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Motivation and Background
We define an important combinatorial parameter that measures the combinatorial complexity of a family of subsets taken from a given universe (learning domain) X. This parameter was originally defined by Vapnik and Chervonenkis (1971) and is thus commonly referred to as Vapnik-Chervonenkis dimension, commonly abbreviated as VC dimension. Subsequently, Dudley (1978, 1979) generalized Vapnik and Chervonenkis’ (1971) results. The reader is also referred to Vapnik’s (2000) book in which he greatly extends the original ideas. This results in a theory which is called structural risk minimization.
As Anthony and Biggs (1992, Page 71) have put it, “The development of this notion is probably the most significant contribution that mathematics has made to Computational Learning Theory.”
Recall that we use | S | and ℘(S)...
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