Exercise 1.12

[henry2015]

Hi,

I thought I could only provide what Hoeffding Inequality's guarantee to my friend.

Ein(g) could be very bad (says 0.5) as my hypothesis set could be bad and so I can only pick g which has smallest Ein(h). And so I can only promise that P[|Eout(g)-Ein(g)| < e] has an upper bound by Hoeffding Inequality...

No?

[jeffjackson]

This is related to my Chapter 1 post about Section 1.3 being fundamentally flawed. You're correct, henry2015: You cannot promise any of the given answers (a) through (c) to Exercise 1.12.

What you can promise is this:

(d) With high probability, you will either produce a hypothesis g that approximates f well out of sample, or you will declare that you have failed.

You can promise this because Hoeffding guarantees that Ein will be close to Eout with high probability. So, with high probability, you will either produce a good-approximating g (Ein is small) or you will declare that you have failed (Ein is large).

Although (d) is similar to the textbook's answer (c), there is an important distinction: (c) promises that almost every time you output a hypothesis g, g is a good approximator to f. (d) does not make any such promise. In fact, (d) allows that it might be the case the every time you output a hypothesis, it is a poor approximator. What (d) does promise is that, if you are in such a learning scenario, you will normally declare that you have failed rather than outputting a poor hypothesis.

[magdon]

You are right. Part (c) should be reworded to say:

With high probability: you will either say you failed or you will produce a good g.

[jeffjackson]

Following up on my earlier reply: In my thread regarding Section 1.3 I have presented an argument for the feasibility of learning that, if accepted, allows us to promise something a good bit stronger than what I offered earlier, which was based only on Hoeffding. The stronger promise is this:

(e) Assuming that you are given sufficient data and/or allowed a sufficiently large error so that the Hoeffding probability is ultra low, such as , you will either produce a hypothesis g that approximates f well out of sample, or you will declare that you have failed.

Put another way, given my argument for feasible learning and given the assumption above, I can in good conscience promise that whenever I produce a hypothesis, it is a good approximation to the target. That is, it is reasonable for me to promise that I will never output a poor-approximating hypothesis.