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Why does estimation error increase with $|H|$ and decrease with $m$ in PAC learning?

I came across this statement in the section 5.2 of the book "understanding machine learning: from theory to algorithms". You just search "increases (logarithmically)" in your browser and then you can find the sentence.

I just can't understand the statement. And there is no proof in the book either. What I would like to do is prove that estimation error $\epsilon_{est}$ increase (logarithmically) with |𝐻| and decrease with 𝑚. Hope you can help me out. A rigorous proof can't be better!

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Definitely, you can find the proof in different resources (for example, in these notes or in the paper that originally proposed PAC learnability, A Theory of the Learnable). However, the intuition behind your question is when the size of the hypothesis increases, if you do not change anything, you can't see more part of the space. Hence, the estimation error will increase. Moreover, when you increase the number of samples, you have more chance to see more part of the hypothesis space, hence, the estimation error decrease.

Also, you can see some lemma about the relation of the PAC learnability and other similar concepts in the Wikipedia article Probably approximately correct learning:

Under some regularity conditions these three conditions are equivalent:

  1. The concept class $C$ is PAC learnable.
  2. The VC dimension of $C$ is finite.
  3. $C$ is a uniform Glivenko-Cantelli class.
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    $\begingroup$ Thanks. But it seems that this paper doesn't provide the proof but the sample complexity. For a finite class H, $m_H(\epsilon,\delta)\le\lceil\frac{log(|H|/\delta)}{\epsilon}\rceil$. But how could we draw the conclusion that we discussed? $\endgroup$ – Ben Sep 16 at 11:45
  • $\begingroup$ @ChenglinBen this part is the definition of the PAC learnability and a definition does not need any proof. $\endgroup$ – OmG Sep 16 at 12:23
  • $\begingroup$ I know that. I means that maybe our discussion about that estimation error increase (logarithmically) with |𝐻| and decrease with 𝑚 in PAC learning can be inspired by this. I'm not intended to prove the definition of PAC learnability. $\endgroup$ – Ben Sep 16 at 12:26
  • $\begingroup$ @ChenglinBen may be the last part of post (it is updated) could help. Indeed, the relation of the PAC learnability with other similar concepts such as VC dimension could help to know better the concept. The proof of the equivalency of these concepts could help to prove what you want. $\endgroup$ – OmG Sep 16 at 12:28
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    $\begingroup$ First, thank you very much. But I'm sorry that maybe I didn't state my question clearly. What I would like to do is prove that estimation error $\epsilon_{est}$ increase (logarithmically) with |𝐻| and decrease with 𝑚. Because the book I mentioned initially stated this conclusion toughly, I would like to figure out the rigorous proof. Hope you can help me out. $\endgroup$ – Ben Sep 16 at 12:32

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