[Haowen]

I have a question regarding the value of M in the multiple-bins Hoeffding bound slides.

M is supposed to be the number of different alternate hypotheses considered by the learning algorithm.

At the same time, H is the space of possible hypotheses that can be considered by the algorithm (e.g., all linear functions, etc).

I keep going back and forth in my mind about whether M=|H|.

Specifically, suppose that for a SPECIFIC training set X, after looking at the data points in X, the algorithm only explored some subset of H, say G with |G| < |H|.

Would it then be correct to set M = |G| and say that for the specific training set X, the probability of the hypothesis being bad is at most 2|G|*the hoeffding bound ? Or would this be incorrect since the theorem only deals with the behavior of the system over all possible X with the distribution P.

Thanks!