Learning how to separate

  • Authors:

  • Sanjay Jain;Frank Stephan

  • Affiliations:

  • School of Computing, National University of Singapore, Singapore 119260, Singapore;Mathematisches Institut, Im Neuenheimer Feld 294, Ruprecht-Karls-Universität Heidelberg, Heidelberg 69120, Germany

  • Venue:
  • Theoretical Computer Science - Special issue: Algorithmic learning theory
  • Year:
  • 2004

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Abstract

The main question addressed in the present work is how to find effectively a recursive function separating two sets drawn arbitrarily from a given collection of disjoint sets. In particular, it is investigated when one can find better learners which satisfy additional constraints. Such learners are the following: confident learners which converge on all data-sequences; conservative learners which abandon only definitely wrong hypotheses; set-driven learners whose hypotheses are independent of the order and the number of repetitions of the data-items supplied; learners where either the last or even all hypotheses are programs of total recursive functions.The present work gives a complete picture of the relations between these notions: the only implications are that whenever one has a learner which only outputs programs of total recursive functions as hypotheses, then one can also find learners which are conservative and set-driven. The following two major results need a nontrivial proof: (1) There is a class for which one can find, in the limit, recursive functions separating the sets in a confident and conservative way, but one cannot find even partial-recursive functions separating the sets in a set-driven way. (2) There is a class for which one can find, in the limit, recursive functions separating the sets in a confident and set-driven way, but one cannot find even partial-recursive functions separating the sets in a conservative way.