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Old 09-11-2014, 05:00 AM
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magdon magdon is offline
Join Date: Aug 2009
Location: Troy, NY, USA.
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Default Re: Exercise 1.12 - Failing to make Ein(g) small enough

Just to elaborate a little on the last point in Yaser's answer.

Suppose your strategy is to use {\mathcal H}_1 \cup {\mathcal H}_2 only if {\mathcal H}_1 fails.

If {\mathcal H}_1 fails and you use {\mathcal H}_1 \cup {\mathcal H}_2 it is natural that you should pay the price for the VC bound implied by {\mathcal H}_1 \cup {\mathcal H}_2.

The interesting case is if {\mathcal H}_1 succeeds and you get low E_{in}. You cannot use the VC bound that applies for {\mathcal H}_1.

It is the option to use {\mathcal H}_1 \cup {\mathcal H}_2 in the event that {\mathcal H}_1 fails that complicates the matter.

This is why to get a correct theoretical bound, you must always specify your entire strategy first. The simplest strategy is to fix a hypothesis set. If it fails, it fails and you are done. If in the back of your mind you are thinking about the possibility of changing hypothesis sets if it fails, then this has to be taken into account in the theoretical analysis from the very begining, in particular, even if the first hypothesis set succeeds.

As mentioned by Yaser, one framework that is useful in analyzing such adaptive strategies is structural risk minimization.

Originally Posted by yaser View Post
Let us say that {\mathcal H}_1 is the hypothesis set that didn't work, and you want now to try another hypothesis set {\mathcal H}_2. The theoretical guarantees would still hold, but for the equivalent hypothesis set {\mathcal H}_1 \cup {\mathcal H}_2.

Just because this uses a "hierarchy" of hypothesis sets (in this case the hierarchy being {\mathcal H}_1 followed by {\mathcal H}_1 \cup {\mathcal H}_2 upon failure of {\mathcal H}_1, folllowed by possibly other expansions if {\mathcal H}_1 \cup {\mathcal H}_2 failed), there is in general an additional theoretical price to pay, but it is low. Look at structural risk minimization if you are further interested.
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