We instantiate the KDE gradient~\eqref{eq:kde-grad-combined} for the logistic sigmoid
kernel used in our implementation.

\paragraph{Kernel definition.}

\begin{equation}
  \sigma(x ; v) = \frac{1}{1 + \exp(-vx)},
  \qquad
  \sigma'(x ; v) = \frac{v\,\exp(-v|x|)}{\bigl(1 + \exp(-v|x|)\bigr)^{2}}
  \label{eq:sigmoid-kernel}
\end{equation}

where $v > 0$ is the bandwidth parameter.  The derivative uses $|x|$ rather than $x$:
since $\sigma'$ is even, both forms are mathematically identical, but $\exp(-v|x|) \to 0$
as $|x| \to \infty$ whereas $\exp(-vx) \to \infty$ for $x \to -\infty$, so the $|x|$
form avoids floating-point overflow when scores fall well below $\tau$.

\paragraph{Forward pass.}

Substituting \eqref{eq:sigmoid-kernel} into the KDE CDF:
%
\begin{equation}
  \hat{F}_{k}(\tau ; \mathbf{s}^{(k)}) =
  \frac{1}{|\mathcal{B}_k|}
  \sum_{\mathbf{x}_j \in \mathcal{B}_k}
  \frac{1}{1 + \exp\!\bigl(-v_k\bigl(\tau - f_\theta(\mathbf{x}_j)\bigr)\bigr)}
  \label{eq:kde-sigmoid-cdf}
\end{equation}
%
The derivative of $\hat{F}_k$ with respect to $\tau$ — used in Newton--Raphson inversion
and in the gradient — is:
%
\begin{equation}
  \frac{\partial \hat{F}_{k}(\tau)}{\partial \tau}
  =
  \frac{1}{|\mathcal{B}_k|}
  \sum_{\mathbf{x}_j \in \mathcal{B}_k}
  \frac{v_k\,\exp\!\bigl(-v_k\bigl|\tau - f_\theta(\mathbf{x}_j)\bigr|\bigr)}
       {\Bigl(1 + \exp\!\bigl(-v_k\bigl|\tau - f_\theta(\mathbf{x}_j)\bigr|\bigr)\Bigr)^{2}}
  \label{eq:kde-sigmoid-pdf}
\end{equation}

\paragraph{Gradient.}

Define the shorthand $\varphi_k(u) = \sigma'(u ; v_k)$:
\begin{equation}
  \varphi_k(u) \;=\;
  \frac{v_k\,\exp(-v_k|u|)}{\bigl(1+\exp(-v_k|u|)\bigr)^{2}}
  \label{eq:sigmoid-phi}
\end{equation}

Substituting \eqref{eq:sigmoid-kernel} into \eqref{eq:kde-grad-combined}:

\begin{equation}
  \frac{\partial \mathcal{L}_{\text{ROLL-TPR@FPR}}}{\partial f_\theta(\mathbf{x}_i)} =
  \begin{cases}
    -\dfrac{\varphi_1\!\left(\tau - f_\theta(\mathbf{x}_i)\right)}{|\mathcal{B}_1|}
    & \text{if } y_i = 1\\[14pt]
    +\,\dfrac{\displaystyle\sum_{\mathbf{x}_j \in \mathcal{B}_1}
                \varphi_1\!\left(\tau - f_\theta(\mathbf{x}_j)\right)}{|\mathcal{B}_1|}
    \;\cdot\;
    \dfrac{\varphi_0\!\left(\tau-f_\theta(\mathbf{x}_i)\right)}
          {\displaystyle\sum_{\mathbf{x}_j \in \mathcal{B}_0}
             \varphi_0\!\left(\tau-f_\theta(\mathbf{x}_j)\right)}
    & \text{if } y_i = 0
  \end{cases}
  \label{eq:kde-sigmoid-grad}
\end{equation}

where $\tau = \hat{F}_0^{-1}(1-\alpha ; \mathcal{B}_0)$.
