Thread: Exercise 1.12
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Old 07-26-2016, 12:02 PM
jeffjackson jeffjackson is offline
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Default Re: Exercise 1.12

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.
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