Hoeffding's inequality with M

[ESRogs]

Hi,

I'm wondering why in Lecture 2 -- when we're trying to modify the right hand side of Hoeffding's inequality to take into account drawing from multiple bins (see page 16 of the slides) -- we sum all the probabilities of samples being non-representative, rather than taking 1 - the product of all the probabilities of each sample being representative.

In other words, (calling the right hand side of the original inequality p for simplicity) why doesn't the right hand side of the modified form become 1-(1-p)^M instead of M*p.

Wouldn't that be a still correct but tighter bound?

Thanks!

[yaser]

Originally Posted by ESRogs

Hi,

I'm wondering why in Lecture 2 -- when we're trying to modify the right hand side of Hoeffding's inequality to take into account drawing from multiple bins (see page 16 of the slides) -- we sum all the probabilities of samples being non-representative, rather than taking 1 - the product of all the probabilities of each sample being representative.

In other words, (calling the right hand side of the original inequality p for simplicity) why doesn't the right hand side of the modified form become 1-(1-p)^M instead of M*p.

Wouldn't that be a still correct but tighter bound?

Nice thought. Multiplying would necessitate an independence assumption. While such independence can be easily satisfied in a bin experiment by making sure that different samples are picked independently of each other, the learning counterpart will not follow suit because the different samples are really the same sample (training set) evaluated with different hypotheses.

[ESRogs]

Oh, I see, that makes sense. The union bound just feels very loose to me, so I was hoping for a tighter one. But it sounds like maybe that's the best we can do without any additional assumptions about the hypothesis set -- is that correct?

Also looking forward to finding out how infinite hypothesis sets are dealt with in future lectures...

[yaser]

Originally Posted by ESRogs

The union bound just feels very loose to me, so I was hoping for a tighter one. But it sounds like maybe that's the best we can do without any additional assumptions about the hypothesis set -- is that correct?

Correct. The theory will use a simple property of the hypthesis set to get a much better bound. Stay tuned!