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#7
04-12-2013, 03:58 PM
 grozhd Junior Member Join Date: Apr 2013 Posts: 4
Re: Is the Hoeffding Inequality really valid for each bin despite non-random sampling

I have the same concern as scottedwards2000 and I still don't understand how it is resolved.

As I understand the bin symbolizes the probability space of all possible inputs . Sample of balls drawn randomly from the bin symbolizes our training set .

Now we pick a hypothesis (suppose we are running PLA). We look at our sample , compute and use Hoeffding's Inequality. We do one step of PLA and come up with new hypothesis which automatically gives us and professor is saying that we can write down Hoeffding inequality for and ?

I guess, we can. But that inequality tells us something about random variable , i.e. about: where is a random sample. But it seems like we are using where is hardly random with regard to since we built using that sample.

Here is an example that illustrates my point: say we tried some random , compared it with target function on our training sample , wrote down Hoeffding's inequality. Now let's construct as follows: and . Let's write down Hoeffding's ineqaulity for this hypothesis. If we are indeed using then here it would be equal to 1 since on and we would have: is small. So somehow we are saying with high probability that does an excellent job out of sample though we didn't change it much from . This example shouldn't be correct, right? If it isn't how is the one with PLA correct?