The learning model of Valiant is extended to allow the number of examples to depend on the particular concept to be learned, instead of requiring a uniform bound for all concepts of a concept class.
This extension, called nonuniform learning, enables learning many concept classes not learnable by the previous definitions. Nonuniformly learnable concept classes are characterized. Some examples (Boolean formulae, recursive and r.e. sets) are shown to be nonuniformly learnable by a polynomial number of examples, but not necessarily in polynomial time. Restricting the learning protocol such that the learner has to commit himself after a finite number of examples does not effect the concept classes which can be learned.
An extension of nonuniform learnability to nonuniform learnability w.r.t. specific distributions is presented.
Unable to display preview. Download preview PDF.
- [AS]Angluin D. and Smith C.H., "A survey of inductive inference: theory and methods", TR 250, Computer Science Department, Yale University TR 250 (1982).Google Scholar
- [BI]Benedek G.M. and Itai A., "Learnability by fixed distributions", TR 468, Department of Computer Science, Technion (1987).Google Scholar
- [BEHW1]Blumer A., Ehrenfeucht A., Haussler D. and Warmuth M., "Classifying learnable geometric concepts with the Vapnik-Chervonenkis dimension", Proc. of 18th Symp. Theory of Comp., 273–282., (1986).Google Scholar
- [BEHW2]Blumer A., Ehrenfeucht A., Haussler D. and Warmuth M., "Occam's razor", Inf. Proc. Letters 24 (1987), 377–380, North-Holland.Google Scholar
- [HU]Hopcroft J.E. and Ullman J.D., "Introduction to automata theory, languages and computation", Addison-Wesley (1979).Google Scholar
- [KLPV]Kearns M., Ling Mi, Pitt L., Valiant L.G., "On the learnability of Boolean formulae", Proc. of 19th Symp. Theory of Comp., 285–295. ACM, New York, (1987).Google Scholar
- [N]Natarajan B. K., "On learning Boolean functions", In Proc. of 19th Symp. Theory of Comp., 296–304. ACM, New York, (1987).Google Scholar
- [PV]Pitt L. and Valiant, L. G., "Computational limitations on learning from examples", Aiken Computation Laboratory, Harward University, Cambridge, MA 02138, (July 1986).Google Scholar
- [VC]Vapnik V.N. and Chervonenkis A.Ya., "On the uniform convergence of relative frequencies of events to their probabilities", Th. Prob. and its Appl., 16(2), 264–80, (1971).Google Scholar
- [V1]Valiant L.G., "A Theory of the Learnable", Comm. ACM, 27(11), 1134–42, (1984).Google Scholar
- [V2]Valiant L.G., "Learning disjunctions of conjunctions", Proceedings of the 9th IJCAI, vol. 1, 560–566, Los Angeles, CA., (August 1985).Google Scholar
- [V3]Valiant L.G., "Deductive learning", Aiken Computational Laboratory, Harvard University, (1984).Google Scholar