The complexity of learning concept classes with polynomial general dimensionAuthors: Johannes Köbler and Wolfgang Lindner
Source:
Theoretical Computer Science Vol. 350, Issue 1,
January 2006, pp. 4962
Abstract. The general dimension is a combinatorial measure that characterizes the number of queries needed to learn a concept class. We use this notion to show that any pevaluatable concept class with polynomial query complexity can be learned in polynomial time with the help of an oracle in the polynomial hierarchy, where the complexity of the required oracle depends on the querytypes used by the learning algorithm. In particular, we show that for subset and superset queries an oracle in Σ_{3}^{p} suffices. Since the concept class of DNF formulas has polynomial query complexity with respect to subset and superset queries with DNF formulas as hypotheses, it follows that DNF formulas are properly learnable in polynomial time with subset and superset queries and the help of an oracle in Σ_{3}^{p}. We also show that the required oracle in our main theorem cannot be replaced by an oracle in a lower level of the polynomialtime hierarchy, unless the hierarchy collapses.

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