Hiding Initial Hypothesis from the Data
 

Suppose I have a set of hypothesis, but I also want to look at the data to refine my choosing of the hypotheses to test.

So I randomly select 1/3 of the data points and see how my initial hypotheses work, and refine them, pick a new set H.

If I test the new H on the next 2/3 of the data, can I disregard the first 1/3 of the data and the first hypotheses that I tested, and therefore get a smaller H when using Hoeffding's bound? Or do I still have to consider all of the hypotheses tested so far?

Re: Hiding Initial Hypothesis from the Data
 

The 2/3 of the data were not involved in the selection of the new hypothesis set, so you can apply the Hoeffding/VC bounds to the 2/3 data and new hypothesis set.

Re: Hiding Initial Hypothesis from the Data
 

I see that Yaser has answered your question; Let me emphasize the answer is yes, you need only consider the size of your smaller trimmed hypothesis set.

Essentially you can treat the 2/3 of the data as your "data". Whatever you do to get a hypothesis set before you look at your "data" does not count against you. The 1/3 of the original data that you used to select the second hypothesis set can be viewed in just the same way that your human experience about (say) the credit default problem is used to select the hypothesis set. You start with all the hypotheses in your mind, then based on your knowledge about the credit problem (aka experience or as far as we care, data) you narrow that down to the perceptron model. Now you look at the data. You are not penalized for the entire hypothesis set in your mind. Just the hypothesis set you started with at the time you look at the data

Now, since you fixed your second hypothesis set, and now you look at your "data" to pick a , you are justified in computing the generalization error bar only based on the second hypothesis.