Hoeffding Inequality

[henry2015]

Hi,

On page 22, it says, "the hypothesis h is fixed before you generate the data set, and the
probability is with respect to random data sets D; we emphasize that the assumption "h is fixed before you generate the data set" is critical to the validity of this bound".

Few questions:
1. Does the "data set" in "generate the data set" refer to the marble (which is the data set D) we pick randomly from the jar? Or it refers to the set of outputs (red/green) of h(x) on D?
2. It keeps mentioning "h is fixed before you generate the data set". Does it mean in machine learning, a set of h should be predefined before seeing any training data and no h can be added to the set after seeing the training data?

Thanks!

[yaser]

Originally Posted by henry2015

1. Does the "data set" in "generate the data set" refer to the marble (which is the data set D) we pick randomly from the jar? Or it refers to the set of outputs (red/green) of h(x) on D?

The target is assumed to be fixed, so since is also fixed, the colors of all marbles are fixed and picking the data set would mean picking the marbles in the sample.

2. It keeps mentioning "h is fixed before you generate the data set". Does it mean in machine learning, a set of h should be predefined before seeing any training data and no h can be added to the set after seeing the training data?

This is the assumption that the theory is based on. If one wants to add hypotheses after seeing the data and still apply the theory, one should take the set of hypotheses to include all potential hypotheses that may be added (whatever the data set may be).

[henry2015]

Thanks for your quick reply Professor!

Now, I wonder why "we cannot just plug in g for h in the Hoeffding inequality". Given g is one of h's and for each h, Hoeffding inequality is valid for the upper bound of P[|Ein(h) - Eout(h)| > E]. Even g is picked after we look at all the outputs of all h's, g is still one of h's. So Hoeffding inequality should be still valid for g. No?

Thanks!

[yaser]

Originally Posted by henry2015

Thanks for your quick reply Professor!

Now, I wonder why "we cannot just plug in g for h in the Hoeffding inequality". Given g is one of h's and for each h, Hoeffding inequality is valid for the upper bound of P[|Ein(h) - Eout(h)| > E]. Even g is picked after we look at all the outputs of all h's, g is still one of h's. So Hoeffding inequality should be still valid for g. No?

Thanks!

This is the main point of this part. Take the coin flipping example, with each of 1000 fair coins flipped 10 times. Hoeffding applies to each coin, right? Now if we pick "g" to be the coin that produced the most heads, we lose the Hoeffding guarantee because the small probability of bad behavior for each coin accumulates into a not-so-small probability of bad behavior of some coin (which we picked deliberately because it behaved badly).

[henry2015]

Hi Professor,

I just have a hard time to understand that how choosing a hypothesis changes a theory -- Hoeffding inequality.

Let's say h1(x) < P1, h2(x) < P2. We choose h2 to be g. Then h2(x) < P2 is no longer true?

I sort of understand your example because we pick the run of coin flipping that produces most heads, and so if we plot the graph, the graph indicates that Hoeffding inequality doesn't apply. But Hoeffding inequality is talking about probability and so the reality might be off a bit.

Maybe I am in a wrong direction?

[yaser]

It's a subtle point. There is "cherry picking" if we fish for a sample that has certain properties after many trials, instead of having a sample that is fairly drawn from a fixed hypothesis.

Statements involving probability are tricky because they don't guarantee a particular outcome, just the likelihood of getting that outcome. Therefore, changing the game to allow more trials or different conditions would change the probabilities.