Wrote stuff?

content/appendixs/appendix_gauss_grad.tex

@@ -1,3 +1,34 @@
+We instantiate the general ROLL framework (\Cref{sec:roll-formulation}) with
+Gaussian-distributed scores.  Each class's scores are modelled as
+$\mathcal{N}(\mu_k, \sigma_k^2)$, fitted by maximum-likelihood estimation.
+
+\paragraph{Forward pass.}
+
+The MLE parameters are:
+\begin{equation}
+  \mu_k = \frac{1}{|\mathcal{B}_k|}\sum_{\mathbf{x}_j \in \mathcal{B}_k} f_\theta(\mathbf{x}_j),
+  \qquad
+  \sigma_k^2 = \frac{1}{|\mathcal{B}_k|}\sum_{\mathbf{x}_j \in \mathcal{B}_k}(f_\theta(\mathbf{x}_j)-\mu_k)^2
+\end{equation}
+
+The CDF and quantile function of $\mathcal{N}(\mu,\sigma^2)$ are:
+\[
+  F(x) = \tfrac{1}{2}\!\left[1 + \operatorname{erf}\!\left(\tfrac{x-\mu}{\sigma\sqrt{2}}\right)\right],
+  \qquad
+  F^{-1}(p) = \mu + \sigma\sqrt{2}\,\operatorname{ierf}(2p-1)
+\]
+where $\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,\mathrm{d}t$ is the Gauss
+error function and $\operatorname{ierf}$ its functional inverse.
+Substituting into the general ROLL objective \eqref{eq:roll-tpr-at-fpr}:
+\begin{align}
+  \tau &= \mu_0 + \sigma_0\sqrt{2}\,\operatorname{ierf}(1-2\alpha) \nonumber \\
+  \mathcal{L}_{\text{ROLL-TPR@FPR}}^{\text{GAUSSIAN}}(f_\theta(\mathcal{B});\,\alpha)
+    &= \tfrac{1}{2}\!\left[1 + \operatorname{erf}\!\left(\tfrac{\tau-\mu_1}{\sigma_1\sqrt{2}}\right)\right]
+  \label{eq:roll-tpr-at-fpr-gaussian}
+\end{align}
+
+\paragraph{Gradient derivation.}
+
 We derive the gradient of \eqref{eq:roll-tpr-at-fpr-gaussian} with respect to
 $f_\theta(\mathbf{x}_i)$, instantiating the general structure~\eqref{eq:roll-gradient}
 for the Gaussian case. Throughout, $\tau = \hat{F}_0^{-1}(1-\alpha) =
@@ -65,4 +96,18 @@ Therefore:
   \label{eq:gauss-grad-y0}
 \end{equation}
 
-Substituting \eqref{eq:gauss-grad-y1} and \eqref{eq:gauss-grad-y0} into \eqref{eq:roll-gradient} yields \eqref{eq:gauss-grad-combined}.
+Substituting \eqref{eq:gauss-grad-y1} and \eqref{eq:gauss-grad-y0} into \eqref{eq:roll-gradient}:
+\begin{equation}
+  \frac{\partial \mathcal{L}_{\text{ROLL-TPR@FPR}}^{\text{GAUSSIAN}}}{\partial f_\theta(\mathbf{x}_i)} =
+  \begin{cases}
+    \dfrac{-1}{|\mathcal{B}_1|\,\sigma_1\sqrt{2\pi}}
+    \exp\!\left(-\dfrac{(\tau-\mu_1)^2}{2\sigma_1^2}\right)
+    \!\left(1 + \dfrac{(\tau-\mu_1)(f_\theta(\mathbf{x}_i)-\mu_1)}{\sigma_1^2}\right)
+      & \text{if } y_i = 1 \\[18pt]
+    \dfrac{1}{|\mathcal{B}_0|\,\sigma_1\sqrt{2\pi}}
+    \exp\!\left(-\dfrac{(\tau-\mu_1)^2}{2\sigma_1^2}\right)
+    \!\left(1 + \dfrac{(\tau-\mu_0)(f_\theta(\mathbf{x}_i)-\mu_0)}{\sigma_0^2}\right)
+      & \text{if } y_i = 0
+  \end{cases}
+  \label{eq:gauss-grad-combined}
+\end{equation}