The VC dimension of hypothesis space H1 is larger than the VC dimension of hypothesis space H2

Question

I have read the definition of VC dimension. But I am not clear that if the VC dimension of hypothesis space H1 is larger than the VC dimension of hypothesis space H2, then does this indicates that the number of examples required for learning a hypothesis in H1 is smaller than the number of examples required for H2 or the reverse is true. As I know only that VC dimension is the size of the largest finite subset of X shattered by hypothesis space H. But I don't know the relation of number of samples required from VC dimension. Please anyone clear me the concept in simple sentences.

Answer 1

If VC dimension of hypothesis space H1 is larger than the VC dimension of hypothesis space H2. Considering an example, VC(circle) = 3 and VC(triangle)= 7. We can inspect that circle can shatter upto 3 points and triangle can shatter upto 7 points. Thus, instance points are larger in triangle classifier than circle. So more training example would be needed to learn the classifier in case of triangle. As:

m=1/2ε * [ln(|H|) + ln(1/δ)]

where m indicates the minimum number of training examples require to learn the classifier. Thus

|H| for circle is 2^3

|H| for triangle is 2^7

taking there are two classes only.