Thesis - wip

content/appendixs/appendix_kde_sigmoid.tex

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