I was reading the book of Kearns and Vazirani and I didn't completely understand the following:
Let C be a concept class and suppose we want to PAC learn C, they say first consider a larger hypothesis set $H$ such that $C\subset H$, where the output of the learner lies in $H$. Then, they go on to show that suppose the VC(H)=d, then it suffices to take $O(d)$ many labelled examples $(x,c(x))$ in order to PAC learn $C$ by outputting a hypothesis in H.
My questions: 1) why do we need this H? In particular, isn't there a second layer of optimization going on; I could possibly pick a H which is the set of all functions, for which the VC(H) would be super large and that doesn't seem to tell us a good upper bound at all, so we need to find a H that includes $C$ but yet not too large. I lack intuition on what's going on here.
2) Is there a nice way to just talk about VC(C) and say, VC(C) many samples suffice to learn C?