Advanced Composition (Differential Privacy) I have a problem understanding the proof of the corollary of Advanced Composition
Theorem Advanced Composition:
For all $\varepsilon,\delta,\delta' \geq 0$ the class of $(\varepsilon,\delta)$-differentially private mechanisms satisfies $(\varepsilon',k\delta+\delta')$-differential privacy under $k $-fold adaptive composition for:
\begin{equation*}
    \varepsilon'= \sqrt{2k\ln(1/\delta')}\varepsilon+k\varepsilon(e^\varepsilon-1)
\end{equation*}
Corollary
Given target privacy parameters $0&lt;\varepsilon'&lt;1$ and $\delta'&gt;0$, to ensure $(\varepsilon',k\delta+\delta')$ cumulative privacy loss over $k$ mechanisms, it suffices that each mechanism is $(\varepsilon,\delta)$-differentially private, where
\begin{equation*}
    \varepsilon=\frac{\varepsilon'}{2\sqrt{2k \ ln(1/\delta')}}
\end{equation*}
Proof:
The theorem of Advanced Composition tells us the composition will be $(\varepsilon^*,k\delta+\delta')$ for all $\delta'$, where $\varepsilon^*=\sqrt{2k \ln(1/\delta')}\cdot \varepsilon+k\varepsilon^2$. When $\varepsilon'&lt;1$ we have that $\varepsilon^*&lt;\varepsilon'$.

I don't see clearly the connections between the proof and the corollary. I tried using the quadratic formula. I think it, and don't see how they get the $\varepsilon$ from the corollary.
Turning arcynia's comment into an answer so this question can be marked as fixed: « The very next paragraph of the book explains this: &quot;Note that the above corollary gives a rough guide for how to set $\varepsilon$ to get desired privacy parameters under composition. When one cares about optimizing constants (which one does when dealing with actual implementations), $\varepsilon$ can be set more tightly by appealing directly to the composition theorem.&quot; That is to say, this value for epsilon does not represent anywhere near a calculated optimal value, but just a fairly good rough guide. »