### Growth function bound related to generalization error

This is an inequality on page 36 of the book Foundations of Machine Learning, but the author only states it without proof. $$ \mathbb{P}\left[\left|R(h)-\widehat{R}_{S}(h)\right|>\epsilon\right] \leq 4 \Pi_{\mathcal{H}}(2 m) \exp \left(-\frac{m \epsilon^{2}}{8}\right) $$ Here The growth function $\Pi_{\mathcal{F}}: \mathbb{N} \rightarrow \mathbb{N}$ for a hypothesis set $\mathcal{H}$ is defined by: $$ \forall m \in \mathbb{N}, \Pi_{\mathcal{F}}(m)=\max _{\left\{x_{1},…

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