An earlier version of the implementation used Newton--Raphson to invert the KDE CDF
instead of bisection (\Cref{sec:roll-kde-forward}).  We document it here as a reference
alternative.

\paragraph{Method.}
To find $\tau = \hat{F}_0^{-1}(1-\alpha;\mathcal{B}_0)$, choose an initial guess $\tau_0$
and iterate:
\begin{equation}
  \tau_{n+1} = \tau_n
    - \frac{\hat{F}_0(\tau_n;\,\mathcal{B}_0) - (1-\alpha)}
           {\dfrac{\partial\hat{F}_0(\tau_n;\,\mathcal{B}_0)}{\partial\tau}},
  \label{eq:kde-nr-step}
\end{equation}
where the denominator is the KDE density evaluated at $\tau_n$:
\[
  \frac{\partial\hat{F}_0(\tau_n;\,\mathcal{B}_0)}{\partial\tau}
  = \frac{1}{|\mathcal{B}_0|}\sum_{\mathbf{x}_j\in\mathcal{B}_0}
    \sigma_0'(\tau_n - f_\theta(\mathbf{x}_j)).
\]
Iteration continues until
$\lvert\hat{F}_0(\tau_n;\,\mathcal{B}_0)-(1-\alpha)\rvert < \varepsilon$.

\paragraph{Limitations.}
Newton--Raphson requires evaluating $\sigma_0'$ at every step and can diverge if
$\tau_0$ is far from the root or if the KDE density is very small near the target
quantile (e.g.\ early in training when scores cluster tightly and $v_0$ is large).
These failure modes motivated the switch to bisection, which requires no derivative
evaluation in the forward pass and is guaranteed to converge given a valid bracket.