Quote:
Originally Posted by Haowen
I have a question regarding the value of M in the multiplebins 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 2G*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!

You raise interesting points. First, indeed
. Second, if the algorithm does not fully explore the hypothesis set
, then
is still a working upper bound as far as generalization from insample to outofsample is concerned. Third, the analysis fixes
before the data set
is presented, and is done independently of the probability distribution
, i.e., the same bound applies regardless of which
is the true distribution.
In some cases, we can find a better (read: smaller) upper bound, such as in regularization which will be studied later in the course.