# Generalization bound (single hypothesis) in "Foundations of Machine Learning"

I have a question about Corollary $$2.2$$: Generalization bound--single hypothesis in the book "Foundations of Machine Learning" Mohri et al. $$2012$$.

Equation $$2.17$$ seems to only hold when $$\hat{R}_S(h) in equation $$2.16$$ because of the absolute operator. Why is this not written in the corollary? Am I missing something important?

Thank you very much for reading this question.

You are right. The relaxed inequality $$R(h) \le \hat{R}_S(h)+ \epsilon.$$ can be replaced with the complete inequality $$\left |\hat{R}_S(h) - R(h) \right| \le \epsilon.$$ Actually, authors use this complete inequality for the follow up examples in the book. Again in Theorem $$2.13$$, they write the relaxed inequality, but prove for the complete inequality.
Let us denote: $$A:=\hat{R}_S(h) - R(h) \le \epsilon$$ $$B:=\hat{R}_S(h) - R(h) \ge -\epsilon$$ thus, $$\left| \hat{R}_S(h) - R(h) \right| \le \epsilon = A \text{ and } B$$ Equation $$(2.16)$$ states: \begin{align*} & {\Bbb P}(\left| \hat{R}_S(h) - R(h) \right| \ge \epsilon) \le \delta \\ & \Rightarrow {\Bbb P}(\left| \hat{R}_S(h) - R(h) \right| \le \epsilon) \ge 1 - \delta \\ & \Rightarrow {\Bbb P}(A \text{ and } B) \ge 1 - \delta \\ \end{align*} knowing that $${\Bbb P}(B) \ge {\Bbb P}(A \text{ and } B)$$, \begin{align*} & {\Bbb P}(B) \ge {\Bbb P}(A \text{ and } B) \ge 1 - \delta \\ & \Rightarrow {\Bbb P}(\hat{R}_S(h) - R(h) \ge -\epsilon) \ge 1 - \delta \end{align*} which is equivalent to $$R(h) \le \hat{R}_S(h) + \epsilon$$ with probability at least $$1-\delta$$, i.e. equation $$(2.17)$$.