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Let $H$ be a hypothesis class of multiclass predictors; namely, each $h\in H$ is a function from $X$ to $[k]$.

Denote the Natarajan dimension of $H$ by $Ndim(H)$. Hope you can give me an intuitive proof of the following lemma .

$|H|\le |X|^{Ndim(H)}\cdot k^{2Ndim(H)}$

The lemma is in the book "Understanding Machine Learning: from theory to algorithms". You just search keywords "Lemma 29.4".

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You can check Natarajan, On learning sets and functions or Haussler and Long, A generalization of Sauer's lemma.

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  • $\begingroup$ Thank you. But those proofs are very complex, making me really sleepy. Could you give me an intuitive proof? $\endgroup$ – Ben Nov 23 '19 at 0:16
  • $\begingroup$ If you don’t like proofs, you can just accept the validity of the statement. I don’t expect there to be simpler proofs around of the lemma. $\endgroup$ – Yuval Filmus Nov 23 '19 at 7:45

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