M=|H|? (Lecture 2 slide 16-17)

[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!

[yaser]

Originally Posted by 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!

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 in-sample to out-of-sample 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.

[Haowen]

Thank you Professor for your reply. It makes sense now. I think trying to consider the space of hypothesis "actually explored" is not that useful, as you said, the space of possible hypothesis is independent of and and the resultant bound is a much more general and useful characterization of the learning model.