We study the complexity of computing the VC Dimension and Littlestone’s Dimension. Given an explicit description of a finite universe and a concept class (a binary matrix whose (x,C)-th entry is 1 iff element x belongs to concept C), both can be computed exactly in quasipolynomial time (n ). Assuming the randomized Exponential Time Hypothesis (ETH), we prove nearly matching lower bounds on the ...

Valiant’s theorem from the previous lecture is meaningless for infinite hypothesis classes, or even classes with more than exponential size. In 1968, Vladimir Vapnik and Alexey Chervonenkis wrote a very original and influential paper (in Russian) [5, 6] which allows us to estimate the sample complexity for infinite hypothesis classes too. The idea is that the size of the hypothesis class is a p...