronic)2640-3498


Conference24th International Conference on Learning Theory, COLT 2011


  • List of keywords

ASJC Scopus subject areas

  • Software
  • Control and Systems Engineering
  • Statistics and Probability
  • Artificial Intelligence


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